Measuring the DC properties of a signal path between transmitter and receiver

[0001] This application is a continuation of U.S. patent application Ser. No. 13/404,896, filed on Feb. 24, 2012, entitled “Techniques for the Measurement of I/Q Impairments and Associated Computational Tools”, invented by Stephen Dark and Christopher Behnke application Ser. No. 13/404,896 is hereby incorporated by reference in its entirety.

[0002] The present invention relates to the field of signal processing, and more particularly to systems and methods for measurement and correction of I/Q impairments in a receiver device or a transmitter device.

[0003] A transmitter receives a complex digital signal I(n)+jQ(n), converts the complex digital signal into an analog signal I(t)+jQ(t), and upconverts the analog signal using an I/Q modulator. The upconverted signal is transmitted onto a channel. Ideally, a pure complex exponential tone supplied to the I/Q modulator will result in a pure tone being transmitted. In reality, however, I/Q impairments in the transmitter will cause the I channel and Q channel to have different gains and different phase displacements. Such distortions imply among other things that the transmitted signal will have unwanted energy at a frequency equal to the negative of the tone frequency. Depending on the communication standard, this unwanted “image” results in potential skew on the constellation diagram or an artificial noise floor. Receivers have a similar problem. When a receiver is stimulated with a pure tone at frequency f, the complex signal appearing at the output of the receiver's I/Q demodulator will include unwanted signal energy at frequency −f in addition to energy at frequency f. In both case (transmitter and receiver), the difficulty arises due to imbalances in gain and phase between the I channel and the Q channel. Thus, there exists a need for mechanisms capable of correcting for I/Q impairments in a transmitter and/or receiver.

[0004] Furthermore, to achieve a high-quality correction for I/Q impairments, one needs to have access to a high-quality measurement of the I/Q impairments. However, quality measurements may be difficult to obtain. For example, to measure the I/Q impairments of a transmitter involves directing the transmitter to transmit a signal to a receiver. The receiver estimates the transmitter's I/Q impairments based on its received signal. However, the receiver's I/Q demodulator corrupts the estimates with its own I/Q impairments. Furthermore, the signal path between the transmitter's I/Q modulator and the receiver's I/Q demodulator also introduces distortion to the estimates. Thus, there exists a need for mechanisms capable of estimating or measuring the I/Q impairments of transmitters and/or receivers, for mechanisms capable of accurately measuring the I/Q impairments implicit in sampled signals, for mechanisms capable of determining the properties of the signal path, and for mechanisms capable of predicting how I/Q impairments are transformed by systems such as the signal path.

[0005] Among other things, the present patent discloses mechanisms capable of compensating for I/Q impairments in a transmitter and/or a receiver. The parameters used to perform the compensation are computed based on measured or estimated values of the I/Q impairments. For example, the parameters used to compensate for the I/Q impairments of a transmitter (or receiver) are computed based on measured or estimated values of those impairments. Any known technique may be used to measure or estimate the I/Q impairments of the transmitter or the receiver or a series combination of the transmitter and receiver, including, but not limited to, the techniques disclosed herein.

[0006] In one embodiment, a system and method for compensating for I/Q impairments of a receiver may involve the following operations.

[0007] An analog input signal is received from a transmission medium. I/Q demodulation is performed on the analog input signal to produce an analog inphase (I) signal and an analog quadrature (Q) signal. The analog I signal and the analog Q signal are then digitized to produce respectively a digital I signal and a digital Q signal. The digital I signal and the digital Q signal are filtered in accordance with a 2×2 matrix of digital filters to produce a filtered digital I signal and a filtered digital Q signal. (The filtering may be performed on a programmable hardware element such as an FPGA, or in dedicated digital circuitry such as an ASIC, or in software on a processor, etc.) The 2×2 matrix of digital filters at least partially compensates for I/Q impairments of the receiver over a range of frequencies. The frequency response of at least one of the diagonal components of the 2×2 matrix is computed based on measurements of the I/Q impairments as a function of frequency and the measurements as a function of the negative of frequency. (The measurements of the receiver's I/Q impairments may be obtained by any known method. The present document describes a number of methods for obtaining such measurements.) Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix is computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency.

[0008] In some embodiments, the receiver's I/Q impairments over positive frequencies and the receiver's I/Q impairments over negative frequencies may be assumed to be functionally related. (A) In one such embodiment, the frequency responses of the 2×2 matrix may be computed as follows. The frequency response of at least one of the diagonal components of the 2×2 matrix at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f). Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix at the frequency f may be computed based only on the measurement of the I/Q impairments at the frequency f (or alternatively, only on the measurement of the I/Q impairments at the frequency −f). (B) In another such embodiment, the gain imbalance is assumed to be even and the phase skew is assumed to be odd. Then both non-diagonal components of the 2×2 matrix may be set to zero; one of the diagonal components may correspond to a pure pass-through filter (i.e., unity frequency response); and a frequency response of the other diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f). (C) In another such embodiment, both diagonal components of the 2×2 matrix may correspond to pure pass-through filters; one of the non-diagonal components may be set to zero; and a frequency response of the other non-diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f).

[0009] In another embodiment, a system and method for configuring a receiver to at least partially compensate for I/Q impairments of the receiver may involve the following operations. Measurements of the I/Q impairments of the receiver over a frequency band are received (or accessed from memory). A 2×2 matrix of digital filters is computed based on the measurements. The 2×2 matrix of digital filters is computed to achieve at least partial compensation for the I/Q impairments of the receiver over the frequency band. The frequency response of at least one of the diagonal components of the 2×2 matrix is computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. In addition, the frequency response of at least one of the non-diagonal components of the 2×2 matrix is computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. A digital circuit is then programmed to implement the 2×2 matrix of digital filters. When so programmed, the digital circuit is configured to at least partially compensate for the I/Q impairments of the receiver over the frequency band. The digital circuit may be realized in any of a wide variety of forms. For example, the digital circuit may be realized by a programmable hardware element, or by dedicated digital circuitry such as an ASIC, or by a processor in response to the execution of program instructions. (The digital circuit may be incorporated as part of the receiver, or as part of another system, e.g., a host computer or controller board).

[0010] In another embodiment, a system and method for operating a transmitter so as to achieve I/Q impairment compensation may involve the following operations.

[0011] A digital inphase (I) signal and a digital quadrature (Q) signal are received. The digital I signal and the digital Q signal is filtered in accordance with a 2×2 matrix of digital filters to produce a filtered digital I signal and a filtered digital Q signal. The 2×2 matrix of digital filters at least partially pre-compensates for I/Q impairments of the transmitter over a range of frequencies. The frequency response of at least one of the diagonal components of the 2×2 matrix is computed based on measurements of the I/Q impairments as a function of frequency and the measurements as a function of the negative of frequency. (The measurements of the transmitter's I/Q impairments may be obtained by any known method. The present document describes a number of methods for obtaining such measurements.) Moreover, the frequency response of at least one of the non-diagonal components of the 2×2 matrix is computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. The filtered digital I and Q signals may then be converted to analog form in order to obtain respective analog I and Q signals. I/Q modulation may be performed on the analog I and Q signals to produce a modulated analog signal.

[0012] In some embodiments, the transmitter's I/Q impairments over positive frequencies and the transmitter's I/Q impairments over negative frequencies may be assumed to be functionally related. (A) In one such embodiment, the computation of the 2×2 matrix of digital filters may be simplified as follows. The frequency response of at least one of the diagonal components of the 2×2 matrix at the arbitrary frequency f in the frequency range may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f). Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix at the frequency f may be computed based only on the measurement of the I/Q impairments at the frequency f (or alternatively, only on the measurement of the I/Q impairments at the frequency −f). (B) In another such embodiment, the gain imbalance is assumed to be even and the phase skew is assumed to be odd. Then both non-diagonal components of the 2×2 matrix may be set to zero; one of the diagonal components may correspond to a pure pass-through filter (i.e., unity frequency response); and a frequency response of the other diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f). (C) In another such embodiment, both diagonal components of the 2×2 matrix may correspond to pure pass-through filters; one of the non-diagonal components may be set to zero; and a frequency response of the other non-diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f).

[0013] In another embodiment, a system and method for configuring a transmitter to at least partially compensate for I/Q impairments of the transmitter may involve the following operations.

[0014] Measurements of the I/Q impairments of the transmitter over a frequency range are received (or accessed from memory). A 2×2 matrix of digital filters is computed based on the measurements. The 2×2 matrix of digital filters is computed to achieve at least partial pre-compensation for the I/Q impairments of the transmitter. The frequency response of at least one of the diagonal components of the 2×2 matrix is computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix is computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. A digital circuit is then programmed to implement the 2×2 matrix of digital filters. When so programmed, the digital circuit is configured to at least partially pre-compensate for the I/Q impairments of the transmitter.

[0015] In another embodiment, a system and method for operating a transmitter so as to achieve at least partial compensation for I/Q impairments of the transmitter at a given frequency f may involve the following operations.

[0016] A digital inphase (I) signal and a digital quadrature (Q) signal are received. The digital I signal and the digital Q signal are transformed in accordance with a 2×2 matrix of constants to produce a resultant digital I signal and a resultant digital Q signal. (In other words, the vector signal comprising the digital I signal and digital Q signal is multiplied by the 2×2 matrix.) The resultant digital I and Q signals are converted to analog form in order to obtain respective analog I and Q signals. I/Q modulation is performed on the analog I and Q signals to produce a modulated analog signal. The 2×2 matrix is configured to at least partially pre-compensate for the I/Q impairments at frequency f. A first of the constants, corresponding a diagonal element of the 2×2 matrix, is computed based on a measurement of the I/Q impairments at frequency f and a measurement of the I/Q impairments at frequency −f. Furthermore, a second of the constants, corresponding to a non-diagonal element of the 2×2 matrix, is computed based on the measurement at frequency f and the measurement at frequency −f.

[0017] In another embodiment, a method for determining (i.e., measuring) I/Q impairments of a transmitter may involve the following actions.

[0018] The method involves performing a set of operations. The set of operations includes: (a) directing that a complex exponential tone at frequency f be supplied to the transmitter; (b) supplying a pre-compensation transformation to a pre-compensation circuit of the transmitter, where the pre-compensation circuit is configured to apply the pre-compensation transformation to the complex exponential tone to obtain an adjusted complex signal, where the pre-compensation transformation is configured to pre-compensate for a current estimate of the I/Q impairments of the transmitter, where the transmitter is configured to transmit a transmit signal based on the adjusted complex signal, where a receiver is configured to receive the transmit signal and capture a sampled complex signal representing the received transmit signal; (c) computing raw I/Q impairments based on the sampled complex signal; (d) transforming the raw I/Q impairments to determine transformed I/Q impairments, where said transforming removes measured I/Q impairments of the receiver from the raw I/Q impairments; (e) removing a current estimate of a signal path from the transformed I/Q impairments to obtain path-compensated I/Q impairments, where the signal path includes a path from an I/Q modulator of the transmitter to a demodulator of the receiver; and (f) updating the current estimate of the I/Q impairments of the transmitter based on the path-compensated I/Q impairments. (The demodulator may be an I/Q demodulator or not, depending the architecture of the receiver.)

[0019] In another embodiment, a method for determining I/Q impairments of a transmitter may involve the following actions.

[0020] The method may include configuring a local oscillator (LO) of the transmitter and a local oscillator (LO) of the receiver to be phase locked to a common reference and so that a frequency of the receiver's LO minus a frequency of the transmitter's LO is equal (e.g., exactly equal) to an amount ΔLO.

[0021] The method may also include performing a set of operations, where the set of operations includes: (a) directing that a complex exponential tone at frequency f be supplied to the transmitter; (b) supplying a pre-compensation transformation to a pre-compensation circuit of the transmitter, where the pre-compensation circuit is configured to apply the pre-compensation transformation to the complex exponential tone in order to obtain an adjusted complex signal, where the pre-compensation transformation is configured to pre-compensate for a current estimate of the I/Q impairments of the transmitter, where the transmitter is configured to transmit a transmit signal based on the adjusted complex signal, where a receiver is configured to receive the transmit signal and to capture a sampled complex signal representing the received transmit signal; (c) frequency shifting the sampled complex signal by the amount ΔLO to obtain a frequency-shifted signal; (d) computing raw I/Q impairments at frequency f based on the frequency-shifted signal; (e) removing a current estimate of a signal path from the raw I/Q impairments at frequency f to obtain path-compensated I/Q impairments at frequency f, where the signal path includes a path from an I/Q modulator of the transmitter to a demodulator of the receiver; and (f) updating the current estimate of the I/Q impairments of the transmitter at frequency f based on the path-compensated I/Q impairments at frequency f. (The demodulator may be an I/Q demodulator or not, depending the architecture of the receiver.)

[0022] In another embodiment, a method for determining (i.e., measuring) I/Q impairments of a receiver may involve the following actions.

[0023] The method may involve directing that an input signal be supplied to the receiver, where the input signal includes an isolated tone at displacement frequency f and includes a void interval around displacement frequency −f (In one embodiment, the receiver includes a calibration tone generator that is configured to generate the input signal.) The receiver is configured to demodulate the input signal in order to obtain a sampled complex signal. The displacement frequencies f and −f are displacements relative to a local oscillator frequency of the receiver.

[0024] The method may also involve computing the I/Q impairments of the receiver at frequency f based on the sampled complex signal.

[0025] The method may also involve repeating the actions of directing and computing for values of the frequency f spanning a specified frequency band.

[0026] The method may also involve storing the I/Q impairments of the receiver for the values of the frequency f in a memory.

[0027] In another embodiment, a method for estimating I/Q impairments associated with a sampled complex signal produced by a receiver may involve the following actions.

[0028] A device is directed to stimulate the receiver with a stimulus signal having an isolated tone at displacement frequency f and a void interval at displacement frequency −f (The displacement frequencies f and −f are displacements with respect to a local oscillator frequency of the receiver. The sampled complex signal may be a baseband signal produced by the receiver.) A Discrete-Time Fourier transform value C_I at frequency f is computed for an I component of the sampled complex signal. A Discrete-Time Fourier transform value C_Q at frequency f is computed for a Q component of the sampled complex signal. A gain imbalance g of the sampled complex signal at frequency f is computed based on magnitudes of the values C_I and C_Q. The gain imbalance g includes at least a gain imbalance of the receiver. A phase skew φ of the sampled complex signal at frequency f is computed based on phases of the values C_I and C_Q, where the phase skew φ includes at least a phase skew of the receiver.

[0029] In another embodiment, a method for estimating a DC scaling of a signal path between an I/Q modulator of a transmitter and an I/Q demodulator of a receiver may involve the following operations. To facilitate this estimation method, the output of the transmitter may be coupled to the input of the receiver, e.g., via a cable.

[0030] The transmitter is directed to supply a zero signal as input to the I/Q modulator. A first response signal that has been captured from the I/Q demodulator in response to said supplying the zero signal is received. The transmitter is directed to supply a constant signal equal to a non-zero complex constant as input to the I/Q modulator. A second response signal that has been captured from the I/Q demodulator in response to said supplying the constant signal is received. The first response signal is averaged to obtain a first average, and the second response signal is averaged to obtain a second average. A difference between the second average and the first average is computed. The DC scaling is computed based on the difference and the non-zero complex constant. Furthermore, a DC rotation of the signal path may be computed based on a phase of the difference and a phase of the non-zero complex constant. The DC scaling and DC rotation are usable to remove an effect of the signal path from measured I/Q impairments at the receiver in order to obtain estimates of the I/Q impairments of the transmitter.

[0031] In one alternative embodiment of the above-described method of DC scaling/rotation estimation, the transmitter has no (or negligible) local oscillator leakage. (Such might be the case, e.g., when the transmitter has an RF architecture other than a direct conversion architecture.) Thus, one may omit the transmission of the zero signal, the capture of the first response signal, the computation of the first average, and the computation of the difference. The DC scaling is then computed based on the second average and the non-zero complex constant. The DC rotation is computed based on a phase of the second average and the phase of the non-zero complex constant.

[0032] In another embodiment, a method for computing I/Q impairments at a complex output (i.e., an I/Q output pair) of an electrical system based on I/Q impairments at a complex input (i.e., an I/Q input pair) of the electrical system may include the following operations.

[0033] A spectrum A(f) is computed according to the expression
H(f)(1+g(f)exp(jφ(f)),
where H(f) is a spectrum of a linear system model of the electrical system, where g(f) is a gain imbalance at the complex input, where φ(f) is a phase skew at the complex input. A spectrum B(f) is computed according to the expression
H(−f)(1−g(f)exp(−jφ(f))).

[0034] A sum of the spectra A(f) and B(f) and a difference of the spectra A(f) and B(f) are computed. A gain imbalance and phase skew at the complex output are computed based on real and imaginary parts of the sum, and real and imaginary parts of the difference.

[0035] In some embodiments, the electrical system being modeled by the spectrum H(f) is the inverse of a signal path from an I/Q modulator of a transmitter to a demodulator of a receiver, e.g., as variously described herein. The gain imbalance and the phase skew at the complex input of the electrical system may represent a gain imbalance and a phase skew at the input (or alternatively, at the output) of the demodulator. The gain imbalance and the phase skew at the complex output of the electrical system may represent a gain imbalance and a phase skew at the output of the I/Q modulator.

[0036] Various embodiments of communication devices and associated methods for reducing I/Q impairments in signals used by the communication devices are described herein. According to one embodiment, a receiver device may receive a transmission signal over a communication medium, and may perform I/Q demodulation on the received transmission signal to produce a pair of analog I (in-phase) and Q (quadrature) signals. The receiver device may perform analog-to-digital conversion of each of the analog I and Q signals to produce respective digital I and Q signals. The resulting digital I and Q signals may have I/Q impairments caused by the I/Q demodulation and/or the analog-to-digital conversion and/or other processing. The receiver device may be configured to perform wideband I/Q impairment correction on the digital I and Q signals to correct the I/Q impairments. The wideband I/Q impairment correction may compensate for frequency-dependent variations of gain imbalance and phase imbalance in the digital I and Q signals, e.g., may compensate for gain imbalances and phase imbalances in the digital I and Q signals at a plurality of frequency offsets across an instantaneous bandwidth of the receiver device.

[0037] Performing the wideband I/Q impairment correction on the digital I and Q signals may comprise filtering one or more of the digital I signal or the digital Q signal to produce a resultant digital I signal and a resultant digital Q signal. The resultant digital I and Q signals represent corrected signals. In some embodiments, the resultant digital I signal is identical to the digital I signal, and the resultant digital Q signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain one or more respective filtered signals and by adding the one or more filtered signals. In other embodiments, the resultant digital Q signal is identical to the digital Q signal, and the resultant digital I signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain one or more respective filtered signals and by adding the one or more filtered signals. In yet other embodiments, the resultant digital I signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain respectively one or more filtered signals and by adding the one or more filtered signals; and the resultant digital Q signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain respectively one or more additional filtered signals and by adding the one or more additional filtered signals.

[0038] In further embodiments a calibration system (or the receiver device itself) may determine correction information by providing a plurality of known test signals to the receiver device and measuring I/Q impairments introduced by the receiver device in response to the known test signals. (In one embodiment, the receiver device may include a calibration tone generator to generate the known test signal.) The wideband I/Q impairment correction may utilize the correction information to compensate for the frequency-dependent variations of gain imbalance and phase imbalance in the digital I and Q signals.

[0039] In some embodiments the calibration system may operate in an offline calibration phase and an online operation phase. Performing the offline calibration phase may include providing a plurality of known test signals to the receiver device, measuring I/Q impairments introduced by the receiver device in response to the known test signals, and determining correction information based on the measured I/Q impairments. Performing the online operation phase may include receiving a transmission signal over a communication medium, performing I/Q demodulation on the received transmission signal to produce analog I and Q signals, performing analog-to-digital conversion of each of the analog I and Q signals to produce digital I and Q signals, and performing wideband I/Q impairment correction on the digital I and Q signals. The wideband I/Q impairment correction may use the correction information determined in the offline calibration phase to compensate for frequency-dependent variations of gain imbalance and phase imbalance in the digital I and Q signals.

[0040] In some embodiments the offline calibration phase may be performed in response to the receiver device being powered on. In some embodiments the receiver device may automatically enter the online operation phase in response to determining that the offline calibration phase is complete. In some embodiments the receiver device may automatically switch from the online operation phase to the offline calibration phase in response to determining that the receiver device is not busy processing received transmission signals in the online operation phase. In some embodiments the offline calibration phase may be initiated in response to user input.

[0041] According to other embodiments, a transmitter device may receive digital I (in-phase) and Q (quadrature) signals to be transmitted. The transmitter device may perform wideband I/Q impairment pre-correction on the digital I and Q signals. The action of performing the wideband I/Q impairment pre-correction may involve filtering one or more of the digital I and Q signals to produce a resultant digital I signal and a resultant digital Q signal to pre-compensate for frequency-dependent variations of gain imbalance and phase imbalance that will be subsequently introduced during synthesis of a transmission signal. The transmission signal may be synthesized using the resultant digital I signal and the resultant digital Q signal.

[0042] The action of synthesizing the transmission signal may include performing digital-to-analog conversion of the resultant digital I signal and the resultant digital Q signal to produce an analog I signal and an analog Q signal, and performing I/Q modulation to produce the transmission signal using the analog I signal and the analog Q signal. The resultant digital I signal and the resultant digital Q signal may pre-compensate for frequency-dependent variations of gain imbalance and phase imbalance caused by one or more of the digital to analog conversion and the I/Q modulation.

[0043] In some embodiments, the resultant digital I signal is identical to the digital I signal, and the resultant digital Q signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain one or more respective filtered signals and by adding the one or more filtered signals. In other embodiments, the resultant digital Q signal is identical to the digital Q signal, and the resultant digital I signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain one or more respective filtered signals and by adding the one or more filtered signals. In yet other embodiments, the resultant digital I signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain respectively one or more filtered signals and by adding the one or more filtered signals; and the resultant digital Q signal is generated by filtering one or more of the digital I signal and the digital Q signal to obtain respectively one or more additional filtered signals and by adding the one or more additional filtered signals.

[0044] In further embodiments, a calibration system may determine correction information by providing a plurality of known digital test signals to the transmitter device and measuring I/Q impairments introduced by the transmitter device in response to the known test signals. The wideband I/Q impairment pre-correction may utilize the correction information to produce the resultant digital signals.

[0045] In some embodiments the transmitter device may operate in an offline calibration phase and an online operation phase. The offline calibration phase may include providing a plurality of known test signals to the transmitter device, measuring I/Q impairments introduced by the transmitter device in response to the known test signals, and determining correction information based on the measured I/Q impairments.

[0046] In some embodiments the offline calibration phase may be performed in response to the transmitter device being powered on. In some embodiments the transmitter device may automatically enter the online operation phase in response to determining that the offline calibration phase is complete. In some embodiments the transmitter device may automatically switch from the online operation phase to the offline calibration phase in response to determining that the transmitter device is not busy transmitting signals in the online operation phase. In some embodiments the offline calibration phase may be initiated in response to user input.

[0047] The online operation phase may include receiving digital I and Q signals to be transmitted, and performing wideband I/Q impairment pre-correction on the digital I and Q signals. The action of performing the wideband I/Q impairment pre-correction may use the correction information determined in the offline calibration phase to filter one or more of the digital I and Q signals to produce a resultant digital I signal and a resultant digital Q signal to pre-compensate for frequency-dependent variations of gain imbalance and phase imbalance that will be subsequently introduced during synthesis of a transmission signal. The transmission signal may be synthesized using the resultant digital I signal and a resultant digital Q signal

[0048] According to another embodiment, a measurement system may include a receiver device and a device under test. The receiver device may be configured to receive a transmission signal including measurement data acquired from the device under test, perform I/Q demodulation on the received transmission signal to produce analog I (in-phase) and Q (quadrature) signals, perform analog-to-digital conversion of each of the analog I and Q signals to produce digital I and Q signals, and perform wideband I/Q impairment correction on the digital I and Q signals. The wideband I/Q impairment correction may compensate for frequency-dependent variations of gain imbalance and phase imbalance in the digital I and Q signals.

[0049] In further embodiments, the measurement system may also include a transmitter device. The transmitter device may be configured to receive digital I and Q signals to be transmitted. The digital I and Q signals may specify information to be transmitted to the device under test. The transmitter device may be further configured to perform wideband I/Q impairment pre-correction on the digital I and Q signals. The action of performing the wideband I/Q impairment pre-correction may involve filtering one or more of the digital I and Q signals to produce a resultant digital I signal and a resultant digital Q signal to pre-compensate for frequency-dependent variations of gain imbalance and phase imbalance that will be subsequently introduced during synthesis of a transmission signal. The transmitter device may synthesize the transmission signal using the resultant digital I signal and a resultant digital Q signal, and transmit the transmission signal to the device under test.

[0125] While the invention is susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to limit the invention to the particular form disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the present invention as defined by the appended claims. Note that the various section headings in the following Detailed Description are for organizational purposes only and are not meant to be used to limit the claims.

[0126] The following is a glossary of terms used in the present application:

[0127] Memory Medium—Any of various types of memory devices or storage devices. The term “memory medium” is intended to include an installation medium, e.g., a CD-ROM, floppy disks 105, or tape device; a computer system memory or random access memory such as DRAM, DDR RAM, SRAM, EDO RAM, Rambus RAM, etc.; a non-volatile memory such as a Flash, magnetic media, e.g., a hard drive, or optical storage; registers, or other similar types of memory elements, etc. The memory medium may comprise other types of memory as well or combinations thereof. In addition, the memory medium may be located in a first computer in which the programs are executed, or may be located in a second different computer which connects to the first computer over a network, such as the Internet. In the latter instance, the second computer may provide program instructions to the first computer for execution. The term “memory medium” may include two or more memory mediums which may reside in different locations, e.g., in different computers that are connected over a network.

[0128] Programmable Hardware Element—includes various hardware devices comprising multiple programmable function blocks connected via a programmable interconnect. Examples include FPGAs (Field Programmable Gate Arrays), PLDs (Programmable Logic Devices), FPOAs (Field Programmable Object Arrays), and CPLDs (Complex PLDs). The programmable function blocks may range from fine grained (combinatorial logic or look up tables) to coarse grained (arithmetic logic units or processor cores). A programmable hardware element may also be referred to as “reconfigurable logic”.

[0129] Computer System—any of various types of computing or processing systems, including a personal computer system (PC), mainframe computer system, workstation, network appliance, Internet appliance, personal digital assistant (PDA), television system, grid computing system, or other device or combinations of devices. In general, the term “computer system” can be broadly defined to encompass any device (or combination of devices) having at least one processor that executes instructions from a memory medium.

[0130] Local Oscillator (LO)—a circuit configured to generate a periodic signal at a specified frequency and amplitude. The periodic signal may be a pure sinusoid, and its frequency and/or amplitude may be programmable. The periodic signal may or may not be phase or frequency locked to another periodic signal.”

[0131] Embodiments of the present invention may be realized in any of various forms. For example, in some embodiments, the present invention may be realized as a computer-implemented method, a computer-readable memory medium, or a computer system. In other embodiments, the present invention may be realized using one or more custom-designed hardware devices such as ASICs. In other embodiments, the present invention may be realized using one or more programmable hardware elements such as FPGAs.

[0132] In some embodiments, a computer-readable memory medium may be configured so that it stores program instructions and/or data, where the program instructions, if executed by a computer system, cause the computer system to perform a method, e.g., any of a method embodiments described herein, or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets.

[0133] In some embodiments, a computer system may be configured to include a processor (or a set of processors) and a memory medium, where the memory medium stores program instructions, where the processor is configured to read and execute the program instructions from the memory medium, where the program instructions are executable to implement any of the various method embodiments described herein (or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets). The computer system may be realized in any of various forms. For example, the computer system may be a personal computer (in any of its various realizations), a workstation, a computer on a card, an application-specific computer in a box, a server computer, a client computer, a hand-held device, a tablet computer, a wearable computer, etc.

[0134] In some embodiments, a set of computers distributed across a network may be configured to partition the effort of executing a computational method (e.g., any of the method embodiments disclosed herein). In some embodiments, a first computer may be configured to receive an O-QPSK modulated signal and to capture samples of that signal. The first computer may send the samples to a second computer through the network. The second computer may operate on the samples according to any of the method embodiments described herein, or, any combination of the method embodiments described herein, or, any subset of any of the method embodiments described herein, or, any combination of such subsets.

[0135] FIG. 1A illustrates one possible application (among many) of the inventive ideas described herein. A mobile device 10 (e.g., mobile phone) communicates wirelessly with a wireless transceiver station 15. The mobile device 10 may include the digital pre-correction described herein to improve the quality of its transmitted signals, i.e., to correct for so-called “I/Q impairments” in its transmission hardware (e.g., in its I/Q modulator). Similarly, the wireless transceiver station 15 may apply digital post-correction to its received signal to correct for I/Q impairments in its reception hardware (e.g., in its I/Q demodulator). Furthermore, the wireless transceiver station and mobile device may apply the same pre-correction and post-correction with roles exchanged, i.e., for transmissions in the opposite direction.

[0136] FIG. 1B illustrates another possible application of the inventive ideas described herein. A test transmitter 20 transmits signals to a receiver under test 25. The test transmitter 20 may perform the digital pre-correction described herein correct for its own I/Q impairments, and thus, improve the quality of its transmissions. For example, the test transmitter 20 may achieve a higher standard on image rejection due to use of the digital pre-correction. Thus, distortions (e.g., I/Q impairments) measured in the signals captured by the receiver may be ascribed to the receiver's imperfections.

[0137] FIG. 1C illustrates yet another possible application of the inventive ideas described herein. A test receiver 35 receives signals transmitted by a transmitter under test 30. The test receiver employs the digital post-correction described herein to correct for its own I/Q impairments. Thus, the receiver may meet a higher standard on image rejection than it would without the post-correction. Therefore, any distortions (e.g., I/Q impairments) measured in the signals captured by the receiver may be clearly assigned to the transmitter's imperfections.

[0138] Wideband Correction Method for Receiver

[0139] In one set of embodiments, a method 100 for compensating for I/Q impairments of a receiver over a range of frequencies may involve the operations shown in FIG. 2A.

[0140] At 110, the receiver may receive an analog input signal. The analog input signal may be received from a transmission medium. The transmission medium is a medium that permits the transmission of signal energy. For example, the transmission medium may be free space, the atmosphere, the earth or some portion of the earth's surface, an electrical cable, a fiber optic cable, a body of water such as an ocean.

[0141] At 115, the receiver may perform I/Q demodulation on the analog input signal to produce an analog inphase (I) signal and an analog quadrature (Q) signal. The process of I/Q demodulation is well understood in the field of communication. Typically, I/Q demodulation involves mixing the analog input signal with a pair of orthogonal carriers. For example, the mixing may be interpreted according to the following model:
I(t)=y(t)cos(ωt)
Q(t)=y(t)sin(ωt).

[0142] In some embodiments, the analog I signal and the analog Q signals may be interpreted as baseband signals, i.e., as the components of a complex baseband signal. In other embodiments, the analog I signal and the analog Q signals may be interpreted as intermediate frequency (IF) signals.

[0143] At 120, the receiver may digitize the analog I signal and the analog Q signal to produce, respectively, a digital I signal and a digital Q signal. (The term “digital signal” is meant to imply a sampled signal, not a two-state signal.) Thus, the receiver may include a pair of analog-to-digital converters (ADCs).

[0144] At 125, the digital I signal and the digital Q signal may be filtered in accordance with a 2×2 matrix of digital filters to produce a filtered digital I signal and a filtered digital Q signal. The filtering may involve applying the 2×2 matrix (h_ij) of digital filters according to the relations:
I_F(n)=h₁₁(n)*I(n)+h₁₂(n)*Q(n)
Q_F(n)=h₂₁(n)*I(n)+h₂₂(n)*Q(n),
where the symbol “*” represents convolution. (Note, elsewhere in this patent disclosure, the symbol “*” may mean convolution or multiplication, depending on the particular context. As a superscript, “*” denotes complex conjugation.)

[0145] The 2×2 matrix of digital filters may compensate (or, at least partially compensate) for I/Q impairments of the receiver over a range of frequencies, e.g., a range of frequencies wide enough to cover the bandwidth of transmitted communication signal or an instantaneous bandwidth of the receiver. (Processes for measuring I/Q impairments are discussed at length later in this patent disclosure.) In other words, the digital filters make the receiver's input-output behavior more closely approximate a perfect receiver that has no I/Q impairments. In response to the application of a pure sinusoidal tone at arbitrary frequency ω as input, a perfect receiver would produce signals I(n) and Q(n) that are equal in amplitude and 90 degrees apart in phase, i.e., no gain imbalance and no phase skew.

[0146] The 2×2 matrix of digital filters may have the following properties. The frequency response of at least one of the diagonal components of the 2×2 matrix may be computed based on measurements of the I/Q impairments as a function of frequency and the measurements of the I/Q impairments as a function of the negative of frequency. For example, if one characterizes the I/Q impairments with a gain imbalance function g(f) and a phase skew function φ(f), with f covering the range of frequencies, the frequency response of the component h₂₂ (or the component h₁₁, or each of the components h₁₁ and h₂₂) may be computed based on the functions g(f), g(−f), φ(f) and φ(−f).

[0147] Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix may be computed based on the measurements of the I/Q impairments as a function of frequency and the measurements of the I/Q impairments as a function of the negative of frequency.

[0148] By saying that the filtering of the digital I signal and the digital Q signal is performed “in accordance with a 2×2 matrix of digital filters” is not meant to suggest that the receiver (or whatever device is used to implement the filtering) must include a filter circuit to implement a trivial multiplication by zero when the corresponding element of the 2×2 matrix is identically zero, or an adder to implement a trivial addition by zero. As an example, if h₁₂=0, then I_F(n) may be computed according to the simplified expression
I_F(n)=h₁₁(n)*I(n)
with only one convolution circuit. Similarly, if one of the components of the 2×2 matrix is a unit impulse at time n=0, then the receiver need not include a multiplier to perform that trivial convolution. For example, if h₁₁(n) is a unit impulse, then I_F(n) may be simply computed according to the expression
I_F(n)=I(n)+h₁₂(n)*Q(n)
with only one convolution unit and one adder. Thus, filtering “in accordance with a 2×2 matrix of digital filters” does not necessarily require a full 2×2 array of convolution circuits in all cases.

[0149] In some embodiments, the filtered digital I signal and the filtered digital Q signal are usable to recover a stream of information bits. The receiver (or another processing agent such as a host computer) may recover the stream of information bits by performing symbol demodulation on the filtered digital I signal and the filtered digital Q signal. In symbol demodulation, the vector signal (I_F(n),Q_F(n)) may be decimated to determine a sequence of complex symbols, and each of the complex symbols may be mapped to the closest constellation point in a given constellation (set of points in the complex plane). The sequence of resulting complex points determines a stream of information bits.

[0150] In some embodiments, the receiver includes a digitizer, where the digitizer performs the above-described actions of digitizing and filtering. The term “digitizer” is meant to imply an instrument that is calibrated to a known standard. For example, the relationship between the analog input and the digital output is calibrated to a known standard for both the I channel and the Q channel.

[0151] In some embodiments, the receiver is a test instrument such as a vector signal analyzer (VSA). (The term “vector signal” is a synonym for complex signal or I/Q signal.) The test instrument may receive the analog input signal from a transmitter, e.g., a transmitter under test. The analog input signal is received in response to the transmitter's action of transmitting a transmit signal onto the transmission medium. The test instrument may be configured to compensate for its own I/Q impairments, but to not compensate for the I/Q impairments of the transmitter. In the context of test and measurement, it is important to be able to accurately measure and report the impairments of a device under test rather than to compensate for the impairments of that device. Thus, for a test instrument, it may be preferable that the measurements of the receiver's I/Q impairments (on which the receiver's impairment compensation is based) do not include I/Q impairments of the transmitter. The present patent disclosure describes methods for measuring receiver-only impairments.

[0152] Test instruments are generally used to perform the testing of devices under test (DUTs) or systems under test (SUTs). Test instruments generally include one or more inputs and outputs for connecting to SUTs. The inputs and outputs may be analog, digital, radio frequency, etc., e.g., at various voltage levels and frequencies. Test instruments are generally able to perform one or more tests or features. For example, test instruments may be configured to capture and analyze waveforms, calculate measured power, generate a tone at a programmed frequency, etc. Test instruments are also typically calibrated in order to achieve a specified level of accuracy on its I/O. Finally, test instruments usually include a user interface in order to specify how the test instrument should behave.

[0153] In other contexts, the receiver may be expected to compensate for the transmitter's impairments and its own impairments. Thus, the 2×2 matrix of digital filters may be computed based on measurements of the I/Q impairments of the transmitter-and-receiver combination. The same principle regarding calculation of frequency responses based on the impairments as a function of f and the impairments as a function −f applies here.

[0154] In some embodiments, the filtering operation 125 may be performed on a programmable hardware element such as an FPGA, or in dedicated digital circuitry such as an application specific integrated circuit (ASIC). The programmable hardware element or dedicated digital circuitry may be supplied with the same sample clock that drives the ADC conversion.

[0155] In some embodiments, the filtering operation 125 may be performed by a processor in response to the execution of program instructions. The processor may be incorporated as part of the receiver, or as part of another system such as a host computer or controller board.

[0156] As described above, at least one of the diagonal components of the 2×2 matrix is computed based on the measurements of the I/Q impairments as a function of f and the measurements of the impairments as a function −f. In some embodiments, the “at least one diagonal” is to be interpreted as “exactly one diagonal”, and the other diagonal component of the 2×2 matrix is a discrete-time unit impulse function (e.g., taking the value one at time zero, and zero elsewhere).

[0157] As described above, at least one of the non-diagonal components of the 2×2 matrix is computed based on the measurements of the I/Q impairments as a function of f and the measurements of the I/Q impairments as a function −f. In some embodiments, the “at least one non-diagonal” is to be interpreted as “exactly one non-diagonal”, and the other non-diagonal component of the 2×2 matrix is the zero function.

[0158] Constraint Between Receiver Impairments at Frequency f and Frequency −f

[0159] In some embodiments, the receiver's I/Q impairments over positive frequencies and the receiver's I/Q impairments over negative frequencies may be assumed to be functionally related. In one such embodiment, the computation of the 2×2 matrix of digital filters may be simplified as follows. The frequency response of one of the diagonal components of the 2×2 matrix at the arbitrary frequency f in the frequency range may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f). For example, if the I/Q impairments are characterized by a gain imbalance function g(f) and a phase skew function φ(f), the frequency response H₂₂(f) of the component h₂₂ may be computed based only on a measurement of g(f) and a measurement of φ(f), where f ranges over the frequencies at which measurements have been obtained. Furthermore, the frequency response of one of the non-diagonal components of the 2×2 matrix at the frequency f may be computed based only on the measurement of the I/Q impairments at the frequency f (or alternatively, only on the measurement of the I/Q impairments at the frequency −f).

[0160] In some embodiments, the I/Q impairments at the frequency f and the I/Q impairments at frequency −f are constrained such that the I/Q impairments at f are determined by the I/Q impairments at −f, or such that the I/Q impairments at frequency −f are determined by the I/Q impairments at f. For example, the gain imbalance at the frequency f and the gain imbalance at frequency −f may be constrained to be equal, and the phase skew at frequency f and the phase skew at frequency −f may be constrained to be equal (or negatives of each other).

[0161] In some embodiments, the gain imbalance is assumed to be even and the phase skew is assumed to be odd. In these embodiments, both non-diagonal components of the 2×2 matrix may be set to zero; one of the diagonal components may correspond to a pure pass-through filter (i.e., unity frequency response); and a frequency response of the other diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f).

[0162] In some embodiments, both diagonal components of the 2×2 matrix may correspond to pure pass-through filters; one of the non-diagonal components may be set to zero; and a frequency response of the other non-diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f).

[0163] Receiver Configured for Wideband Correction

[0164] In one set of embodiments, a receiver 200 may be configured as shown in FIG. 2B. (Receiver 200 may include any subset of the features described above in connection with method 100.) Receiver 200 may include an I/Q demodulator 210, a digitization unit 215 and a digital circuit 220.

[0165] The I/Q demodulator 210 may be configured to receive an analog input signal y(t) and perform I/Q demodulation on the analog input signal to produce an analog inphase (I) signal and an analog quadrature (Q) signal, denoted I(t) and Q(t). The I/Q demodulator may receive a pair of orthogonal carriers from a local oscillator circuit.

[0166] The digitization unit 215 may be configured to digitize the analog I signal and the analog Q signal to produce, respectively, a digital I signal and a digital Q signal, which are denoted I(n) and Q(n). The digitization unit 215 may receive a conversion clock from a clock generation circuit. The digitization unit includes an I-channel ADC and a Q-channel ADC, each being driven by the same conversion clock.

[0167] The digital circuit 220 may be configured to filter the digital I signal and the digital Q signal in accordance with a 2×2 matrix of digital filters (as described above) to produce a filtered digital I signal and a filtered digital Q signal. The 2×2 matrix of digital filters may be configured to compensate (or, at least partially compensate) for I/Q impairments of the receiver over a range of frequencies. The digital circuit, when programmed with the 2×2 matrix of digital filters, makes the receiver 200 behave more like a mathematically perfect receiver, i.e., one having a perfect I/Q demodulator and perfect digitization unit.

[0168] In some embodiments, the digital circuit 220 is realized by (or, as part of) a programmable hardware element, or dedicated digital circuitry such as an ASIC.

[0169] In some embodiments, the digital circuit 220 is (or includes, or is realized by) a processor that is configured to execute program instructions. In one embodiment, the processor is part of a computer system such as a host computer system or controller board.

[0170] In some embodiments, the receiver 200 may include a means for recovering a stream of information bits by performing symbol demodulation on the filtered digital I signal and the filtered digital Q signal. The recovering means may include any one or more of the following: a processor executing on the receiver, a processor executing on a host computer, a processor executing on a controller board (e.g., a controller board installed in an instrumentation chassis along with the receiver), a programmable hardware element, an ASIC.

[0171] In some embodiments, the receiver 200 is (or includes) a test instrument. See the above discussion of the notion of a test instrument.

[0172] Method for Configuring a Receiver to Perform Impairment Correction

[0173] In one set of embodiments, a method 300 for configuring a receiver may involve the operations shown in FIG. 3. The method 300 may be used to configure the receiver to at least partially compensate for I/Q impairments of the receiver. The method 300 may be implemented by a computer system in response to the execution of program instructions. (The method 300 may include any subset of the features described above.)

[0174] At 310, the computer system may receive measurements of the I/Q impairments of the receiver over a frequency band. (“Over a frequency band” means that the measurements include measurements at a plurality of different frequencies within the frequency band, e.g., uniformly or non-uniformly covering the frequency band.) The receiver may include an I/Q demodulator, a pair of analog-to-digital converters (ADCs) and a digital circuit, e.g., as described above. The I/Q demodulator may be configured to generate an analog I signal and an analog Q signal from an analog input signal. The ADCs may be configured to sample the analog I signal and the analog Q signal to respectively obtain a digital I signal and a digital Q signal. The digital circuit may be configured to filter the digital I signal and the digital Q signal to obtain a filtered digital I signal and a filtered digital Q signal. (See the above discussion for various ways of realizing the digital circuit.)

[0175] At 315, the computer system may compute a 2×2 matrix of digital filters based on the measurements. The 2×2 matrix of digital filters may be computed to achieve at least partial compensation for the I/Q impairments of the receiver over the frequency band. A frequency response of at least one of the diagonal components of the 2×2 matrix may be computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. Furthermore, a frequency response of at least one of the non-diagonal components of the 2×2 matrix may be computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency.

[0176] At 320, the computer system may program the digital circuit to implement the 2×2 matrix of digital filters, where the digital circuit, when so programmed, is configured to at least partially compensate for the I/Q impairments of the receiver over the frequency band. The action of programming the digital circuit involves transferring the 2×2 matrix of digital filters (or parameters specifying those filters) to the digital circuit or to a memory used by the digital circuit.

[0177] Wideband Correction Method for Transmitter

[0178] In one set of embodiments, a method 400 for compensating for I/Q impairments of a transmitter may involve the operations shown in FIG. 4.

[0179] At 410, a digital inphase (I) signal and a digital quadrature (Q) signal may be received. The digital I signal and the digital Q signal may be interpreted as the components of a complex-valued signal I(n)+jQ(n). The digital I signal and the digital Q signal may carry one or more streams of information bits, e.g., as the result of symbol modulation according to a given constellation. In some embodiments, the digital I signal and the digital Q signal may be interpreted as the components of a complex-valued baseband signal or intermediate frequency (IF) signal.

[0180] At 415, the digital I signal and the digital Q signal may be filtered in accordance with a 2×2 matrix of digital filters to produce a filtered digital I signal and a filtered digital Q signal. (The filtering operation may be performed by the transmitter or some other agent.) The filtering operation may involve applying the 2×2 matrix (h_ij) of digital filters according to the relations:
I_F(n)=h₁₁(n)*I(n)+h₁₂(n)*Q(n),
Q_F(n)=h₂₁(n)*I(n)+h₂₂(n)*Q(n).

[0181] The 2×2 matrix of digital filters may pre-compensate (or, at least partially pre-compensate) for the I/Q impairments of the transmitter over a range of frequencies, e.g., over a frequency range broad enough to cover the bandwidth of a communication signal to be transmitted.

[0182] The 2×2 matrix of digital filters may have the following properties. The frequency response of at least one of the diagonal components of the 2×2 matrix may be computed based on measurements of the I/Q impairments as a function of frequency and the measurements of the I/Q impairments as a function of the negative of frequency. For example, if the I/Q impairments are characterized by a gain imbalance function g(f) and a phase skew function φ(f), with f covering the range of frequencies, the frequency response of the digital filter h₂₂ (or the digital filter h₁₁, or each of the digital filters h₁₁ and h₂₂) may be computed based on g(f), g(−f), φ(f) and φ(−f).

[0183] Furthermore, a frequency response of at least one of the non-diagonal components of the 2×2 matrix may be computed based on the measurements of the I/Q impairments as a function of frequency and the measurements of the I/Q impairments as a function of the negative of frequency.

[0184] In the description of the receiver 100, we were careful to qualify the meaning of filtering “in accordance with a 2×2 matrix of digital filters”. Those same qualifications apply here for the transmitter compensation.

[0185] At 420, the transmitter may convert the filtered digital I signal and the filtered digital Q signal to analog form in order to respectively obtain an analog I signal and an analog Q signal.

[0186] At 425, the transmitter may perform I/Q modulation on the analog I and Q signals to produce a modulated analog signal. The modulated analog signal may be transmitted onto a transmission medium, e.g., a transmission medium as described above. A receiver may receive the modulated analog signal, likely in a noise-perturbed and channel-distorted form.

[0187] Above, we described the 2×2 matrix of digital filters as “pre-compensating” for I/Q impairments of the transmitter. That is because the I/Q impairments occur after the application of the digital filters, especially in the I/Q modulation stage. Thus, the 2×2 matrix may be interpreted as applying an inverse distortion that together with the following distortions will give an approximation to the identity map overall.

[0188] In some embodiments, the filtering operation 415 may be performed in a programmable hardware element (PHE) such as an FPGA, or in dedicated digital circuitry such as an application-specific integrated circuit (ASIC).

[0189] In some embodiments, the filtering operation 415 may be performed by a processor in response to the execution of program instructions, e.g., a processor of a host computer system or an instrumentation controller broad.

[0190] In some embodiments, the transmitter is a test instrument (e.g., an arbitrary waveform generator or a vector signal generator). The test instrument may transmit the modulated analog signal to a receiver, e.g., a receiver under test. In the context of test and measurement, it may be important for the test instrument to correct for its own impairments but to not correct for the impairments of the receiver. Thus, in this context, the above-described measurements of the transmitter's I/Q impairments (on which the transmitter's pre-compensation is based) preferably do not include I/Q impairments of the receiver. This patent disclosure describes methods for measuring transmitter-only impairments (cleanly separated from receiver impairments).

[0191] In some contexts, the transmitter may be expected to correct for the receiver's impairment and its own impairments. Thus, the 2×2 matrix of digital filters may be computed based on measurements of the I/Q impairments of the transmitter-and-receiver combination. The same principle regarding calculation of frequency responses based on the impairments as a function off and the impairments as a function −f applies here.

[0192] Constraint Between Transmitter Impairments at Frequency f and Frequency −f

[0193] In some embodiments, the transmitter's I/Q impairments over positive frequencies and the transmitter's I/Q impairments over negative frequencies may be assumed to be functionally related. In one such embodiment, the computation of the 2×2 matrix of digital filters may be simplified as follows. The frequency response of at least one of the diagonal components of the 2×2 matrix at the arbitrary frequency f in the frequency range may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f). For example, if the I/Q impairments are characterized by a gain imbalance function g(f) and a phase skew function φ(f), the frequency response value H₂₂(f) of the component h₂₂ may be computed based only on the a measurement of g(f) and a measurement of φ(f), where f ranges over the frequencies at which measurements have been obtained. Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix at the frequency f may be computed based only on the measurement of the I/Q impairments at the frequency f (or alternatively, only on the measurement of the I/Q impairments at the frequency −f).

[0194] In some embodiments, the I/Q impairments at the frequency f and the I/Q impairments at frequency −f are constrained such that the I/Q impairments at f are determined by the I/Q impairments at −f, or such that the I/Q impairments at frequency −f are determined by the I/Q impairments at f. For example, the gain imbalance at the frequency f and the gain imbalance at frequency −f may be constrained to be equal, and the phase skew at frequency f and the phase skew at frequency −f may be constrained to be equal (or alternatively, negatives of each other).

[0195] In some embodiments, the gain imbalance is assumed to be even and the phase skew is assumed to be odd. Then both non-diagonal components of the 2×2 matrix may be set to zero; one of the diagonal components may correspond to a pure pass-through filter (i.e., unity frequency response); and a frequency response of the other diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f).

[0196] In some embodiments, both diagonal components of the 2×2 matrix may correspond to pure pass-through filters; one of the non-diagonal components may be set to zero; and a frequency response of the other non-diagonal component at the arbitrary frequency f may be computed based only on a measurement of the I/Q impairments at the frequency f (or alternatively, only on a measurement of the I/Q impairments at the frequency −f).

[0197] Transmitter Configured for Wideband Correction

[0198] In one set of embodiments, a transmitter 500 may be configured as shown in FIG. 5. (Transmitter 500 may incorporate any subset of the features described above in connection with method 400.) Transmitter 500 may include a digital circuit 510, a digital-to-analog conversion (DAC) unit 515 and an I/Q modulator 520.

[0199] The digital circuit 510 may be configured to receive a digital inphase (I) signal and a digital quadrature (Q) signal, and filter the digital I signal and the digital Q signal with a 2×2 matrix of digital filters to produce a filtered digital I signal and a filtered digital Q signal. (The filtering may be performed as variously described above.) The digital I signal and the digital Q signal may carry one or more streams of information bits.

[0200] The 2×2 matrix of digital filters may be computed to pre-compensate (or, at least partially pre-compensate) for I/Q impairments of the transmitter over a range of frequencies. A frequency response of at least one of the diagonal components of the 2×2 matrix may be computed based on measurements of the I/Q impairments as a function of frequency and the measurements of the I/Q impairments as a function of the negative of frequency. Furthermore, a frequency response of at least one of the non-diagonal components of the 2×2 matrix may be computed based on the measurements of the I/Q impairments as a function of frequency and the measurements of the I/Q impairments as a function of the negative of frequency.

[0201] The digital circuit 510 is said to “pre-compensate” for the I/Q impairment of the transmitter because the I/Q impairments occur in transmitter stages after the digital circuit, especially in the I/Q modulator 520. Thus, the digital circuit (by applying the 2×2 matrix of digital filters) introduces a pre-distortion to the complex signal I(n)+jQ(n) so that the net effect of the pre-distortion followed by the subsequent impairments will approximate an ideal transmitter having no I/Q impairments. In other words, the digital circuit applies an inverse distortion that in combination with the subject distortion approximates the identity map (i.e., frequency response function identically equal to unity).

[0202] The DAC unit 515 may be configured to convert the filtered digital I and Q signals to analog form in order to obtain respective analog I and Q signals. The DAC unit 515 may receive a conversion clock from a clock generation unit. The digital circuit 510 may receive the same conversion clock so that it generates the complex samples (I_F(n),Q_F(n)) at the same rate that the DAC unit converts the samples into analog form (I(t),Q(t)).

[0203] The I/Q modulator 520 may be configured to perform I/Q modulation on the analog I and Q signals to produce a modulated analog signal. The modulated analog signal may be transmitted to a receiver through a transmission medium. The notion of I/Q modulation is well understood in the field of communication. For example, the I/Q modulation may be modeled by the expressions:

[0204] $x\left(t\right)=I\left(t\right)\cos \left(\omega t\right)-Q\left(t\right)\sin \left(\omega t\right)=\mathrm{Re}\left\{\left(I\left(t\right)+jQ\left(t\right)\right)\exp \left(\mathrm{j\omega }t\right)\right\},$
where ω is the carrier frequency.

[0205] In some embodiments, the digital circuit 510 is realized by (or, as part of) a programmable hardware element, or dedicated digital circuitry such as an ASIC.

[0206] In some embodiments, the digital circuit 510 is (or includes, or is realized by) a processor that is configured to execute program instructions. In one embodiment, the processor is part of a computer system such as a host computer system or controller board.

[0207] In some embodiments, the transmitter 500 may be a test instrument. See the above discussion of test instrument in the context of method 400.

[0208] Method for Configuring a Transmitter for Impairment Correction

[0209] In one set of embodiments, a method 600 for configuring a transmitter may involve the operations shown in FIG. 6. The method 600 may be used to configure the transmitter to at least partially compensate for I/Q impairments of (or introduced by) the transmitter. The method 600 may be performed by a computer system in response to the execution of program instructions.

[0210] At 610, the computer system may receive measurements of the I/Q impairments of the transmitter over a frequency range. (“Over a frequency range” implies that the measurements of the I/Q impairments are obtained at a plurality of frequencies within the frequency range, e.g., frequencies covering the frequency range uniformly or non-uniformly.) The transmitter may include a digital circuit, a pair of digital-to-analog converters (DACs) and an I/Q modulator. The digital circuit may be configured to filter a digital I signal and a digital Q signal to respectively obtain a filtered digital I signal and a filtered digital Q signal. The pair of DACs may be configured to convert the filtered digital I signal and the filtered digital Q signal to analog form in order to respectively obtain an analog I signal and an analog Q signal. The I/Q modulator may be configured to modulate a carrier signal with the analog I and Q signals to obtain a modulated carrier signal. The modulated carrier signal may be transmitted to a receiver through a transmission channel.

[0211] At 615, the computer system may compute a 2×2 matrix of digital filters for the digital circuit based on the measurements. The 2×2 matrix of digital filters may be computed to achieve pre-compensation (or, at least partial pre-compensation) for the I/Q impairments of the transmitter over the frequency range. The frequency response of at least one of the diagonal components of the 2×2 matrix may be computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency. Furthermore, the frequency response of at least one of the non-diagonal components of the 2×2 matrix may be computed based on the measurements as a function of frequency and the measurements as a function of the negative of frequency.

[0212] At 620, the computer system may program the digital circuit to implement the 2×2 matrix of digital filters, where the digital circuit, when so programmed, is configured to at least partially pre-compensate for the I/Q impairments over the frequency range. The action of programming the digital circuit involves transferring the digital filters (or parameters specifying the filters) to the digital circuit or to parameter memory used by the digital circuit.

[0213] In various embodiments, the digital circuit may be a programmable hardware element, an application specific integrated circuit (ASIC), a processor executing under the control of program instructions, or any combination thereof.

[0214] Derivation of Digital Filters for Wideband Impairment Compensation

[0215] As described above, a 2×2 matrix of digital filters may be used to compensate for I/Q impairments at a receiver or a transmitter. (Indeed, both the transmitter and the receiver may employ the matrix compensation, each using its own 2×2 compensation matrix. The transmitter's compensation matrix may be computed based on the transmitter's I/Q impairments, while the receiver's compensation matrix may be computed based on the receiver's I/Q impairments.) This section derives the frequency responses for the digital filters in the special case where the 2×2 matrix has the special form shown in FIG. 7.

[0216] Since gain imbalance g and phase skew φ are relative measurements, we have the freedom to model the gain imbalance and phase skew as being due to distortions on only one channel (I or Q), the other channel being ideal. FIG. 7 represents the choice of modeling both the gain imbalance and the phase skew as being due to distortions on the Q channel only. FIG. 8 illustrates the opposite choice. (Thus, frequency responses H₁₁ and H₁₂ are used to effect the compensation, while H₂₂=1 and H₂₁=0.) One might also model the gain imbalance as being due to amplitude distortion on one channel only, and the phase skew as being due to phase distortion on the opposite channel only. As yet another alternative, one might model the gain imbalance and/or phase skew as being due to partial distortions on both channel, e.g., as suggested by FIG. 9. Thus, the digital compensation may be performed using all four frequency responses H₁₁, H₁₂, H₂₁ and H₂₂. After appreciating the following derivation based on FIG. 7, one of ordinary skill in the art will find it straightforward to apply the same mathematical principles to all other cases.

[0217] FIG. 7 may be interpreted as the filtering operation performed by the receiver's digital circuit 220 or the filtering operation performed by the transmitter's digital circuit 510. Thus, the following derivation applies both to the transmitter's compensation matrix and to the receiver's compensation matrix.

[0218] While the compensation is applied digitally, for the sake of simplicity, the following derivation will be expressed in terms of continuous time t. To achieve compensation, we seek frequency responses U(ω) and V(ω) so that the distorted signal
cos(ωt)+jg(ω)sin(ωt+φ(ω))
gets transformed to corrected signal cos(ωt)+jsin(ωt) for all frequencies ω in a frequency band (e.g., a frequency band that is symmetric about zero), or at least at selected frequencies where measurements of the impairments g(ω) and φ(ω) are available. g(ω) is the gain imbalance corresponding to frequency ω, and φ(ω) is the phase skew corresponding to frequency ω. Thus, we obtain the equation:
u(t)*cos(ωt)+v(t)*g(ω)sin(ωt+φ(ω))=sin(ωt),
where “*” denotes convolution, where u(t) and v(t) are the impulse responses corresponding respectively to the frequency responses U(ω) and V(ω).

[0219] By making the substitutions
cos(θ)=(½){exp(jθ)+exp(−jθ)}
sin(θ)=(−j/2){exp(jθ)−exp(−jθ)},
we obtain the equation

[0220] $\left(1\text{/}2\right)U\left(\omega \right)\exp \left(\mathrm{j\omega }t\right)+\left(1\text{/}2\right)U\left(-\omega \right)\exp \left(-\mathrm{j\omega }t\right)+\left(-j/2\right)V\left(\omega \right)g\left(\omega \right)\exp \left(\mathrm{j\varphi }\left(\omega \right)\right)\exp \left(\mathrm{j\omega }t\right)+\left(j/2\right)V\left(-\omega \right)g\left(\omega \right)\exp \left(-\mathrm{j\varphi }\left(\omega \right)\right)\exp \left(-\mathrm{j\omega }t\right)=\left(-j/2\right)\left[\exp \left(\mathrm{j\omega }t\right)-\exp \left(-\mathrm{j\omega }t\right)\right\}.$

[0221] Due to the linear independence of exp(jωt) and exp(−jωt), we obtain the following two equations:
jU(ω)+V(ω)g(ω)exp(jφ(ω))=1  (a):
jU(−ω)−V(−ω)g(ω)exp(−jφ(ω))=−1.  (b):
Because equation (b) holds for all ω, we can replace ω with −ω, thus obtaining equation (b′) below.
jU(ω)−V(ω)g(−ω)exp(−jφ(−ω))=−1.  (b′):

[0222] Equations (a) and (b′) specify a matrix equation in vector unknown [U(ω),V(ω)]^T, whose solution is given by:

[0223] $U\left(\omega \right)=j\frac{g\left(\omega \right)\exp \left\{j\phi \left(\omega \right)\right\}-g\left(-\omega \right)\exp \left\{-j\phi \left(-\omega \right)\right\}}{g\left(\omega \right)\exp \left\{j\phi \left(\omega \right)\right\}+g\left(-\omega \right)\exp \left\{-j\phi \left(-\omega \right)\right\}},V\left(\omega \right)=\frac{2}{g\left(\omega \right)\exp \left\{j\phi \left(\omega \right)\right\}+g\left(-\omega \right)\exp \left\{-j\phi \left(-\omega \right)\right\}}.$

[0224] Observe that U(ω) and V(ω) each depend on g(ω), g(−ω), φ(ω) and φ(−ω). This property of the frequency responses (of the digital filters) depending on impairment information at ω and −ω applies more generally than to the special matrix form of FIG. 7. Indeed, it applies to any form of the compensation matrix. Also observe that U and V are conjugate symmetric with respect to frequency: U(−ω)=U(ω)* and V(−ω)=V(ω)*, as expected for filters whose impulses responses are entirely real.

[0225] To simplify the process of designing digital filters (impulse responses) corresponding to the frequency responses U(ω) and V(ω), it may be useful to express those frequency responses in terms of their even and odd parts:
U(ω)=A(ω)+B(ω)
A(ω)=(½){U(ω)+U(−ω)}
B(ω)=(½){U(ω)−U(−ω)}
V(ω)=C(ω)+D(ω)
C(ω)=(½){V(ω)+V(−ω)}
D(ω)=(½){V(ω)−V(−ω)}.

[0226] In the time domain, the corresponding expressions are:
u(t)=a(t)+b(t)
a(t)=(½){u(t)+u(−t)}
b(t)=(½){u(t)−u(−t)}
v(t)=c(t)+d(t)
c(t)=(½){v(t)+v(−t)}
d(t)=(½){v(t)−v(−t)},
where u, a, b, v, c, and d are the impulse responses corresponding respectively to frequency responses U, A, B, V, C and D.

[0227] Using the above-derived expressions for U(ω) and V(ω), it follows that:

[0228] $A\left(\omega \right)=\frac{-2g\left(\omega \right)g\left(-\omega \right)\sin \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}{g^2\left(\omega \right)+g^2\left(-\omega \right)+2g\left(\omega \right)g\left(-\omega \right)\cos \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}$ $B\left(\omega \right)=\frac{j\left\{g^2\left(\omega \right)-g^2\left(-\omega \right)\right\}}{g^2\left(\omega \right)+g^2\left(-\omega \right)+2g\left(\omega \right)g\left(-\omega \right)\cos \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}$ $C\left(\omega \right)=\frac{2g\left(\omega \right)\cos \left\{\phi \left(\omega \right)\right\}+2g\left(-\omega \right)\cos \left\{\phi \left(-\omega \right)\right\}}{g^2\left(\omega \right)+g^2\left(-\omega \right)+2g\left(\omega \right)g\left(-\omega \right)\cos \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}$ $D\left(\omega \right)=\frac{2j\left\{-g\left(\omega \right)\sin \left(\phi \left(\omega \right)\right)+g\left(-\omega \right)\sin \left(\phi \left(-\omega \right)\right)\right\}}{g^2\left(\omega \right)+g^2\left(-\omega \right)+2g\left(\omega \right)g\left(-\omega \right)\cos \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}.$

[0229] The above expressions may be used to compute frequency responses U and V based on measured or estimated impairment functions g and φ. These expressions apply equally to post-compensation at the receiver or pre-compensation at the transmitter. In other words, the frequency responses U(ω) and V(ω) for pre-correcting I/Q impairments g(f) and φ(f) are the same as the frequency responses for post-correction those same I/Q impairments.

[0230] The computed frequency responses U and V may be used to determine corresponding impulses responses u(n) and v(n) using any of various known filter design algorithms.

[0231] Note Regarding Filters with Odd Frequency Response

[0232] Given a filter with odd frequency response B(ω), it is basic fact that the function E_B(ω) given by
E_B(ω)=jB(ω)sgn(ω)
is even and has the property that:
b(t)*x(t)=HT(e_B(t)*x(t)),
where HT is the Hilbert transform operator, where b(t) is the impulse response corresponding to B(ω), and x(t) is an arbitrary input function, where sgn(ω) is 1 if ω is greater than zero and −1 if ω is less than zero.

[0233] If we apply this fact to the odd functions B(ω) and D(ω) from the discussion above, we arrive at the corresponding even functions:

[0234] $E_B\left(\omega \right)=\frac{\left\{g^2\left(-w\right)-g^2\left(\omega \right)\right\}\mathrm{sgn}\left(\omega \right)}{g^2\left(\omega \right)+g^2\left(-\omega \right)+2g\left(\omega \right)g\left(-\omega \right)\cos \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}$ $E_D\left(\omega \right)=\frac{2\mathrm{sgn}\left(\omega \right)\left\{g\left(\omega \right)\sin \left(\phi \left(\omega \right)\right)-g\left(-\omega \right)\sin \left(\phi \left(-\omega \right)\right)\right\}}{g^2\left(\omega \right)+g^2\left(-\omega \right)+2g\left(\omega \right)g\left(-\omega \right)\cos \left\{\phi \left(\omega \right)+\phi \left(-\omega \right)\right\}}.$

[0235] Note Regarding Special Case of Even g(ω) and Odd φ(ω)

[0236] In many circumstances, the gain imbalance function may be modeled as being even and the phase skew function may be modeled as being odd, i.e., g(ω)=g(−ω) and φ(ω)=−φ(−ω). Under these constraints, U(ω)=0 and V(ω) is complex.

[0237] Note Regarding Special Case of Even g(ω) and Even φ(ω)

[0238] The above-derived expressions for U(ω) and V(ω) are typically complex valued. However, when the gain imbalance and phase skew functions are even, i.e., g(ω)=g(−ω) and φ(ω)=φ(−ω), it follows that U(ω) and V(ω) become real valued:
U(ω)=−tan(φ(ω))
V(ω)=1/{g(ω)cos φ(ω)}.

[0239] Constant Matrix to Post-Correct Receiver Impairments at a Single Frequency

[0240] In one set of embodiments, a method 1000 for operating a receiver (or operating a system including a receiver) may involve the operations shown in FIG. 10.

[0241] At 1010, the receiver may receive an analog input signal. The analog input signal may be received from a transmission medium, e.g., as described above.

[0242] At 1015, the receiver may perform I/Q demodulation on the analog input signal to produce an analog inphase (I) signal and an analog quadrature (Q) signal, e.g., as described above.

[0243] At 1020, the receiver may digitize the analog I signal and the analog Q signal to produce respectively a digital I signal and a digital Q signal.

[0244] At 1025, the receiver may transform the digital I signal and the digital Q signal in accordance with a 2×2 matrix c=(c_ij) of constants to produce a resultant digital I signal and a resultant digital Q signal. The transformation may be performed by applying the following matrix multiplication:

[0245] $\left[\begin{array}{c}{I}_R\left(n\right)\\ Q_R\left(n\right)\end{array}\right]=\left[\begin{array}{cc}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}\right]\left[\begin{array}{c}I\left(n\right)\\ Q\left(n\right)\end{array}\right],$
where I_R(n) and Q_R(n) represent the resultant digital I signal and the resultant digital Q signal respectively. The 2×2 matrix c may be configured to post-compensate (or, at least partially post-compensate) for measured I/Q impairments of the receiver at a particular frequency f.

[0246] The matrix c may have the following properties. At least one of the diagonal elements c₁₁ and c₂₂ may be computed based on the measured I/Q impairments of the receiver at frequency f. For example, the coefficient c₂₂ may be computed as a function of the measured value g(f) and/or the measured value φ(f), where g is the gain imbalance function and φ is the phase skew function. Similarly, at least one of the non-diagonal elements c₁₂ and c₂₁ may be computed based on the measured I/Q impairments of the receiver at frequency f. For example, the coefficient c₂₁ may be computed as a function of the measured value g(f) and/or the measured value φ(f). In some embodiments, each of the four matrix elements is similarly computed (i.e., based on the measured impairments at f).

[0247] See the section “Performing Traditional Impairment Compensation at a Single Frequency” for one possible embodiment of the matrix c.

[0248] Let c_ij(f) denote the functional expression that is used to determine the coefficient c_ij from the I/Q impairments at frequency f. Due to the continuity of the functional expressions c_ij(f) with respect frequency f, the matrix c(f) is a good approximation for the matrix c(f+Δf) as long as Δf is sufficiently small. Thus, when the receiver uses the matrix c(f) to perform transform operation 1025, the receiver will achieve at least partial compensation over a neighborhood of frequencies around f. The quality of the compensation will typically degrade as Δf increases in absolute value.

[0249] In some embodiments, the analog input signal is a pure sinusoidal tone, e.g., a tone at frequency f or a tone at frequency f+f_LO, where f_LO is the frequency of the receiver's local oscillator. In other embodiments, the analog input signal is a communication signal that carries a stream of binary information.

[0250] In some embodiments, the matrix c has the additional property that one of its diagonal elements is one. In some embodiments, the matrix c has the additional property that one of its non-diagonal elements is zero. In some embodiments, the matrix c has one of the following special forms:

[0251] $c=\left[\begin{array}{cc}1& 0\\ c_{21}& c_{22}\end{array}\right],c=\left[\begin{array}{cc}{c}_{11}& c_{12}\\ 0& 1\end{array}\right].$

[0252] As described above, the transform operation 1025 is performed “in accordance with a 2×2 matrix”. That qualifying phrase is not meant to require that the receiver include a multiplier (or adder) to implement a trivial multiplication by one (or a trivial addition by zero). For example, in the first special form given above, the resultant digital I signal is equal to the digital I signal: I_R(n)=I(n). This requires no computation at all. One can simply pass the I(n) input to the I_R(n) output.

[0253] In one set of embodiments, a receiver 1100 may be configured as shown in FIG. 11. (The receiver 1100 may incorporate any subset of the features described above in connection with method 1000.) The receiver 1100 may include an I/Q demodulator 1110, a digitization unit 1115 and a digital circuit 1120.

[0254] The I/Q demodulator 1110 may be configured to receive an analog input signal, and perform I/Q demodulation on the analog input signal to produce an analog inphase (I) signal and an analog quadrature (Q) signal. The analog input signal may be received from a transmission medium as described above.

[0255] The digitization unit 1115 may be configured to digitize the analog I signal and the analog Q signal to produce respectively a digital I signal and a digital Q signal.

[0256] The digital circuit 1120 may be configured to transform the digital I signal and the digital Q signal in accordance with a 2×2 matrix of constants to produce a resultant digital I signal and a resultant digital Q signal. The 2×2 matrix may be configured to at least partially compensate for I/Q impairments of the receiver at a particular frequency f. A first of the constants, corresponding to a first diagonal element of the 2×2 matrix, may be computed based on measured I/Q impairments of the receiver at frequency f. Furthermore, a second of the constants, corresponding to a first non-diagonal element of the 2×2 matrix, may be computed based on the measured I/Q impairments at frequency f. In some embodiments, each of the four constants is similarly computed (i.e., based on the measured impairments at f).

[0257] In one set of embodiments, a method 1200 for configuring a receiver may involve the operations shown in FIG. 12. The method 1200 may be used to configure the receiver to at least partially compensate for I/Q impairments of the receiver at a given frequency f. The method 1200 may be implemented by a computer system in response to the execution of program instructions. (The method 1200 may include any subset of the features described above in connection with FIGS. 10 and 11.)

[0258] At 1210, the computer system may receive measured I/Q impairments of the receiver at frequency f. The receiver may include an I/Q demodulator, an analog-to-digital conversion (ADC) unit and a digital circuit, e.g., as described above in connection with FIGS. 10 and 11. The I/Q demodulator may be configured to generate an analog I signal and an analog Q signal from an analog input signal. The ADC unit may be configured to sample the analog I signal and the analog Q signal to respectively obtain a digital I signal and a digital Q signal. The digital circuit may be configured to transform the digital I signal and the digital Q signal to obtain a resultant digital I signal and a resultant digital Q signal. (See the discussions above for various ways of realizing the digital circuit.)

[0259] At 1215, the computer system may compute a 2×2 matrix of constants based on the measured I/Q impairments at frequency f. The 2×2 matrix may be computed to achieve at least partial compensation for the measured I/Q impairments at frequency f. At least one of the diagonal components of the 2×2 matrix may be computed based on the measured I/Q impairments at frequency f. Furthermore, at least one of the non-diagonal components of the 2×2 matrix may be computed based on the measured I/Q impairments at frequency f.

[0260] At 1220, the computer system may program the digital circuit to implement the 2×2 matrix of constants, where the digital circuit, when so programmed, is configured to at least partially compensate for the measured I/Q impairments at frequency f. The action of programming the digital circuit involves transferring the 2×2 matrix (or information specifying the matrix) to the digital circuit or to a memory used by the digital circuit.

[0261] True Matrix Pre-Correction at a Single Frequency

[0262] In one set of embodiments, a method 1300 for compensating for the I/Q impairments of a transmitter at a particular frequency f may involve the operations shown in FIG. 13.

[0263] At 1310, a digital inphase (I) signal and a digital quadrature (Q) signal (e.g., as variously described above) may be received. In some embodiments, the digital I signal and the digital Q signal together may represent a complex exponential tone at the frequency f. In other embodiments, the digital I signal and the digital Q signal may carry respective streams of binary information. The digital I and Q signals may be the components of a complex baseband signal or of a complex intermediate-frequency signal.

[0264] At 1315, the digital I signal and the digital Q signal may be transformed in accordance with a 2×2 matrix c=(c_ij) of constants to produce a resultant digital I signal and a resultant digital Q signal. (The transformation may be performed by the transmitter or some other agent.) The transformation may be described by the following matrix multiplication:

[0265] $\left[\begin{array}{c}{I}_R\left(n\right)\\ Q_R\left(n\right)\end{array}\right]=\left[\begin{array}{cc}{c}_{11}& c_{12}\\ c_{21}& c_{22}\end{array}\right]\left[\begin{array}{c}I\left(n\right)\\ Q\left(n\right)\end{array}\right],$

[0266] where I_R(n) and Q_R(n) represent the resultant digital I signal and the resultant digital Q signal respectively. The 2×2 matrix may be configured to pre-compensate (or, at least partially pre-compensate) for the I/Q impairments of the transmitter at the frequency f. See the above discussion regarding the nature of “pre-compensation”. In brief, the application of the transform introduces an inverse distortion that, in combination with the distortions of the following transmitter stages, makes the transmitter appear more ideal in its input-output behavior.) Note that the above discussion regarding the meaning of transforming “in accordance with a 2×2 matrix” applies here as well.

[0267] The 2×2 matrix c may have the following properties. At least one of the diagonal elements c₁₁ and c₂₂ may be computed based on a measurement of the I/Q impairments at frequency f and a measurement of the I/Q impairments at frequency −f. For example, the diagonal element c₂₂ may be computed based on a measured value of each of g(f), g(−f), φ(f) and φ(−f), where g is the gain imbalance function and φ is the phase skew function. For example, the diagonal element c₂₂ may be computed based on a measured value of each of g(f), g(−f), φ(f) and φ(−f), where g is the gain imbalance function and φ is the phase skew function. Furthermore, at least one of the non-diagonal elements c₁₂ and c₂₁ may be computed based on the measurement at frequency f and the measurement at frequency −f. In some embodiments, each of the four coefficients may be computed based on the measurement at frequency f and the measurement at frequency −f. See the section “Calculating True Single Point Vector Calibration Constants” for one possible embodiment of the matrix c.

[0268] The measured impairments may be impairments measured at the output of the corruption (i.e. the I/Q modulator) and may be different than the impairments if they could be measured at the input. Alternatively, the method may include transforming the output impairments at +f and −f to input impairments at only +f and then computing the matrix constants according to simplified formulas using the input impairments at only +f. The transformation may be derived follows. First, derive specialized expressions for U(f) and V(f) based on the output impairments at +f and −f using equations (7.9) and (7.10) with g_in(f)=g_in(−f)=1 and φ_in(f)=φ_in(−f)=0. Then compute the input impairments g_in(f) and φ_in(f) based on equation (7.7) with g_out(f)=1 and φ_out(f)=0:
g_in(f)exp(jφ_in(f))={1−jU(f)}/V(f).
The matrix constants may then be determined based on g_in(f) and φ_in(f), e.g., according to the relations α=g_in(f)sin(φ_in(f)) and β=g_in(f)cos(φ_in(f)).

[0269] The quality of the compensation achieved by operation 1315 will be limited by the quality of the impairment measurements. The present patent disclosure describes methods for obtaining quality measurements of the transmitter's I/Q impairments at any given frequency, or, over a whole range of frequencies.

[0270] Let c_ij(f) denote the functional expression that is used to determine the coefficient c_ij from the I/Q impairments at frequency f and the I/Q impairments at frequency −f. Due to the continuity of the functional expressions c_ij(f) with respect frequency f, the matrix c(f) is a good approximation for the matrix c(f+Δf) as long as Δf is sufficiently small. Thus, when the transmitter uses the matrix c(f) to perform transform operation 1315, the transmitter will achieve at least partial compensation over a neighborhood of frequencies around f. The quality of the compensation will typically degrade as Δf increases in absolute value.

[0271] At 1320, the transmitter may convert the resultant digital I and Q signals to analog form in order to obtain respective analog I and Q signals.

[0272] At 1325, the transmitter may perform I/Q modulation on the analog I and Q signals to produce a modulated analog signal, e.g., as described above.

[0273] In some embodiments, the matrix c has one of the following special forms:

[0274] $c=\left[\begin{array}{cc}1& 0\\ c_{21}& c_{22}\end{array}\right],c=\left[\begin{array}{cc}{c}_{11}& c_{12}\\ 0& 1\end{array}\right].$

[0275] In the first special form above, the constants c₂₁ and c₂₂ may be computed based on the values A(f), E_B(f), C(f) and E_D(f) as described in the section “Calculating True Single Point Vector Calibration Constants”, especially at equations (1.81) and (1.82).

[0276] In some embodiments, the transformation 1315 may be performed on a programmable hardware element such as an FPGA, or in dedicated digital circuitry such as an application specific integrated circuit (ASIC). The programmable hardware element or dedicated digital circuitry may be supplied with the same sample clock that drives the ADC conversion.

[0277] In some embodiments, the transformation 1315 may be performed by a processor in response to the execution of program instructions. The processor may be incorporated as part of the transmitter, or as part of another system such as a host computer or controller board.

[0278] In one set of embodiments, a transmitter 1400 may be configured as shown FIG. 14. (Transmitter 1400 may include any subset of the features described above in connection with method 1300.) Transmitter 1400 may include a digital circuit 1410, a DAC unit 1415 and an I/Q modulator 1420.

[0279] The digital circuit 1410 may be configured to receive a digital inphase (I) signal and a digital quadrature (Q) signal, and transform the digital I signal and the digital Q signal in accordance with a 2×2 matrix of constants to produce a resultant digital I signal and a resultant digital Q signal. The digital circuit 1410 may be realized in any of various forms, e.g., as variously described above in connection with method 1300.

[0280] The DAC unit 1415 may be configured to convert the resultant digital I signal and the resultant digital Q signal to analog form in order to respectively obtain an analog I signal and an analog Q signal.

[0281] The I/Q modulator 1420 may be configured to perform I/Q modulation on the analog I and Q signals to produce a modulated analog signal. The 2×2 matrix is configured to at least partially pre-compensate for I/Q impairments of the transmitter at frequency f. A first of the constants, corresponding a first diagonal element of the 2×2 matrix, may be computed based on a measurement of the I/Q impairments at frequency f and a measurement of the I/Q impairments at frequency −f. A second of the constants, corresponding to a first non-diagonal element of the 2×2 matrix, may be computed based on the measurement at frequency f and the measurement at frequency −f.

[0282] The Meaning of “Impairments at Frequency f”

[0283] The present disclosure repeatedly uses the term “the I/Q impairments at frequency f”. Whether that term is applied to the transmitter, the receiver, or the series combination comprising the transmitter, the transmission path and the receiver, it includes within its scope of meaning the I/Q impairments that result from stimulating the system in question with a complex exponential tone exp(j2πft)=cos(2πft)+jsin(2πft) at frequency f, as shown in FIG. 15. The real and imaginary outputs of the system may be represented as:
I′(t)=g_I(f)cos(2πft+φ_I(f))
Q′(t)=g_Q(f)sin(2πft+φ_Q(f)).

[0284] The I/Q impairments at frequency f may include the gain imbalance g(f) and the phase skew φ(f) given by:
g(f)=g_Q(f)/g_I(f)
φ(f)=φ_Q(f)−φ_I(f).

[0285] Here we adopt the convention of using the I channel as the reference for both the gain imbalance and the phase skew. However, the inventive principles described herein apply equally to any other referencing convention. For example, one might just as well use the opposite convention (i.e., the choice of the Q channel as reference for both the gain imbalance and the phase skew), or a convention where the gain imbalance is referenced to one channel and the phase skew to the other.

[0286] Because we are interested in compensating for imbalances of gain and differences in phase between the two channels, we can model the gain imbalance and the phase skew as appearing all on the I channel or all on the Q channel. For example, FIG. 16 illustrates the latter choice. Thus, the Q channel output has the form:
Q′(t)=g(f)sin(2πft+φ(f)).

[0287] The Physical Consequence of I/Q Impairments

[0288] The consequence of I/Q impairments at frequency f is the appearance of unwanted signal energy at the frequency −f. To see this, we analyze the complex output signal as follows:

[0289] $\begin{array}{c}{I}^{\prime }\left(t\right)+j{Q}^{\prime }\left(t\right)=\cos \left(\omega t\right)+jg\left(\omega \right)\sin \left(\omega t+\phi \left(\omega \right)\right)\\ =\left(1/2\right)\left\{\exp \left(j\omega t\right)+\exp \left(-\mathrm{j\omega }t\right)\right\}+\\ \left(1/2\right)g\left(\omega \right)\left\{\exp \left(\mathrm{j\omega }t+\phi \left(\omega \right)\right)-\exp \left(-\mathrm{j\omega }t-\phi \left(\omega \right)\right)\right\}\\ =\left(1/2\right)\left\{1+g\left(\omega \right)\exp \left[\phi \left(\omega \right)\right]\right\}\exp \left(\mathrm{j\omega }t\right)+\\ \left(1/2\right)\left\{1-g\left(\omega \right)\exp \left[-\phi \left(\omega \right)\right]\right\}\exp \left(-\mathrm{j\omega }t\right)\\ =A_{\mathrm{TONE}}\left(\omega \right)\exp \left(\mathrm{j\omega }t\right)+A_{\mathrm{IMAGE}}\left(\omega \right)\exp \left(-\mathrm{j\omega }t\right).\end{array}$
(We have switched from f to ω=2πf merely for the sake of notational brevity.) Thus, in response to the stimulus signal exp(jωt), the system produces a complex exponential tone at frequency ω with complex amplitude A_TONE(ω) and a complex exponential tone at frequency −ω with complex amplitude A_IMAGE(ω).
A_TONE=(½){1+g(ω)exp[φ(ω)]}
A_IMAGE=(½){1−g(ω)exp[−φ(ω)]}.

[0290] The complex exponential tone at frequency ω is often referred to simply as “the tone”, while the complex exponential tone at frequency −ω is often referred to as “the image”. As expected, A_TONE(ω)→1 and A_IMAGE→0 as g(ω)→1 and φ(ω)→0. It is desirable to have g(ω) as close to one as possible and φ(ω) as close to zero as possible. (Linear scale is being assumed here for the gain imbalance. Gain imbalance may also be expressed in a logarithmic scale, e.g., in units of dB, in which case 0 dB represents the case of no gain imbalance.)

[0291] From the above discussion, one can readily see that the series combination of two systems, the first having gain imbalance g₁(ω) and phase skew φ₁(ω), and the second having gain imbalance g₂(ω) and phase skew φ₂(ω), does not give a net gain imbalance of g(ω)=g₁(ω)g₂(ω) and a net phase skew of φ(ω)=φ₁(ω)+φ₂(ω). (This is because the second system is not being stimulated by a pure complex exponential exp{jωt}.) The true relations are more complicated.

[0292] Image Rejection

[0293] Image rejection is a measure of the relative magnitudes of complex amplitudes A_TONE(ω) and A_IMAGE(ω). For example, according to one conventional definition:
Image_Rejection=20*log(|A_IMAGE|/A_TONE|).

[0294] Because |A_IMAGE| is typically smaller than |A_TONE|, the image rejection is typically negative. The more negative the image rejection, the better.

[0295] Post-Compensation vs. Pre-Compensation

[0296] The notion of post-compensation involves coupling a compensation block to the output of a system that exhibits I/Q impairments. The compensation block is configured so that the series combination of the system followed by the compensation block behaves (or, approximates the behavior of) an ideal model having unity gain imbalance and zero phase skew. When the system is stimulated by a complex exponential tone at frequency co, it will generate a distorted complex signal that can be modeled as
cos(ωt)+jg(ω)sin(ωt+φ(ω)),
where g(ω) and φ(ω) are the I/Q impairments of the system at frequency ω. The compensation block operates on the distorted complex signal to generate a corrected output signal equal to the original complex exponential tone at frequency ω. Thus, the compensation block is said to “compensate” or “post-compensate” for the I/Q impairments of the system at frequency ω. Wideband post-compensation of I/Q impairments means post-compensating for I/Q impairments at each frequency ω in a frequency range or frequency band.

[0297] The notion of pre-compensation involves placing a compensation block before the system, i.e., the output of the compensation block couples to the input of the system. The compensation block is configured so that the series combination of the compensation block followed by the system behaves (or, approximates the behavior of) an ideal model having unity gain imbalance and zero phase skew. In response to a complex exponential tone at frequency ω, the compensation block will produce a pre-distorted complex signal. The system receives the pre-distorted complex signal and further distorts that signal (by introducing I/Q impairments), thereby producing a complex output signal. The compensation block generates the pre-distorted complex signal so that the complex output signal from the system is equal to the original complex exponential tone at frequency ω. Thus, the compensation block is said to “compensate” or “pre-compensate” for the I/Q impairments of the system at frequency ω. Wideband pre-compensation of I/Q impairments means pre-compensating for I/Q impairments at each frequency ω in a frequency range or frequency band.

[0298] Performing Traditional Impairment Compensation at a Single Frequency

[0299] If one is interested in post-compensating for I/Q impairments at a particular frequency ω₀, one might use the block diagram of FIG. 17 with real constants α and β. By appropriate choice of the constants, the disturbed complex input signal
cos(ω₀t)+jg(ω₀)sin(ω₀t+φ(ω₀))
will be mapped to the corrected output signal cos(ω₀t)+j sin(ω₀t), as desired. The appropriate values are:
α=−tan(φ(ω₀))
β=1/{g(ω₀)cos(φ(ω₀))}.

[0300] This method of compensation is referred to herein as “traditional single-point compensation”.

[0301] Due to the continuity of the gain imbalance g and phase skew φ with respect to frequency ω, the real constants α and β will achieve partial compensation for I/Q impairments at frequencies in a neighborhood of ω₀, with the quality of compensation degrading with increasing distance from ω₀. However, because g(ω₀) is typically different from g(−ω₀), and φ(ω₀) is typically different from φ(−ω₀), the appropriate pair of values (α,β) for compensating I/Q impairments at frequency ω₀ is typically different from the appropriate pair of values for compensating I/Q impairments at frequency −ω₀. Thus, unfortunately, one typically cannot find a single pair of values that will work for both ω₀ and −ω₀.

[0302] While the above-derived values of α and β work perfectly for post-compensation of I/Q impairments at the single frequency ω₀, they may also be used for pre-compensation of I/Q impairments at the single frequency ω₀ with generally less than perfect results. (Various ones of the methods described herein may employ such pre-compensation, even though it gives less than perfect results, in part because it does not require knowledge of the I/Q impairments at frequency −ω₀.) To achieve perfect pre-compensation of I/Q impairments at a single frequency, see the section “Calculating True Single Point Vector Calibration Constants”.

[0303] Wideband I/Q Impairment Equalization

[0304] FIG. 18 depicts a basic model of a system H that will be used repeatedly throughout this patent disclosure, e.g., to represent the equalization filtering performed by the receiver and the equalization filtering performed by the transmitter. (Equalization is used here as a synonym for I/Q impairment compensation.)

[0305] In the case where system H represents the receiver's equalization filtering, the complex input signal I(t)+jQ(t) represents a distorted signal supplied by a preceding system G, as illustrated in FIG. 19. The system G generates the distorted signal
I(t)+jQ(t)=cos(2πft)+jg(f)sin(2πft+φ(f))
in response to being stimulated by the complex exponential signal i(t)+jq(t)=exp(j2πft) at frequency f. The gain imbalance g(f) and the phase skew φ(f) are the I/Q impairments of the system G at frequency f. System G may represent the baseband equivalent of the receiver's front end, i.e., the portions of the receiver from its RF input to the output of the I/Q digitization unit. Alternatively, in the situation where the receiver is expected to compensate for the transmitter's I/Q impairments as well as for its own, system G may represent the path from the input of the transmitter's I/Q DAC unit to the output the receiver I/Q digitization unit.

[0306] The system H operates on the distorted input to produce a corrected output signal I′(t)+jQ′(t)=exp(j2πft), for all f in a desired frequency band. Note, however, that the set B given by
{exp(j2πft):f in given frequency range}
forms a basis for the space of functions {x(t)} that are band limited to the given frequency range. Because the series combination of G followed by H is an identity map on each function of the basis set B, by linearity it will be an identity map on all band-limited functions x(t).

[0307] The equalization system H may be realized by the receiver's digital circuit 220 as variously described above.

[0308] In the case where system H represents the transmitter's equalization filtering, we interpret H as receiving a basis function
I(t)+Q(t)=cos(2πft)+jg(f)sin(2πft+φ(f)),
and, in response to that basis function, generating a pre-compensated complex signal
I′(t)+jQ′(t)=exp(j2πft),
as shown in FIG. 20A. Note that the set X given by
{cos(2πft)+jg(f)sin(2πft+φ(f)):f in given frequency range}
also forms a basis for the space of functions {x(t)} that are band limited to the given frequency range.

[0309] The pre-compensated signal that gets distorted by the following system G. The system G generates distorted signal
i(t)+jq(t)=cos(2πft)+jg(f)sin(2πft+φ(f)),
where g(f) and φ(f) represent the gain imbalance and phase skew of the system G at frequency f. Because the series combination of H followed by G is an identity map on each function of the basis set X, it will be an identity map on all band-limited functions x(t). Thus, when stimulated by a complex exponential tone exp(j2πft) at any frequency f in the frequency band, the series combination will produce that same complex exponential tone at its final output, as shown in FIG. 20B.

[0310] The system G may represent the baseband equivalent of the transmitter's RF front end, i.e., the portions of the transmitter from the input of the transmitter's DAC unit to the RF output. Alternatively, in situations where the transmitter is expected to compensate for the receiver's I/Q impairments as well as for its own, the system G may represent the path from the input of the transmitter's DAC unit to the output of the receiver's digitization unit. The system H may be realized by the digital circuit 510 as variously described above.

[0311] Complex exponentials are used throughout the present analysis since any band-limited signal can be represented as the superposition of an ensemble of complex exponentials via Fourier analysis. I/Q impairments may include an imbalance of the gain when comparing the in-phase (I) channel with the quadrature-phase (Q) channel, and a skew in the phase that occurs due to imperfect quadrature mixing. (The phase skew perturbs the ideal 90-degree phase relationship between the I and Q channels). While phase skew is typically modeled as an imperfection in the quadrature mixing, it can also be modeled as the phase skew between the I(t) and Q(t) signals. In both cases discussed above, the input to the distortion model G is a complex exponential signal. Since I/Q impairments are relative, we may assume that the I/Q impairments appear entirely at the Q(t) output while the I(t) output is ideal. While other assumptions could be made, this assumption will simplify the following mathematical derivations.

[0312] The equalization system H may be modeled by a 2×2 frequency response matrix H(f)=(H_ij(f)), or equivalently, by a 2×2 matrix h(t)=(h_ij(t)) of real-valued impulse responses. However, under the above-identified assumption on how the impairments are expressed at the output of the distortion model G, the matrix H may be simplified to the structure shown in FIG. 21, i.e., H₁₁(f)=1 and H₁₂(f)=0. For notational efficiency, we define U(f)=H₂₁(f) and V(f)=H₂₂(f). Thus,
I′(t)=I(t)
Q′(t)=u(t)*I(t)+v(t)*Q(t),
where u(t) and v(t) are the impulse responses corresponding to U(f) and V(f) respectively.

[0313] Any real-valued filter must necessarily have a symmetric magnitude response and an anti-symmetric phase response. In other words, x(t) being real implies
I′(t)=I(t)
Q′(t)=u(t)*I(t)+v(t)*Q(t),
for all f, where X(f) is the Fourier transform of x(t). As a result, the frequency response V(f) cannot apply independent impairment corrections at frequencies f and −f. In the typical situation g(f) and g(−f) are different, and φ(f) and φ(−f) are different. Thus, the filter V acting by itself (i.e., with U identically equal to zero) is not sufficient to provide correction at f and −f. If the goal was to only correct for the wideband I/Q impairments over positive frequencies only or over negative frequencies only, the filter V would suffice. (Note: V acting by itself can correct for +f and −f impairments as long as the impairments are constrained to g(f)=g(−f) and φ(f)=−φ(−f), as demonstrated in the “Adding Constraints” section.) However, since it is desirable to correct for both sides of the spectrum, the second filter U(f) is introduced. Applying another filter from in-phase component and summing it into the quadrature-phase channel provides the needed degrees of freedom to control both sides of the complex spectrum. This is due to the fact that the in-phase component I(t)=cos(2πft) is that same for frequencies f and −f, while the quadrature-phase component changes its phase by 180 degrees when changing from f to −f.

[0314] In order to solve for U(f) and V(f), their respective output signals need to be known. In order to simplify the mathematical derivation, both U(f) and V(f) are broken into their even and odd parts as shown in FIG. 22. Thus, A(f) and B(f) are the even and odd parts of U(f), while C(f) and D(f) are the even and odd parts of V(f).

[0315] Since any real-valued filter must necessarily have a symmetric magnitude response, we can reduce complexity by solving only for the positive-frequency portion of each spectrum A, B, C and D. However, to achieve impairment compensation for negative frequencies as well as positive frequencies, one cannot simply ignore the inputs I(t)+jQ(t) corresponding to negative frequencies. Rather, we account for such inputs by expressing them as equivalent positive-frequency inputs, relying on the odd symmetry of the sine function and the even symmetry of the cosine function:

[0316] $I\left(t\right)+jQ\left(t\right)=\cos \left(-2\pi \mathrm{ft}\right)+g\left(-f\right)\sin \left(-2\pi \mathrm{ft}+\phi \left(-f\right)\right)=\cos \left(2\pi \mathrm{ft}\right)-g\left(-f\right)\sin \left(2\pi \mathrm{ft}-\phi \left(-f\right)\right).$

[0317] Thus, we shall develop two equations for the positive-frequency portions of A, B, C and D, the first based on the input
I₁(t)+jQ₁(t)=cos(2πft)+g(f)sin(2πft+φ(f)),
and the second based on the input
I₂(t)+jQ₂(t)=cos(2πft)−g(−f)sin(2πft−φ(−f)),
with f>0 for both equations.

[0318] If a filter is constrained to have symmetric impulse response, then the filter will exhibit a symmetric magnitude response and a zero phase response. Such is the case for filters a(t) and c(t). If, however, a filter's impulse response is anti-symmetric, then it will exhibit a symmetric magnitude response but a phase response that equals
−(π/2)sgn(f).

[0319] Thus, an anti-symmetric impulse response is equivalent to an even impulse response followed by a Hilbert transform. Such is the case for the filters b(t) and d(t). As a result, filter b(t) can be expressed as an even impulse response e_B(t) followed by a Hilbert transform (HT), as shown in FIG. 23. Similarly, filter d(t) can be expressed as an even impulse response e_D(t) followed by a Hilbert transform (HT). E_B(f) and E_D(f) are the frequency responses corresponding to e_B(t) and e_D(t), respectively. Now the exact outputs of the original filters A, B, C and D can easily be determined. FIG. 24A shows the outputs of the four filters A, B, C and D in response to signal I₁(t)+jQ₁(t). FIG. 24B shows the outputs of the four filter in response to the signal I₂(t)+jQ₂(t).

[0320] Each of FIGS. 24A and 24B can be directly translated into a corresponding linear equation in A(f), E_B(f), C(f) and E_D(f) for positive f (or non-negative f). We use the following notation:
g₁(f)=g(f) for f>0
g₂(f)=g(−f) for f>0
φ₁(f)=φ(f) for f>0
φ₂(f)=φ(−f) for f>0.

[0321] FIGS. 24A and 24B give rise, respectively, to equations (1.1) and (1.2), which are shown in FIG. 25. FIGS. 26A and 26B present the corresponding phasor diagrams. (Recall that cos(2πft) maps to 1, and sin(2πft) maps to −j in the phasor diagram.)

[0322] The horizontal projections of the vectors in FIG. 26A give equation (1.3) below; the vertical projections give equation (1.4). Similarly, the horizontal projections of the vectors in FIG. 26B give equation (1.5), while the vertical projections give equation (1.6):
0=A(f)+C(f)g₁(f)sin(φ₁(f))−E_D(f)g₁(f)cos(φ₁(f))  (1.3)
−1=−E_B(f)−C(f)g₁(f)cos(φ₁(f))−E_D(f)g₁(f)sin(φ₁(f))  (1.4)
0=A(f)+C(f)g₂(f)sin(φ₂)(f))+E_D(f)g₂(f)cos(φ₂(f))  (1.5)
1=−E_B(f)+C(f)g₂(f)cos(φ₂(f))−E_D(f)g₂(f)sin(φ₂(f))  (1.6)

[0323] This system of equations is a 4×4 linear system in unknown vector (A,E_B,C,E_D):

[0324] $\begin{array}{cc}\left[\begin{array}{c}0\\ -1\\ 0\\ 1\end{array}\right]=\left[P\right]\left[\begin{array}{c}A\left(f\right)\\ E_B\left(f\right)\\ C\left(f\right)\\ E_D\left(f\right)\end{array}\right],& \left(1.7\right)\end{array}$

[0325] where

[0326] $\begin{array}{cc}P=\left[\begin{array}{cccc}1& 0& w& -x\\ 0& -1& -x& -w\\ 1& 0& y& z\\ 0& -1& z& -y\end{array}\right]& \left(1.8\right)\end{array}$

[0327] and
w=g₁(f)sin(φ₁(f))  (1.9)
x=g₁(f)cos(φ₁(f))  (1.10)
y=g₂(f)sin(φ₂(f))  (1.11)
z=g₂(f)cos(φ₂(f)).  (1.12)

[0328] The determinate of matrix P is given by:
Det(P)=w²+x²+y²+z²−2wy+2xz.  (1.13)
Det(P)=g₁²(f)+g₂²(f)+2g₁(f)g₂(f)cos(φ₁(f)+φ₂(f))  (1.14)
As long as
g₁²(f)+g₂²(f)+2g₁(f)g₂(f)cos(φ₁(f)+φ₂(f))≠0,  (1.15)
there exists a unique solution vector (A(f),E_B(f),C(f),E_D(f)). As an example, the equations cannot be solved when both the phase skew φ(f) and gain imbalance g(f) are completely odd. However, it does not make sense for the gain imbalance g(f) to be completely odd since the gain imbalance is typically close to one for all f, or at least bounded below by a positive constant.

[0329] Using Cramer's Rule, we find that
A(f)=−2(wz+xy)/Det(P).  (1.16)
E_B(f)=(−w²−x²+y²+z²)/Det(P)  (1.17)
C(f)=2(x+z)/Det(P)  (1.18)
E_D(f)=2(w−y)/Det(P).  (1.19)

[0330] Substituting equation (1.9) through (1.14) into equations (1.16) through (1.19) yields equations (1.20) through (1.23), shown in FIG. 27.

[0331] Adding Constraints

[0332] In many cases, the gain imbalance and the phase skew approximate common constraints. This section simplifies equations (1.20) through (1.23) for some typical real world conditions. For the most perfect compensation, the equations (1.20-1.23) may be used. But if the compensation performance can be relaxed, then adding some constraints can decrease the computational requirements.

[0333] Case 1: Odd Phase Skew

[0334] In the case of odd phase skew, i.e., φ(f)=φ₁(f)=−φ₂(f) for f>0, equations (1.20) through (1.23) specialize to:
A(f)=0  (1.24)
E_B(f)={g₂(f)−g₁(f)}/{g₁(f)+g₂(f)}  (1.25)
C(f)=2 cos(φ(f))/{g₁(f)+g₂(f)}  (1.26)
E_D(f)=2 sin(φ(f))/{g₁(f)+g₂(f)}.  (1.27)

[0335] Case 2: Even Gain Imbalance

[0336] In the case of even gain imbalance, i.e., g(f)=g₁(f)=g₂(f) for f>0, equations (1.20) through (1.23) specialize to:

[0337] $\begin{array}{cc}A\left(f\right)=\frac{-\sin \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)}{1+\cos \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)}& \left(1.28\right)\\ E_B\left(f\right)=0& \left(\mathrm{1.29`}\right)\\ C\left(f\right)=\frac{\cos \left({\phi }_1\left(f\right)\right)+\cos \left({\phi }_2\left(f\right)\right)}{g\left(f\right)\left\{1+\cos \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)\right\}}& \left(1.30\right)\\ E_D\left(f\right)=\frac{\sin \left({\phi }_1\left(f\right)\right)-\sin \left({\phi }_2\left(f\right)\right)}{g\left(f\right)\left\{1+\cos \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)\right\}}.& \left(1.31\right)\end{array}$

[0338] Case 3: Odd Phase Skew and Even Gain Imbalance

[0339] In the case of odd phase skew and even gain imbalance, equations (1.20) through (1.23) specialize to:
A(f)=0  (1.32)
E_B(f)=0  (1.33)
C(f)=cos(φ(f))/g(f)  (1.34)
E_D(f)=sin(φ(f))/g(f).  (1.35)

[0340] Case 4: Zero Phase Skew and Arbitrary Gain Imbalance

[0341] In the case of zero phase skew and arbitrary gain imbalance, equations (1.20) through (1.23) specialize to:
A(f)=0  (1.36)
E_B(f)={g₂(f)−g₁(f)}/{g₂(f)+g₁(f)}  (1.37)
C(f)=2/{g₂(f)+g₁(f)}  (1.38)
E_D(f)=0.  (1.39)

[0342] Case 5: Arbitrary Phase Skew and Unity Gain Imbalance

[0343] In the case of arbitrary phase skew and unity gain imbalance, equations (1.20) through (1.23) specialize to:

[0344] $\begin{array}{cc}A\left(f\right)=\frac{-\sin \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)}{1+\cos \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)}=-\tan \left(\frac{{\phi }_1\left(f\right)+{\phi }_2\left(f\right)}{2}\right)& \left(1.40\right)\\ E_B\left(f\right)=0& \left(1.41\right)\\ C\left(f\right)=\frac{\cos \left({\phi }_1\left(f\right)\right)+\cos \left({\phi }_2\left(f\right)\right)}{1+\cos \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)}& \left(1.42\right)\\ E_D\left(f\right)=\frac{\sin \left({\phi }_1\left(f\right)\right)-\sin \left({\phi }_2\left(f\right)\right)}{1+\cos \left({\phi }_1\left(f\right)+{\phi }_2\left(f\right)\right)}.& \left(1.43\right)\end{array}$

[0345] Case 6: Constant Gain Imbalance and Phase Skew

[0346] In the case where the gain imbalance and phase skew functions are constant functions, i.e., g(f)=g and φ(f)=φ over all f, equations (1.20) through (1.23) specialize to:

[0347] $\begin{array}{cc}A\left(f\right)=\frac{-\sin \left(2\phi \right)}{1+\cos \left(2\phi \right)}=-\tan \left(\phi \right)& \left(1.44\right)\\ E_B\left(f\right)=0& \left(1.45\right)\\ C\left(f\right)=\frac{2\cos \left(\phi \right)}{g\left\{1+\cos \left(2\phi \right)\right\}}=\frac{1}{g\cos \left(\phi \right)}& \left(1.46\right)\\ E_D\left(f\right)=0.& \left(1.47\right)\end{array}$

[0348] Filter Design

[0349] In one embodiment, symmetric linear-phase FIR filters â(n) and ĉ(n) are designed based respectively on the magnitude responses |A(f)| and |C(f)|, while antisymmetric linear-phase FIR filters {circumflex over (b)}(n) and {circumflex over (d)}(n) are designed based respectively on the magnitude responses |B(f)| and |D(f)|. Note that |B(f)|=|E_B(f)| and |D(f)|=|E_D(f)| for all f. The Remez algorithm may be used to design these filters. The equalization system of FIG. 22 may then be implemented using the filters â(n), {circumflex over (b)}(n), ĉ(n) and {circumflex over (d)}(n). By creating four filters, each with either symmetric or anti-symmetric filter taps, and summing the filters as shown in FIG. 22, we can effectively match the two arbitrary frequency responses U(f) and V(f). (Note: Depending on the filter design tool, the summation might actually be a subtraction. The definition of Hilbert Transform used by many filter design tools to create anti-symmetric filters differ from the definition we have used by a negation. The filter design tools often use +(π/2)sgn(f) for the phase.)

[0350] In another embodiment, symmetric linear-phase FIR filters â(n), ê_B(n), ĉ(n) and ê_D(n) are designed based respectively on the magnitude responses |A(f)|, |E_B(f)|, |C(f)| and |E_D(f)|. Again, the Remez algorithm may be used to design these filters. The equalization system of FIG. 23 may then be realized using the filters â(n), ê_B(n), ĉ(n) and ê(n).

[0351] In yet another embodiment, filters û(n) and {circumflex over (v)}(n) may be designed based on the frequency responses U(f) and V(f). An L_p-norm design method may be used to design these filters based on the magnitude and phase responses of U(f) and the magnitude and phase responses of V(f). The equalization system of FIG. 21 may then be implemented using the filters û(n) and {circumflex over (v)}(n).

[0352] Corruption I/Q Impairments

[0353] As described above, FIG. 15 illustrates a system that introduces I/Q impairments, i.e., gain imbalance g(f) and phase skew φ(f), into a received complex exponential signal exp(j2πft). In general, a 2×2 frequency response matrix H characterizing the system may be derived from the impairment functions g(f) and φ(f). To simplify this derivation, we model the gain imbalance g(f) and phase skew φ(f) of the system as appearing entirely on the Q channel output, as shown in FIG. 28. This model makes it convenient to use the special form of the matrix shown in FIG. 29, where U(f) and V(f) are frequency responses corresponding to real filters u(t) and v(t). Response U(f) may be represented as the sum of its even part A(f) and odd part B(f), as shown in FIG. 30. Similarly, V(f) may be represented as the sum of its even part C(f) and odd part D(f). The filter with odd spectrum B(f) may be represented by a subsystem with even spectrum E_B(f) followed by a Hilbert Transform HT, as shown in FIG. 31. (See the above “Note Regarding Filters with Odd Frequency Response”.) Similarly, the filter with odd spectrum D(f) may be represented by a subsystem with even spectrum E_D(f) followed by a Hilbert Transform HT. Note that the magnitude responses of B and E_B are identical, |B(f)|=|E_B(f)|, as are the magnitude responses of D(f) and E_D(f).

[0354] We will develop equations for the positive-frequency portions of A(f), E_B(f), C(f) and E_D(f) since the negative-frequency portions are determined by the respective positive frequency portions. One equation will come from stimulating the system with a positive frequency input I₁(t)+jQ₁(t)=exp(j2πft) for f>0, as shown in FIG. 32A. Another equation will come from stimulating the system with a negative frequency input that is expressed in terms of an equivalent positive frequency input:

[0355] $I_2\left(t\right)+j{Q}_2\left(t\right)=\exp \left(-j2\pi \mathrm{ft}\right)=\cos \left(-2\pi \mathrm{ft}\right)+\mathrm{jsin}\left(-2\pi \mathrm{ft}\right)=\cos \left(2\pi \mathrm{ft}\right)-\mathrm{jsin}\left(2\pi \mathrm{ft}\right)$
for f>0, as shown in FIG. 32B.

[0356] FIG. 33 shows the two equations. Equation (1.48) is based on FIG. 32A. Equation (1.49) is based on FIG. 32B.

[0357] FIGS. 34A and 34B show the corresponding phasor diagrams, relying on the following notation:
g₁(f)=g(f) for f>0
g₂(f)=g(−f) for f>0
φ₁(f)=φ(f) for f>0
φ₂(f)=φ(−f) for f>0.

[0358] The phasor diagrams give the following equations:
g₁(f)sin(φ₁(f))=A(f)−E_D(f)  (1.50)
g₁(f)cos(φ₁(f))=E_B(f)+C(f)  (1.51)
g₂(f)sin(φ₂(0)=A(f)+E_D(f)  (1.52)
g₂(f)cos(φ₂(f))=C(f)−E_B(f)  (1.53)

[0359] These equations comprise a 4×4 matrix equation (1.54) in the unknowns A(f), E_B(f), C(f) and E_D(f), as shown in FIG. 35. By inverting the 4×4 coefficient matrix, we obtain the solution. See the matrix equation (1.55) in FIG. 36. It follows that:
A(f)=(½){g₁(f)sin(φ₁(f))+g₂(f)sin(φ₂(f))}  (1.56)
E_B(f)=(½){g₁(f)cos(φ₁(f))−g₂(f)cos(φ₂(f))}  (1.57)
C(f)=(½){g₁(f)cos(φ₁(f))+g₂(f)cos(φ₂(f))}  (1.58)
E_D(f)=(½){−g₁(f)sin(φ₁(f))+g₂(f)sin(φ₂(f))}.  (1.59)

[0360] Since A, E_B, C and E_D are even functions of frequency f. Thus, their negative-frequency portions are specified by the even symmetry. Furthermore, the odd frequency responses B(f) and D(f) are given by:
B(f)=−jE_B(f)sgn(f) and
D(f)=−jE_D(f)sgn(f).

[0361] Special Case: Even Gain Imbalance and Odd Phase Skew

[0362] In the case where the gain imbalance is an even function and the phase skew is an odd function, i.e., g(f)=g(−f) and φ(f)=−φ(−f), then equations (1.56) through (1.59) specialize to:
A(f)=0
E_B(f)=0
C(f)=g(f)cos(φ(f))
E_D(f)=−g(f)sin(φ(f)).

[0363] Special Case: Even Gain Imbalance and Even Phase Skew

[0364] In the case where the gain imbalance and phase skew are even functions, i.e., g(f)=g(−f) and φ(f)=φ(−f), then equations (1.56) through (1.59) specialize to:
A(f)=g(f)sin(φ(f))
E_B(f)=0
C(f)=g(f)cos(φ(f))
E_D(f)=0.

[0365] Special Case: Constant Gain Imbalance and Phase Skew

[0366] In the case where the gain imbalance and phase skew are constant functions, i.e., g(f)=g and φ(f)=φ, then equations (1.56) through (1.59) specialize to:
A(f)=g sin(φ)  (1.60)
E_B(f)=0  (1.61)
C(f)=g cos(φ)  (1.62)
E_D(f)=0.  (1.63)

[0367] Calculating the Mapping Between the RX and Tx

[0368] In some embodiments, a transmitter pre-distorts digital I/Q signals in order to compensate for its own I/Q impairments as variously described above. To accomplish this compensation, one must have an estimate of the transmitter's I/Q impairments. The quality of the compensation will be limited by the quality of the estimate (the extent to which it matches the truth). While a high-quality estimate is desirable, it is difficult to measure the transmitter's I/Q impairments directly. Rather, the measurements are obtained indirectly, e.g., using a receiver as shown in FIG. 37.

[0369] FIG. 37 shows a transmitter 3700 that couples to a receiver 3725 via a channel (e.g., a cable 3720 or a wireless channel). The transmitter may include a digital compensation unit 3702, a DAC unit 3705, an I/Q modulator 3710 and a front end 3715. The compensation unit 3702 may perform pre-compensation (pre-distortion) on the digital signal I(n)+jQ(n) to obtain a pre-compensated signal I′(n)+jQ′(n), e.g., as variously described above. DAC unit 3705 may convert the pre-compensated signal into an analog signal s(t)=I′(t)+jQ′(t). The analog signal s(t) may be upconverted to RF using the I/Q modulator 3710. The upconverted signal may be conditioned by the TX front end 3715 to obtain a transmit signal. The transmit signal may be delivered to the receiver by the cable 3720.

[0370] The receiver 3725 may include a front end 3730, an I/Q demodulator 3735 and a digitization unit 3740. The front end 3830 may receive the transmitted signal from the cable 3720 and operate on the received signal to produce a conditioned signal. The conditioned signal may be down-converted by the I/Q demodulator to produce a complex down-converted signal. The complex down-converted signal may be sampled by the digitization unit 3740 to obtain a sampled complex signal. The sampled complex signal may be used to make I/Q impairment measurements. In some embodiments, the receiver is a spectrum analyzer, e.g., a vector signal analyzer.

[0371] It is important to understand how measurements of I/Q impairments taken at the receiver 3725 relate to the transmitter's I/Q impairments. They are not the same. This is because the I/Q impairments at the transmitter, e.g., at the output of the I/Q modulator, are obscured (distorted) by the signal path including the TX front end 3715, the cable 3720 and the receiver front end 3730. The signal path may be characterized by a frequency response H(f)=m(f)exp(jθ(f)), where H(f) is a complex number. The amplitude m(f) is referred to herein as the “scaling” of the signal path at frequency f. The phase θ(f) is referred to herein as the “rotation” of the signal path at frequency f.

[0372] The problem of estimating the transmitter's I/Q impairments from receiver-based measurements is not trivial. Its solution is disclosed in this patent disclosure. (See the iterative methods disclosed below.) Part of the solution includes obtaining an initial estimate for the signal path response function H(f). This section will focus on obtaining that initial estimate in the form of H(0), i.e., the frequency response of the signal path at DC (zero frequency). The amplitude m(0) of H(0) is referred to as the “DC scaling” of the signal path. The phase θ(0) of H(0) is referred to as the “DC rotation” of the signal path.

[0373] One way to estimate the transmitter's I/Q impairments involves performing an iterative procedure using a spectrum analyzer. (A spectrum analyzer is a device configured to measure the magnitude of an input signal versus frequency within the frequency range of the instrument.) The spectrum analyzer measures the I/Q impairments of its demodulated signal, and then compensation is applied at the transmitter based on that measurement. The measurement may only crudely approximate the transmitter's I/Q impairments, but it may be good enough to achieve at least partial compensation. The spectrum analyzer then makes a second measurement of the I/Q impairments of its demodulated signal. This second measurement may be used to make an adjustment to the compensation being applied at the transmitter, and so on. The sequence of measurements may converge, i.e., the measured gain imbalance may converge to one and the measured phase skew may converge to zero, indicating that appropriate compensation has been achieved at the transmitter. Because the spectrum analyzer does not capture phase information, multiple iterations may be required to achieve convergence.

[0374] In some embodiments, the transmitter's I/Q impairments may be determined using a measurement device (such as a vector signal analyzer) that can make phase measurements and can lock the phase of the measurements to the transmitter. In this case, the transmitter's I/Q impairments may be determined with two measurements or less at the measurement device.

[0375] The method described below makes two measurements, but requires that the transmitter's I/Q modulator and the receiver's I/Q demodulator be locked together in frequency (via phase locked loops with common reference). Unlike other methods, this method is resistant to synchronous spurs (i.e., spurs such as the LO leakage that are phase locked to the LO of the transmitter). While this technique can be used at any frequency, the main application is to determine the DC scaling m(0) and the DC rotation θ(0) of the signal path in order to calibrate out the transmitter's LO leakage impairment.

[0376] In FIG. 38, vector A=A_I+jA_Q represents the transmitter's LO leakage when the transmitter is stimulated with a constant zero signal: I′(n)=Q′(n)=0. (The term “vector” is used here as a synonym for “complex number”.) The amplitude and phase of vector A represent the amplitude and phase of the LO leakage. As this LO leakage signal moves from the transmitter's I/Q modulator to the receiver's I/Q demodulator, it gets scaled by m(0) and rotated by θ(0) so that vector A is transformed into vector A′ at the receiver. See FIG. 39. The vector A′ is measured, e.g., by averaging the sampled complex signal captured from the output of the I/Q demodulator.

[0377] Then we stimulate the transmitter with a known non-zero vector B:
I′(n)=B_I
Q′(n)=B_Q.
(Vector B need not be real as shown in FIG. 38. However, it does need to be non-zero.) This intentionally-applied LO leakage B is superimposed on top of the transmitter's intrinsic LO leakage A so that the transmitter's total leakage is vector C. (The choice of B, primarily its magnitude, can affect the accuracy of the measurement. The optimum size will depend on the specific hardware. If it is too small, noise will influence the measurement more; if it is too big, the hardware could be put into a non-linear region of operation.) This total leakage signal experiences the same scaling m(0) and rotation θ(0) as it traverses the signal path so that vector C is transformed to vector C′ at the receiver. Referring to FIG. 39, observe that vector C′ is the sum of vector A′ and B′. Vector B′ is the vector that would have resulted if vector B could have traversed the signal path by itself.

[0378] At the receiver, vector C′ is measured, e.g., by averaging the sampled complex signal captured from the output of the I/Q demodulator during the stimulation by vector B. Since both A′ and C′ are known by measurement, vector B′ can be calculated by subtraction. The DC scaling m(0) and the DC rotation θ(0) may be calculated from the vector B′ and the vector B:
Map=m(0)exp(jθ(0))=B′/B.

[0379] Similarly, the inverse map that will undo the effect of the signal path can be determined from the inverse expression:
InverseMap=exp(−j(θ(0))/m(0)=B/B′.

[0380] Then the LO leakage vector A may be computed by multiplying vector A′ by the inverse map. In practice, it may be advisable to keep the magnitude of B to within an order of magnitude of A. It is also a good practice to not transmit vector B by itself, but to transmit the sum of the vector B and another signal K, where the signal K has larger energy than the vector B signal and frequency content bounded away from DC since the transmitter's LO leakage can potentially change with power in the instantaneous bandwidth. For example, the signal K may be a tone.

[0381] In some embodiments, the sampled complex signals are windowed. If a window is not applied, there are restrictions on the frequency of the tone K. In addition to tone K, if there are other signal tones present, they could also potentially leak into the measurement. Thus, if not using a window, the tones (intentional or not) are preferably restricted to certain frequencies to avoid leakage.

[0382] Method for Determining the Transmitter's LO Leakage

[0383] 1. Stimulate the transmitter with a constant zero signal.

[0384] 2. Measure the vector A′ produced at the receiver.

[0385] 3. Stimulate the transmitter with non-zero complex constant B.

[0386] 4. Measure vector C′ at the receiver.

[0387] 5. Calculate the transmitter's LO leakage vector A from the following equations:
B′=C′−A′  (1.64)
InvMap=B/B′  (1.65)
A=A′*InvMap.  (1.66)

[0388] Once the transmitter's LO leakage vector A has been computed, the transmitter may remove (or substantially compensate for) the LO leakage by applying the translation vector −A to transmitted signals
I′(n)=I(n)−A_I
Q′(n)=Q(n)−A_Q.

[0389] The compensation unit 3702 may apply this translation in addition to the I/Q impairment pre-compensation described above. For example, the complex signal (I(n),Q(n)) may be subjected to the 2×2 matrix of digital filters to pre-compensate for I/Q impairments, and then translated to pre-compensate for the LO leakage.

[0390] In some embodiments, the calculation of the DC mapping may include the following extra calculation. As described herein, the iterative method may diverge if the estimation error of the phase rotation is too large. In the case that the phase skew is large, this extra step may be used to get a more accurate estimate and enable the iterative method to converge: (1) Calculate the mapping from RX to TX as is already described. (2) Make a measurement of the phase skew. (3) Calculate using the method “Altering the gain imbalance and phase skew through a linear system” using the mapping from #1. (4) Add the rotation measurement of #1 to the computed phase skew of #3 to attain a more accurate rotation estimate.

[0391] Method for Computing a DC Mapping and DC Rotation for Signal Path

[0392] In one set of embodiments, a method 4000 may involve the actions shown in FIG. 40. The method 4000 may be used to estimate a DC scaling m(0) of a signal path between an I/Q modulator of a transmitter and a demodulator of a receiver. (The method 4000 may incorporate any subset of the features described above in the section “Calculating the Mapping Between the RX and Tx”.) The method 4000 is described below as being performed by a “processing agent”. The processing agent may be any system of digital circuitry, e.g., a processor (executing under the control of program instructions), a programmable hardware element, an ASIC, or any combination thereof.

[0393] In some embodiments, the receiver conforms to a direct-conversion architecture, and the demodulator is an analog I/Q demodulator. In other embodiments, the receiver may conform to a different architecture (e.g., a superheterodyne architecture) which performs analog down-conversion followed by digital I/Q demodulation. Thus, in this case the demodulator is realized by digital circuitry, e.g., on a programmable hardware element, in dedicated digital circuitry, in software on a processor, or any combination thereof.

[0394] At 4010, the processing agent may direct the transmitter to supply a zero signal as input to the I/Q modulator. The zero signal is a constant zero signal. The zero signal may be a digital zero signal that is supplied to the complex input of the transmitter's DAC unit (e.g., DAC unit 3705 of FIG. 37). Thus, I′(n)=0 and Q′(n)=0.

[0395] At 4015, the processing agent may receive a first response signal that has been captured from the demodulator in response to the action of supplying the zero signal. The first response signal may be captured from the output of the receiver's ADC unit. (See, e.g., digitization unit 215 of FIG. 2B.)

[0396] At 4020, the processing agent may direct the transmitter to supply a constant signal equal to a non-zero complex constant B=B_I+jB_Q as input to the I/Q modulator. Again, the constant signal may be supplied to the complex input of the transmitter's DAC unit. Thus, I′(n)=B_I and Q′(n)=B_Q. In some embodiments, B is entirely real, i.e., B_Q=0.

[0397] At 4025, the processing agent may receive a second response signal that has been captured from the demodulator in response to the action of supplying the constant signal. The second response signal may be captured from the output of the receiver's ADC unit.

[0398] At 4030, the processing agent may average the first response signal to obtain a first average and averaging the second response signal to obtain a second average. The averaging helps to reduce noise in the measurements.

[0399] At 4035, the processing agent may compute a difference between the second average and the first average, e.g., according to the expression:
Diff=SecondAvg−FirstAvg.

[0400] At 4040, the processing agent may compute the DC scaling based on the difference and the non-zero complex constant, e.g., as described above. The processing agent may store the DC scaling in a memory.

[0401] In some embodiments, the method 4000 may also include computing a DC rotation θ(0) of the signal path based on a phase of the difference and a phase of the non-zero complex constant B, e.g., according to the expression
θ(0)=Phase(Diff)/Phase(B).

[0402] In some embodiments, the DC scaling and DC rotation are used to remove an effect of the signal path from measured I/Q impairments at the receiver in order to obtain estimates of the I/Q impairments of the transmitter.

[0403] In some embodiments, the signal path includes a cable coupling between the transmitter and the receiver. In other embodiments, the signal path includes a wireless channel between the transmitter and the receiver.

[0404] As an alternative to computing a difference of averages, the processing agent may alternatively compute a difference signal by subtracting the first response signal from the second response signal, and then average the difference signal. The DC scaling may then be computed based on average value and the non-zero complex constant.

[0405] In one set of embodiments, a computer system for estimating a DC scaling m(0) of a signal path between an I/Q modulator of a transmitter and a demodulator of a receiver, the computer system comprising a processor and memory. The memory storing program instructions, where the program instructions, when executed by the processor, cause the processor to: direct the transmitter to supply a zero signal as input to the I/Q modulator; receive a first response signal that has been captured from the demodulator in response to said supplying the zero signal; direct the transmitter to supply a constant signal equal to a non-zero complex constant as input to the I/Q modulator; receive a second response signal that has been captured from the demodulator in response to said supplying the constant signal; average the first response signal to obtain a first average and averaging the second response signal to obtain a second average; compute a difference between the second average and the first average; compute the DC scaling based on the difference and the non-zero complex constant. The program instructions may incorporate any subset of the features described above in the section “Calculating the Mapping Between the RX and Tx” and in connection with method 4000.

[0406] Altering the Gain Imbalance and Phase Skew through a Linear System

[0407] When calibrating the transmitter or measuring the I/Q impairments of the transmitter, the method of this section may be used to remove the effects of the signal path between the transmitter's I/Q modulator and the receiver's I/Q demodulator from the receiver's I/Q impairment measurements. Those effects may include the effects of the transmitter's front end, the transmission channel and the receiver's front end. For example, the transmitter's front end may include an RF filter that contributes to the frequency response of the signal path. Similarly, the receiver's front end may include an RF filter that contributes to the frequency response of the signal path.

[0408] In some embodiments, the magnitude response m(f) of the signal path may be calibrated out, while the phase rotation θ(f) is not calibrated out. (Calibration may be achieved by performing pre-compensation at the transmitter using digital circuit 510 and/or post-compensation at the receiver using digital circuit 220.) The calculations of this section allow for the correct measurement of the transmitter's I/Q impairments without first calibrating out the phase response of the signal path.

[0409] In some embodiments, the total frequency response (including both magnitude and phase rotation) of the signal is calibrated out.

[0410] Given a system with frequency response H(f) and an input signal s_input(f,t) having gain imbalance g(f) and phase skew φ(f) as shown in FIG. 41, we will develop equations that allow us to determine the gain imbalance g′(f) and phase skew φ′(f) at the system output s_output(f,t). We assume the input gain imbalance g(f) and input phase skew φ(f) appear entirely on the Q input channel. However, we cannot simultaneously make the same assumption at the system output. In general, the output components I′(t) and Q′(t) will have the form:
I′(t)=g₁(f)cos(2Σft+φ_I(f))
Q′(t)=g_Q(f)sin(2Σft+φ(f)).

[0411] The output gain imbalance g′(f) and output phase skew φ′(f) may then determined by:
g′(f)=g_Q(f)/g_I(f)
φ′(f)=φ_Q(f)−φ_I(f).

[0412] The development starts with equations (1.60) through (1.62) given in FIG. 42, relying on the fact that s_output(f,t)=h(t)*s_input(f,t), where h(t) is the impulse response corresponding to H(f). Equations 1.61 and 1.62 imply that:
g_I(f)exp(jφ_I(f))+g_Q(f)exp(jφ_Q(f))=H(f){1+g(f)exp(jφ(f))}  (1.63)
g_I(f)exp(−jφ_I(f))−g_Q(f)exp(−jφ_Q(f))=H(−f){1−g(f)exp(−jφ(f))}  (1.64)

[0413] Define A(f) and B(f) to be respectively the right-hand sides of equation (1.63) and (1.64):
A(f)=H(f){1+g(f)exp(jφ(f))}  (1.65)
B(f)=H(−f){1−g(f)exp(−jφ(f))}.  (1.66)

[0414] Also, define w(f), x(f), y(f) and z(f) based on the left-hand sides of equations (1.63) and (1.64):
g_I(f)exp(jφ_I(f))=w(f)+jx(f)  (1.67)
g_Q(f)exp(jφ_Q(f))=y(f)+jz(f)  (1.68)
It follows that
w(f)+jx(f)+y(f)+jz(f)=A(f)  (1.69)
w(f)−jx(f)−y(f)+jz(f)=B(f),  (1.70)
and thus,
w(f)=(½)Re{A(f)+B(f)}
x(f)=(½)Im{A(f)−B(f)}
y(f)=(½)Re{A(f)−B(f)}
z(f)=(½)Im{A(f)+B(f)}.

[0415] Note that if H(f) has an even magnitude response and an odd phase response, i.e., H(−f)=H(f)*, then the impulse response h(t) corresponding to H(f) is entirely real. As a result, in this specific case, the filter H(f) does not change the measurement of the I/Q impairments:

[0416] $A\left(f\right)+B\left(f\right)=H\left(f\right)+H^{*}\left(f\right)+g\left\{H\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)-H^{*}\left(f\right)\exp \left(-\mathrm{j\phi }\left(f\right)\right)\right\}=2\mathrm{Re}\left(H\right)+2jg\mathrm{Im}\left\{H\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)\right\}A\left(f\right)-B\left(f\right)=H\left(f\right)-H^{*}\left(f\right)+g\left\{H\left(f\right)\exp \left(\mathrm{j\varphi }\left(f\right)\right)+H^{*}\left(f\right)\exp \left(-\mathrm{j\varphi }\left(f\right)\right)\right\}$ $A\left(f\right)-B\left(f\right)=H\left(f\right)-H^{*}\left(f\right)+g\left\{H\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)+H^{*}\left(f\right)\exp \left(-\mathrm{j\phi }\left(f\right)\right)\right\}=2j\mathrm{Im}\left\{H\left(f\right)\right\}+2g\mathrm{Re}\left\{H\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)\right\}$ $w\left(f\right)=\mathrm{Re}\left(H\left(f\right)\right),x\left(f\right)=\mathrm{Im}\left(H\left(f\right)\right)$ $g_I\left(f\right)\exp \left({\mathrm{j\phi }}_I\left(f\right)\right)=w\left(f\right)+jx\left(f\right)=H\left(f\right)$ $y\left(f\right)=g\mathrm{Re}\left\{H\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)\right\}$ $z\left(f\right)=g\mathrm{Im}\left\{H\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)\right\}$ $g_Q\left(f\right)\exp \left({\mathrm{j\phi }}_Q\left(f\right)\right)=y\left(f\right)+jz\left(f\right)=\mathrm{gH}\left(f\right)\exp \left(\mathrm{j\phi }\left(f\right)\right)$ $g_Q\left(f\right)\exp \left({\mathrm{j\phi }}_Q\left(f\right)\right)/g_I\left(f\right)\exp \left({\mathrm{j\phi }}_I\left(f\right)\right)=g\exp \left(\mathrm{j\phi }\left(f\right)\right).$
The method below describes how to iteratively measure the TX impairments when the magnitude and phase of the signal path transfer function H(f) are only approximately known. Part of that iterative measurement method involves using the equations derived in this section to compute the I/Q impairments at the output of the transmitter's I/Q modulator based on the I/Q impairments at the input (or alternatively, at the output) of the receiver's I/Q demodulator. To perform this computation, the frequency response H(f) is set equal to the inverse of an estimate of the frequency response of the signal path. Different estimates of the signal path frequency response may be used in different circumstances.

[0417] Transforming I/Q Impairments through a Linear System H(f)

[0418] In one set of embodiments, a method 4300 may involve the operations shown in FIG. 43. The method 4300 may be used to compute I/Q impairments at a complex output z_OUT of an electrical system based on I/Q impairments at a complex input z_IN of the electrical system. A complex input is an input that includes an inphase channel and a quadrature channel. Likewise a complex output is an output that includes an inphase channel and a quadrature channel. (Method 4300 may include any subset of the features described above in the section “Altering the Gain Imbalance and Phase Skew through a Linear System”.) Method 4300 may be performed by a processing agent as described above.

[0419] At 4310, the processing agent may compute a spectrum A(f) according to the expression
H(f){1+g(f)exp(jφ(f))},
where H(f) is a spectrum of a linear system model of the electrical system, where g(f) is a gain imbalance at the complex input z_IN, where φ(f) is a phase skew at the complex input z_IN.

[0420] At 4315, the processing agent may compute a spectrum B(f) according to the expression
H(−f){1−g(f)exp(−jφ(f))}.

[0421] At 4320, the processing agent may compute a sum of the spectra A(f) and B(f), and a difference of the spectra A(f) and B(f), e.g., according to the relations:
Sum(f)=A(f)+B(f),
Diff(f)=A(f)−B(f).

[0422] At 4325, the processing agent may compute a gain imbalance and phase skew at the complex output z_OUT based on real and imaginary parts of the sum, and real and imaginary parts of the difference. In particular, as described above, the functions g_I(f), g_Q(f), φ_I(f) and φ_Q(f) may be computed based on the sum spectrum and the difference spectrum, and then the gain imbalance and phase skew at the complex output z_OUT may be computed based on g_I(f), g_Q(f), φ_I(f) and φ_Q(f) as shown in FIG. 41. The output gain imbalance and output phase skew constitute useful information in part because they may be used to perform I/Q impairment compensation or calibration as variously described herein.

[0423] The processing agent may store the output gain imbalance and the output phase skew in a memory.

[0424] In some embodiments, the electrical system being modeled by the spectrum H(f) is the inverse of a signal path from an I/Q modulator of a transmitter to a demodulator of a receiver, e.g., as variously described herein. The gain imbalance and the phase skew at the complex input z_IN of the electrical system may represent a gain imbalance and a phase skew at the input (or alternatively, at the output) of the demodulator. The gain imbalance and the phase skew at the complex output z_OUT of the electrical system may represent a gain imbalance and a phase skew at the output of the I/Q modulator.

[0425] In some embodiments, the receiver conforms to a direct-conversion architecture, and the demodulator is an analog I/Q demodulator. In other embodiments, the receiver may conform to a different architecture (e.g., a superheterodyne architecture) which performs analog down-conversion followed by digital I/Q demodulation. Thus, in this case the demodulator is realized by digital circuitry, e.g., on a programmable hardware element, in dedicated digital circuitry, in software on a processor, or any combination thereof.

[0426] In some embodiments, the processing agent may also include computing an inverse of a spectrum of the signal path to determine the spectrum H(f), e.g., as variously described herein.

[0427] In some embodiments, the spectrum H(f) may be determined (or estimated) based on a DC scaling and a DC rotation of the signal path, e.g., according to the relation
H(f)=exp{−jθ(0)}/m(0).

[0428] In some embodiments, the processing agent may compute the DC scaling and the DC rotation by: supplying a zero signal as input to the I/Q modulator, capturing a first response signal from the I/Q demodulator in response to said supplying the zero signal; supplying a constant signal equal to a non-zero complex constant as input to the I/Q modulator; capturing a second response signal from the I/Q demodulator in response to said supplying the constant signal; averaging the first response signal to obtain a first average and averaging the second response signal to obtain a second average; computing a difference between the second average and the first average; and computing the DC scaling based on the difference and the non-zero complex constant.

[0429] In some embodiments, the processing agent may also measure (e.g., direct the measurement of) the gain imbalance g(f) and the phase skew φ(f) of an electronic device at a plurality of frequencies. The electronic device may be a transmitter, a receiver, or the series combination of a transmitter and receiver, as variously described herein.

[0430] In some embodiments, the processing agent may be a programmable hardware element. In other embodiments, the processing agent may be a processor that is configured to perform the method 4300 in response to execution of program instructions.

[0431] Determination of Transmitter I/Q Impairments Using Shared LOs

[0432] In one set of embodiments, a method 4400 for determining I/Q impairments of a transmitter may involve the actions shown in FIG. 44. (Furthermore, the method 4400 may include any subset of the features described in the section “Iterative Technique for Measuring Tx Impairments, in the section “Iterative Estimation of Transmitter Impairments Using Shared LOs”, and in the section “Iterative Estimation of Transmitter Impairment Using Shared LO—Optimized”.) The method 4400 may be enacted by a processing agent, e.g., a processing agent as variously described above.

[0433] At 4410, the processing agent may perform a set of operations. The set of operations may include the operations 4415 through 4440, as shown in FIG. 44.

[0434] At 4415, the processing agent may direct that a complex exponential tone at frequency f be supplied to the transmitter. For example, the processing agent may issue commands causing the complex exponential tone to be supplied (or generated by) the transmitter. The frequency f may be interpreted as a displacement frequency relative to the transmitter's local oscillator frequency. The frequency f may be non-zero.

[0435] At 4420, the processing agent may supply a pre-compensation transformation to a pre-compensation circuit of the transmitter. The pre-compensation circuit may be configured to apply the pre-compensation transformation to the complex exponential tone to obtain an adjusted complex signal. (For example, the pre-compensation circuit may be the digital circuit 510 of FIG. 5 or the compensation unit 3702 of FIG. 37.) The pre-compensation transformation may be configured to pre-compensate for a current estimate of the I/Q impairments of the transmitter. The transmitter may be configured to transmit a transmit signal based on the adjusted complex signal, e.g., as variously described above. The receiver may be configured to receive the transmit signal and capture a sampled complex signal representing the received transmit signal, e.g., as variously described above. (The action of “sampling” a complex signal involves sampling its I component and sampling its Q component. Thus, a “sampled complex signal” includes a sampled I signal and a sampled Q signal.)

[0436] At 4425, the processing agent may compute raw I/Q impairments based on the sampled complex signal. For example, the raw I/Q impairments may include a gain imbalance and phase skew of the sampled complex signal. See the section “Precise Measurement Technique” for information on how to compute the raw I/Q impairments.

[0437] At 4430, the processing agent may transform the raw I/Q impairments to determine transformed I/Q impairments. The transform may remove measured I/Q impairments of the receiver from the raw I/Q impairments. See the section “Removing Receiver Impairments from Measured Output Impairments” for more information on how perform this transform.

[0438] As an alternative to operations 4425 and 4430, the processing agent may apply a 2×2 matrix of digital filters to the sampled complex signal to remove the effect of the receiver's measured I/Q impairments, e.g., as described above in connection with FIGS. 2A, 2B and 3, and in the sections “Wideband I/Q Impairment Equalization” and “Filter Design”. The application of the 2×2 matrix of digital filters to the sampled complex signal produces a filtered complex signal. The filtered complex signal may be used to compute the transformed I/Q impairments. The method described in the section “Precise Measurement Technique” may be used to determine the transformed I/Q impairments based on the filtered complex signal.

[0439] At 4435, the processing agent may remove a current estimate of a signal path from the transformed I/Q impairments to obtain path-compensated I/Q impairments, where the signal path includes a path from an I/Q modulator of the transmitter to a demodulator of the receiver. (The signal path estimate may be removed by using the method described in the section “Altering the Gain Imbalance and Phase Skew Through a Linear System”.) The path-compensated I/Q impairments may represent an estimate for residual I/Q impairments of the transmitter, i.e., “residual” in the sense that they are remaining impairments after the partial correction realized by the pre-compensation transformation of 4420.

[0440] In some embodiments, the receiver may conform to a direct-conversion architecture, and the demodulator may be an analog I/Q demodulator, in which case the sampled complex signal may be captured by digitizing the complex analog output of the analog I/Q demodulator. In other embodiments, the receiver may conform to a different kind of architecture, e.g., a superheterodyne architecture. Thus, the receiver may generate a real analog signal (e.g., a real intermediate-frequency signal) that represents the received transmit signal. The real analog signal may be digitized to obtain a sampled real signal. The sampled complex signal may then be generated computationally, e.g., by digitally mixing the sampled real signal with an orthogonal pair of digital sinusoids to obtain respectively the I and Q components of the sampled complex signal.

[0441] At 4440, the processing agent may update the current estimate of the I/Q impairments of the transmitter based on the path-compensated I/Q impairments, e.g., by combining the path-compensated I/Q impairments with the respective impairments of the current estimate.

[0442] In some embodiments, the method 4400 may include repeating the set of operations to determine a converged estimate (stable estimate) of the I/Q impairments of the transmitter at the frequency f. (This converged estimate comprises a measurement of the transmitter's I/Q impairments at the frequency f.) The set of operations may be repeated until a quality measure based on the path-compensated I/Q impairments is larger than a threshold. The converged estimate may be used to at least partially compensate for the I/Q impairments of the transmitter at frequency f, e.g., as variously described herein.

[0443] In some embodiments, the above-described action of repeating the set of operations may itself be performed a plurality of times to determine the converged estimate at a plurality of different values for the frequency f. The above-described action of repeating the set of operations to determine the converged estimate at frequency f may be referred to herein as a “transmitter I/Q impairment measurement at frequency f”. Thus, a plurality of transmitter I/Q impairment measurements may be made so as to cover the plurality of frequency values.

[0444] In some embodiments, the plurality of frequency values are symmetric about zero. Furthermore, the transmitter I/Q impairment measurements may be made so that the frequency values are visited in manner that alternates in sign and is non-decreasing in absolute value, e.g., as variously described herein.

[0445] In some embodiments, a local oscillator of the transmitter and a local oscillator of the receiver are phase locked to the same frequency reference (infers frequency lock).

[0446] In some embodiments, the current estimate of the signal path is based on a DC scaling and a DC rotation of the signal path, at least for a first of the transmitter I/Q impairment measurements.

[0447] In some embodiments, the DC scaling and the DC rotation may be determined by: supplying a zero vector signal to the transmitter; supplying a non-zero DC vector signal to the transmitter; and computing the DC scaling and the DC rotation based on a first DC vector response and a second DC vector response, where the first DC vector response is measured at the receiver in response to the zero vector signal, where the second DC vector response is measured at the receiver in response to the non-zero DC vector signal. For more information on how to compute the DC scaling and DC rotation, see the section “Calculating the Mapping Between RX and TX”.

[0448] In some embodiments, the pre-compensation transformation has the form of a 2×2 matrix, where at least a first diagonal element of the matrix is computed from the current estimate of the I/Q impairments of the transmitter at frequencies f and −f, and where at least a first non-diagonal element of the matrix is computed from the current estimate of the I/Q impairments of the transmitter at frequencies f and −f.

[0449] In some embodiments, the current estimate of the signal path includes a measured amplitude of the sampled complex signal at frequency f. The amplitude may be measured as described in the section “Precise Measurement Technique”.

[0450] In some embodiments, the current estimate of the signal path also includes a measured rotation of the sampled complex signal at frequency f.

[0451] Determination of Transmitter I/Q Impairments with Offset LOs

[0452] In one set of embodiments, a method 4500 for determining I/Q impairments of a transmitter may involve the actions shown in FIG. 45. (Furthermore, the method 4500 may include any subset of the features described in the section “Iterative Techniques for Measuring Tx Impairments.) The method 4500 may be performed by a processing agent, e.g., a processing agent as variously described above.

[0453] At 4510, the processing agent may configure a local oscillator (LO) of the transmitter and a local oscillator (LO) of the receiver to be phase locked to a common reference and so that a frequency of the receiver's LO minus a frequency of the transmitter's LO frequency is equal to a non-zero amount ΔLO. The amount ΔLO may be positive or negative.

[0454] At 4520, the processing agent may perform a set of operations S_O. The set S_O may include operations 4525 through 4550, as shown in FIG. 45.

[0455] At 4525, the processing agent may direct that a complex exponential tone at frequency f be supplied to the transmitter. (The frequency f may be interpreted as a displacement relative to the transmitter's LO frequency.) The complex exponential tone may be supplied in digital form, e.g., as variously described above. In some embodiments, the transmitter may couple to (or include) a programmable hardware element configured to generate the complex exponential tone. To facilitate this generation, the PHE may receive the sample clock used by the transmitter's DAC unit.

[0456] At 4530, the processing agent may supply a pre-compensation transformation to a pre-compensation circuit of the transmitter. The pre-compensation circuit may be configured to apply the pre-compensation transformation to the complex exponential tone in order to obtain an adjusted complex signal. (For example, the pre-compensation circuit may be the digital circuit 510 of FIG. 5 or the compensation unit 3702 of FIG. 37.) The pre-compensation transformation may be configured to pre-compensate for a current estimate of the I/Q impairments of the transmitter. The transmitter may be configured to transmit a transmit signal based on (derived from) the adjusted complex signal, e.g., as variously described above. A receiver may be configured to receive the transmit signal and to capture a sampled complex signal representing the received transmit signal, e.g., as variously described above. The transmitter may transmit the transmit signal onto a transmission channel (e.g., a cable), and the receiver may receive the transmit signal from the channel.

[0457] At 4535, the processing agent may frequency shift the sampled complex signal by the amount ΔLO to obtain a frequency-shifted signal, e.g., by multiplying the sampled complex signal by a discrete-time complex exponential signal running at frequency ΔLO.

[0458] At 4540, the processing agent may compute raw I/Q impairments at frequency f based on the frequency-shifted signal. The raw I/Q impairments may include a gain imbalance g_R(f) and a phase skew φ_R(f). (The process of computing the I/Q impairments from a complex signal is described above.)

[0459] At 4545, the processing agent may remove a current estimate of a signal path from the raw I/Q impairments at frequency f to obtain path-compensated I/Q impairments at frequency f (e.g., as described above in the section “Transforming I/Q Impairments through a Linear System”, or in the section “Altering the Gain Imbalance and Phase Skew through a Linear System”). The signal path may include a path from an I/Q modulator of the transmitter to a demodulator of the receiver. The path-compensated I/Q impairments at frequency f may represent an estimate for residual I/Q impairments of the transmitter at frequency f.

[0460] In some embodiments, the receiver may conform to a direct-conversion architecture, and the demodulator may be an analog I/Q demodulator, in which case the sampled complex signal may be captured by digitizing the complex analog output of the analog I/Q demodulator. In other embodiments, the receiver may conform to a different kind of architecture, e.g., a superheterodyne architecture. Thus, the receiver may generate a real analog signal (e.g., a real intermediate-frequency signal) that represents the received transmit signal. The real analog signal may be digitized to obtain a sampled real signal. The sampled complex signal may then be generated computationally, e.g., by digitally mixing the sampled real signal with an orthogonal pair of digital sinusoids to obtain respectively the I and Q components of the sampled complex signal.

[0461] At 4550, the processing agent may update the current estimate of the I/Q impairments of the transmitter at frequency f based on the path-compensated I/Q impairments at frequency f.

[0462] In some embodiments, the method 4500 may include repeating the set of operations S_O to determine a converged estimate (or stable estimate) of the I/Q impairments of the transmitter at frequency f. (This converged estimate may be interpreted as a measurement of the transmitter's I/Q impairments at frequency f.) For example, the set of operations may be repeated until a quality measure based on the path-compensated I/Q impairments is larger than a threshold. (The quality measure may be the negative of image rejection at frequency f.) The converged estimate is usable to at least partially compensate for the I/Q impairments of the transmitter at frequency f. The above-described action of frequency shifting may be performed using a frequency-shift signal that is phase continuous between successive repetitions of the set of operations.

[0463] In some embodiments, the method 4500 may also include performing said repeating (of the set of operations S_O) a plurality of times to determine the converged estimate at a plurality of different values for the frequency f, e.g., values covering a desired transmission (or communication) band.

[0464] In some embodiments, the set of operations S_O may also include removing measured I/Q impairments of the receiver at frequency f-ΔLO from the sampled complex signal prior to the frequency shifting operation. The measured I/Q impairments of the receiver at frequency f-ΔLO may be removed by multiplying the sampled complex signal by a 2×2 matrix M=(m_ij) of constants, e.g., according to the relations:

[0465] $\left[\begin{array}{c}{I}^{\prime }\left(n\right)\\ Q^{\prime }\left(n\right)\end{array}\right]=\left[\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right]\left[\begin{array}{c}I\left(n\right)\\ Q\left(n\right)\end{array}\right],$
where I(n) and Q(n) denote respectively the inphase and quadrature components of the sampled complex signal. In one embodiment, the matrix M may have the special form

[0466] $M=\left[\begin{array}{cc}1& 0\\ m_{21}& m_{22}\end{array}\right],$
and the constants m₂₁ and m₂₂ may be determined from a receiver's gain imbalance g_RX(f-ΔLO) and receiver's phase skew φ_RX(f-ΔLO) at frequency f-ΔLO based on the expressions:

[0467] $m_{21}=-\tan \left\{{\phi }_{\mathrm{RX}}\left(f-\Delta \mathrm{LO}\right)\right\}.m_{22}=\frac{1}{g_{\mathrm{RX}}\left(f-\Delta \mathrm{LO}\right)\cos \left\{{\phi }_{\mathrm{RX}}\left(f-\Delta \mathrm{LO}\right)\right\}}.$

[0468] See the section entitled “Performing Traditional Impairment Compensation at a Single Frequency”.

[0469] In an alternative embodiment, the constants m₂₁ and m₂₂ may be determined based on the receiver's measured I/Q impairments at frequency f-ΔLO and its negative −(f-ΔLO), as described in the section “Calculating True Single Point Vector Calibration Constants”, and especially at equations (1.81) and (1.82).

[0470] In some embodiments, the receiver's I/Q impairments may be measured as part of method 4500, i.e., measured based on the sampled complex signal prior to frequency shifting. For example, the set of operations S_O may include measuring the I/Q impairments of the receiver at frequency f-ΔLO based on the sampled complex signal. One technique for performing this measurement involves: (a) computing a Discrete-Time Fourier transform value C₁ at frequency f-ΔLO of an I component of the sampled complex signal; (b) computing a Discrete-Time Fourier transform value C_Q at frequency f-ΔLO of a Q component of the sampled complex signal; (c) computing the receiver gain imbalance at frequency f-ΔLO based on the magnitudes of the values C_I and C_Q; and (d) computing the receiver phase skew at frequency f-ΔLO based on the phases of the values C_I and C_Q. For more information on embodiments of this technique, see the “Precise Measurement Technique” section.

[0471] In some embodiments, the method 4500 may also include applying a time-domain window to the sampled complex signal prior to computing the values C_I and C_Q. The time-domain window may be a rectangular (uniform) window or any of a variety of standard non-uniform windows. For more information on use of the rectangle window, see the section “Rectangle Window Optimization”.

[0472] In some embodiments, the above-described measurement of the receiver's I/Q impairments and the estimation of the transmitter's I/Q impairments may be performed at least partially in parallel. For example, in one embodiment, a programmable hardware element (or perhaps a multicore processor) may be configured to perform the measurement of the receiver's I/Q impairments in parallel with the frequency shift operation on the sampled complex signal.

[0473] In some embodiments, the set of operations may include measuring the receiver's I/Q impairments at frequency f-ΔLO as described above, computing the 2×2 matrix of correction constants based on the measured I/Q impairments as described above, and then applying the 2×2 matrix to the sampled complex signal prior to the frequency shift operation. In other words, the frequency shift operation is applied to the modified complex signal (I′(n),Q′(n)) resulting from the application of the 2×2 matrix.

[0474] In some embodiments, it is assumed that the I/Q impairments of the receiver have already been measured over the frequency band of interest prior to the execution of method 4500. Thus, a 2×2 matrix of digital filters may be designed based on the receiver's I/Q impairments, as described above in connection with FIGS. 2A, 2B and 3, and in the sections “Wideband I/Q Impairment Equalization” and “Filter Design”. The set of operations may include the operation of applying the 2×2 matrix of digital filters to the sampled complex signal prior to the frequency shifting operation. The resulting filtered complex signal may then be subjected to the frequency shifting.

[0475] In some embodiments, the pre-compensation transformation has the form of a 2×2 matrix, and the matrix has the property that at least one of the diagonal elements of the matrix is computed based on the current estimate of the I/Q impairments of the transmitter at frequency f and a current estimate of the I/Q impairments of the transmitter at frequency −f, and the property that at least one of the non-diagonal elements of the matrix is computed based on the current estimate of the I/Q impairments of the transmitter at frequency f and the current estimate of the I/Q impairments of the transmitter at frequency −f. In some embodiments, each of the four matrix elements is computed in this fashion.

[0476] As described above, the processing agent may remove a current estimate of a signal path from the raw I/Q impairments at frequency f to obtain path-compensated I/Q impairments at frequency f. In some embodiments, the current estimate of the signal path may include a measured amplitude of the frequency-shifted signal at frequency f. In one embodiment, the current estimate of the signal path may also include a measured rotation of the frequency-shifted signal at frequency f.

[0477] In some embodiments, the current estimate of the signal path may be based on a DC scaling and a DC rotation of the signal path. Such an estimate may be used in at least a first performance of said set of operations.

[0478] In some embodiments, the method 4500 may also include determining the DC scaling and the DC rotation by: supplying a zero vector signal to the transmitter; supplying a non-zero DC vector signal to the transmitter; computing the DC scaling and the DC rotation based on a first DC vector response and a second DC vector response, where the first DC vector response is measured at the receiver in response to the zero vector signal, where the second DC vector response is measured at the receiver in response to the non-zero DC vector signal. For more information on determination of the DC scaling and DC rotation, see the section “Calculating the Mapping Between the RX and Tx” and the section “Method for Computing a DC Mapping and DC Rotation for Signal Path”.

[0479] Determining I/Q Impairments of a Receiver

[0480] In one set of embodiments, a method 4600 for determining I/Q impairments of a receiver may include the operations shown in FIG. 46. The method 4600 may be performed by a processing agent as described above.

[0481] At 4610, the processing agent may direct that an input signal be supplied to the receiver. In other words, the processing agent may issue commands to cause the input signal to be supplied to (or generated by) the receiver. The input signal may include an isolated tone at displacement frequency f and a void interval (i.e., an interval containing only noise) around displacement frequency −f (To say that a tone is “isolated” at a given frequency means that the tone is the only energy source except for noise in a frequency neighborhood of the given frequency (e.g., in an interval of frequencies centered on the given frequency). If the noise energy is too large, the measurement quality will degrade. The tone is preferably the only significant source of energy in the frequency neighborhood.) The receiver may be configured to demodulate the input signal in order to obtain a sampled complex signal, e.g., as variously described above. The displacement frequencies f and −f may be displacements relative to a local oscillator frequency of the receiver.

[0482] At 4615, the processing agent may compute the I/Q impairments of the receiver at frequency f based on the sampled complex signal.

[0483] At 4620, the processing agent may repeat the actions of directing (4610) and computing (4615) for values of the frequency f spanning a specified frequency band, e.g., the currently-selected input band of the receiver or a standardized communication band.

[0484] At 4625, the processing agent may store the receiver's I/Q impairments for each of the values of the frequency f in a memory.

[0485] In some embodiments, the input signal is supplied by a transmitter whose local oscillator frequency is offset by a non-zero value from the local oscillator frequency of the receiver, e.g., as variously described above.

[0486] In some embodiments, the input signal is supplied by a calibration tone synthesizer. A calibration tone synthesizer is a system configured to create quality tones for the purpose of calibrating other systems. In some embodiments, the term “quality tone” implies stability over amplitude, frequency, temperature, or time. In one embodiment, the receiver includes a calibration tone synthesizer to facilitate self calibration.

[0487] In some embodiments, the action of computing the I/Q impairments of the receiver at frequency f includes: computing a Discrete-Time Fourier transform value C_I at frequency f of an I component of the sampled complex signal; computing a Discrete-Time Fourier transform value C_Q at frequency f of a Q component of the sampled complex signal; computing a gain imbalance of the receiver at frequency f based on magnitudes of the values C_I and C_Q; and computing a phase skew of the receiver at frequency f based on phases of the values C_I and C_Q.

[0488] In some embodiments, the method 4600 may also include applying a time-domain window to the sampled complex signal prior to said computing of the values C_I and C_Q, e.g., as describe below in the section “Precise Measurement Technique”.

[0489] Measuring I/Q Impairments Associated with Complex Signal

[0490] In one set of embodiments, a method 4700 may include the operations shown in FIG. 47. The method 4700 may be used to measure I/Q impairments associated with a sampled complex signal produced by a receiver. The method 4600 may be performed by a processing agent (e.g., a computer system executing under the control of program instructions).

[0491] At 4710, the processing agent may direct a device to stimulate the receiver with a stimulus signal having an isolated tone at displacement frequency f and a void interval at displacement frequency −f The displacement frequencies f and −f may be interpreted as displacements with respect to a local oscillator frequency of the receiver. The sampled complex signal may be a baseband signal produced by the receiver in response the action of stimulating with the stimulus signal.

[0492] At 4715, the processing agent may compute a Discrete-Time Fourier transform value C_I at frequency f of an I component of the sampled complex signal;

[0493] At 4720, the processing agent may compute a Discrete-Time Fourier transform value C_Q at frequency f of a Q component of the sampled complex signal.

[0494] At 4725, the processing agent may compute a gain imbalance g of the sampled complex signal at frequency f based on magnitudes of the values C_I and C_Q, where the gain imbalance g includes a gain imbalance of the receiver.

[0495] At 4730, the processing agent may compute a phase skew φ of the sampled complex signal at frequency f based on phases of the values C_I and C_Q, where the phase skew φ includes a phase skew of the receiver.

[0496] In some embodiments, the processing agent may apply a time-domain window to the sampled complex signal prior to said computing of the values C_I and C_Q.

[0497] In some embodiments, the device that provides the input signal is a calibration tone generator.

[0498] In some embodiments, the device is a transmitter whose local oscillator frequency is (intentionally) offset from the receiver's local oscillator frequency by a non-zero amount. In one such embodiment, the sampled complex signal has been subjected to frequency shifting to remove a difference between the local oscillator frequencies, in which case the gain imbalance g and phase skew φ may be partially dependent on the transmitter's I/Q impairments. In particular, the gain imbalance g and the phase skew co may represent a composite effect of the transmitter's I/Q impairments, the distortion introduced by the signal path (between the transmitter's I/Q modulator and the receiver's demodulator), and the receiver's I/Q impairments. In another such embodiment, the sampled complex signal is a raw signal from the demodulator that has not been subjected to the above-described frequency shifting, and thus, the gain imbalance g and the phase skew φ may be interpreted as including only impairments introduced by the receiver.

[0499] In some embodiments, the method 4700 may also include applying a time-domain window to the sampled complex signal prior to computing the value C_I and computing the value C_Q, e.g., as described below.

[0500] In some embodiments, the receiver is a vector signal analyzer.

[0501] In some embodiments, one or more of the operations 4715 through 4730 may be performed by a programmable hardware element.

[0502] In some embodiments, one or more of the operations 4715 through 4730 may be performed in dedicated digital circuitry.

[0503] In some embodiments, one or more of the operations 4715 through 4730 may be performed by a processor in response to the execution of program instructions.

[0504] Offset LO Calibration Technique

[0505] The offset local oscillator (LO) method allows for I/Q impairment measurement and carrier leakage measurement of both the receiver (RX) and the transmitter (TX) simultaneously. This method uses independently tunable LOs for the transmitter and receiver, e.g., as shown in FIG. 48. In some embodiments, the step size of the transmitter LO and/or the step size of the receiver LO can be fractional or integer in nature. In some embodiments, the step size of the transmitter and/or the step size of the receiver LO should be a small percentage of the total instantaneous bandwidth.

[0506] The transmitter includes an I/Q modulator 4810 and a front end 4815. A complex exponential tone at non-zero displacement frequency f is provided to the I/Q modulator 4810. The I/O modulator 4810 modulates a carrier signal (also referred to as “local oscillator signal”) with the tone to obtain a modulated signal. The carrier signal is provided by the transmitter LO 4805. The modulated signal is transmitted onto a transmission medium (e.g., a cable 4820) by the transmitter front end 4815.

[0507] The front end 4830 of the receiver receives the transmitted signal and conditions the received signal to obtain a conditioned signal. The I/Q demodulator 4835 demodulates the conditioned signal using a carrier signal provided by the receiver LO 4840, resulting in a demodulated signal having components denoted as RX I and RX Q.

[0508] As shown in FIG. 49, shifting the RX and TX carriers off of each other causes the tone, the receiver's image of the tone, the transmitter's image of the tone, the transmitter's carrier leakage and the receiver's carrier leakage to appear at different frequencies. The illustrated spectrum is based on the demodulated signal at the receiver. The transmitter produces the tone at 31 MHz. The spectrum includes two different carrier leakages, one due to the transmitter's LO leakage and one due to the receiver's LO leakage. The spectrum also includes two different primary images of the tone, one due to the transmitter's I/Q impairments and one due to the receiver's I/Q impairments. Additionally, the spectrum includes the receiver's image of the transmitter's image, and the receiver's image of the transmitter's carrier leakage, both due to the receiver's I/Q impairments. In this example the receiver's carrier is placed 6 MHz below that of the transmitter's carrier. This makes the tone, the transmitter's image and the transmitter's leakage appear 6 MHz higher in frequency at the receiver than at the transmitter. Then, as a result of the I/Q demodulator's impairments, each of these three signals (tone, TX image, and TX leakage) created by the transmitter has a corresponding image after the I/Q demodulator, in addition to the receiver's leakage.

[0509] Knowing the frequency offset between the transmit and receive LOs and the frequency of the tone produced at the transmitter before the modulator, the exact spectral locations of all impairment artifacts can be completely determined. If we let
FreqOffset=TxCarrierFrequency−RXCarrierFrequency,  (1.75)
the frequency locations (as seen by the receiver) of the spectral features in the received spectrum are:

[0510] $\begin{array}{cc}\mathrm{RxTone}=\mathrm{TxTone}+\mathrm{FreqOffset}& \left(1.76\right)\\ \mathrm{TxLeakage}=\mathrm{FreqOffset}& \left(1.77\right)\\ \mathrm{TxImage}=\mathrm{FreqOffset}-\mathrm{TxTone}& \left(1.78\right)\\ \mathrm{RxImage}=-\mathrm{TxTone}-\mathrm{FreqOffset}& \left(1.79\right)\\ \mathrm{RxLeakage}=0\mathrm{Hz}& \left(1.80\right)\\ \mathrm{RxImageofTxImage}=\mathrm{TxTone}-\mathrm{FreqOffset}=\mathrm{RxTone}-2\mathrm{FreqOffset}& \left(1.81\right)\\ \mathrm{RxImageOfTxLeakage}=-\mathrm{FreqOffset}.& \left(1.82\right)\end{array}$

[0511] Measuring the receiver's I/Q impairments and carrier leakage is performed in the same way as done in the “Precise Measurement Technique” section. However, measuring the transmitter's impairments is generally more involved since there are multiple things to consider. Measuring the transmitter's impairments may involve removing the receiver impairments. FIG. 50 shows the received spectrum after removal of the receiver's I/Q impairments. After that removal, the spectrum may be frequency shifted by −FreqOffset, as shown in FIG. 51. Now the frequency location of the “tone” in the shifted spectrum is the same as the frequency f at which the tone was originally produced at the transmitter. In addition, the transmitter's leakage (TX leakage) and transmitter's image (TX image) are at the correct frequency locations (−f and zero, respectively) to use the algorithm found in the “Precise Measurement Technique” section once the rotation is calculated and removed. (The rotation may be calculated using the method described in the section “Calculating the Mapping Between and RX and TX”.) This algorithm will give an estimate for the transmitter's I/Q impairments and the transmitter's LO leakage vector. This method for measuring the transmitter's I/Q impairments will work as long as the signal path (including the transmitter's front end and the receiver's front end) has an even magnitude response and odd phase response. In reality, this is not the case, and even small perturbations in magnitude or phase cause serious problems for the measurements. The iterative algorithm removes this issue. The iterations of the iterative algorithm involve performing pre-correction (e.g., using the method of the section “Calculating True Single Point Vector Calibration Constants) based on the current estimate of the transmitter's impairments, and removing the best available estimate of the signal path from the impairments measured at the receiver (using the method of the section “Altering the Gain Imbalance and Phase Skew Through a Linear System”). The iterative algorithm allows the transmitter impairments to be measured even though there is error in the initial estimate of those impairments.

[0512] Measuring the transmitter's impairments can be optimized one further step by doing everything in the method above except removing the receiver's impairments. Shown in FIG. 52 is the frequency-shifted spectrum without first removing the receiver's impairments. By leaving these impairments in the spectrum, the measured impairment values at frequency f (i.e., 31 MHz in this example) are not exactly equal to the transmitter's I/Q impairments since the receiver's impairments have distorted the measurements. However, the same iterative algorithm used to remove distortions of the RF front ends can also remove the distortions due to the receiver's impairments. While ideally it is better to remove the receiver's impairments, in practice this takes extra time during calibration.

[0513] Restrictions

[0514] While this method is highly desirable in that multiple measurements can be made in parallel, it does come with restrictions. The primary restriction is that is cannot be used to measure amplitude as it measures the combination of receiver amplitude and transmitter amplitude without any way to separate the two without another measurement. However, if either the receiver amplitude or the transmitter amplitude are known, then the two can be separated. In most cases, the amplitude changes slower with respect to frequency than the I/Q impairments. Therefore, a separate measurement procedure can be used to measure either receiver amplitude or transmitter amplitude with a coarser frequency step size than the step size used to determine the I/Q impairments over the instantaneous bandwidth. As a result, the total measurement time including amplitude is still much faster than the alternative.

[0515] Another minor problem with the offset LO method is that it places restrictions on the calibration frequency plan. Depending on the value of the LO offset ΔLO, it is possible to get corrupt measurements at various measurement offsets. As shown in FIG. 49, there are seven locations in the frequency spectrum where energy appears in response to the transmission of the tone. In order to correctly measure all of the impairments for both the transmitter and the receiver, all seven of these signals must remain orthogonal, i.e. no two of the signals cannot occur at the same frequency location. For example, if the receiver's LO is set to 2.400 GHz and the transmitter's LO is set to 2.396 GHz, then measurement corruptions will occur when the transmitted baseband tone is: at 4 MHz since this would place the tone at exactly the RX leakage; at −4 MHz since this would place the TX image at the RX leakage; or at 8 MHz since this would place the RX image at the TX leakage. In order to avoid these problems, the transmitted tone (TxTone) cannot be located at the frequencies
{N*FreqOffset:N=−3,−2,−1,0,1,2,3}.

[0516] In addition, there are bandwidth limitations. The total measureable bandwidth is (TotalBW-LO_StepSize), while the total symmetric measureable bandwidth is (TotalBW-2*LO_StepSize). This is why the LO step size must be a fraction of the total instantaneous bandwidth, preferably a small fraction. For example, if the instantaneous bandwidth is 100 MHz, and the LO step size is only 25 MHz, then 75 MHz of the bandwidth is theoretically measureable. In reality, since we typically desire symmetric bandwidth (i.e. +/−25 MHz rather than −25 MHz-50 MHz), our symmetric measureable bandwidth is only 50 MHz. In addition, there is also less measureable bandwidth due to roll-off effects at band-edges.

[0517] Calculating True Single Point Vector Calibration Constants

[0518] This section shows how to calculate constants for a true single-point calibration that will perfectly pre-correct at a single location (i.e., perfectly pre-compensate for I/Q impairments at a single frequency f) given knowledge of the I/Q impairments at f and −f, as indicated in FIGS. 53A and 53B. The single-point vector calibration correction 5310 precedes a double-point vector corruption model 5320. Thus, a complex exponential tone at frequency f, which is provided as input to the single-point vector calibration correction, is pre-distorted to produce a complex signal
cos(2πft)+jΓ sin(2πft+θ).

[0519] The pre-distorted signal is further distorted by the corruption model 5320, resulting in a corrected output signal that is equal to the original complex exponential tone.

[0520] From the “Corruption I/Q Impairments” section, we know how to develop a 2×2 frequency response matrix H that represents the I/Q impairments of a system. In that section, we found that A(f), E_B(f), C(f) and D(f) are determined by the “double point I/Q impairments”, i.e., by the I/Q impairments at f and the I/Q impairments at −f In addition, from the “Adding Constraints” section (i.e., Case 6, where A and C are constants, and E_B and E_D are zero) the structure of the single-point correction is known. Using this information, the true single-point calibration coefficients α and β can be determined.

[0521] The goal is to determine the values a and p given the values A(f), E_B(f), C(f) and E_D(f). A(f), E_B(f), C(f) and E_D(f) are known from equations (1.56) through (1.59) given the double-point I/Q impairments, i.e., the gain imbalance values g₁(f)=g(f) and g₂(f)=g(−f) and phase skew values φ₁=φ(f) and φ₂(f)=φ(−f). The values α and β can be determined from Γ and θ as shown in the following expressions:
α=Γ sin(θ)  (1.75)
β=Γ cos(θ).  (1.76)

[0522] Using the phasor diagram of FIG. 54, sum along the x-axis to obtain equation (1.77), and sum along the y-axis to obtain equation (1.78):
CΓ sin(θ)−E_DΓ cos(θ)=−A  (1.77)
CΓ cos(θ)+E_DΓ sin(θ)=1−E_B  (1.78)

[0523] We have relied on the facts that:
HT{sin(t)}=−cos(t),
HT{cos(t)}=sin(t),
where HT denotes the Hilbert Transform. Equations (1.77) and (1.78) imply:

[0524] $\begin{array}{cc}\left[\begin{array}{cc}-E_D& C\\ C& E_D\end{array}\right]\left[\begin{array}{c}\mathrm{\Gamma cos}\left(\theta \right)\\ \mathrm{\Gamma sin}\left(\theta \right)\end{array}\right]=\left[\begin{array}{c}-A\\ 1-E_B\end{array}\right]& \left(1.79\right)\\ \left[\begin{array}{c}\mathrm{\Gamma cos}\left(\theta \right)\\ \mathrm{\Gamma sin}\left(\theta \right)\end{array}\right]=\frac{1}{C^2+E_D^2}\left[\begin{array}{cc}-E_D& C\\ C& E_D\end{array}\right]\left[\begin{array}{c}-A\\ 1-E_B\end{array}\right]& \left(1.80\right)\\ \alpha =\mathrm{\Gamma sin}\left(\theta \right)\frac{-\mathrm{AC}+E_D\left(1-E_B\right)}{C^2+E_D^2}& \left(1.81\right)\\ \beta =\mathrm{\Gamma cos}\left(\theta \right)=\frac{{\mathrm{AE}}_D+C\left(1-E_B\right)}{C^2+E_D^2}.& \left(1.82\right)\\ \Gamma =\sqrt{\frac{A^2+{\left(1-E_B\right)}^2}{C^2+E_D^2}}& \left(1.83\right)\\ \theta =\arctan \left(\frac{-\mathrm{AC}+E_D\left(1-E_B\right)}{{\mathrm{AE}}_D+C\left(1-E_B\right)}\right).& \left(1.84\right)\end{array}$

[0525] While not needed to solve for α and β, solving for Γ and θ tell us the new gain and phase of the waveform needed to exactly cancel the impacts of the I/Q corruption.

[0526] It should be noted that the correction coefficients α and β given by (1.81) and (1.82) are generally different from the α and β used in traditional single-point compensation as described in the section “Performing Traditional Impairment Compensation at a Single Frequency”. (Thus, the traditional single-point compensation values will generally give less than perfect compensation when used as a pre-compensation, i.e., when used in FIGS. 53A and 53B.) However, there is a special case in which the two coefficient pairs coincide. As explained in the “Corruption I/Q Impairments” section, when the gain imbalance and phase skew functions are even functions, the corruption model values reduce to:
A(f)=g(f)sin(φ(f))
E_B(f)=0
C(f)=g(f)cos(φ(f))
E_D(f)=0.

[0527] Thus, equations (1.81) and (1.82) will specialize to:
α=−tan(φ(f))
β=1/{g(f)cos(φ(f))},
which are the same values used by the traditional single-point compensation.

[0528] Iterative Technique for Measuring TX Impairments

[0529] With reference now to FIG. 55A, the problems of measuring the amplitude response of receive filter 5525 and the I/Q impairments of the receiver is simplified (relative to the corresponding problems for the transmitter) because the I/Q impairments that result from the I/Q demodulator 5530 occur after the distortion effects of the receive filter 5525. For example, if a pure tone is the input signal to the receive path, then the distortion of the receive filter will alter only the magnitude and phase of the single tone. Then this altered pure tone will be distorted by the I/Q demodulator, creating the I/Q impairments. When calibrating the receiver, we can first remove the receiver's I/Q impairments, leaving only the filter's amplitude and phase response effects and then correct for the filter's amplitude and phase distortion in an additional step if desired.

[0530] However, this is not the case for the transmitter. Shown in FIG. 55B are is the signal path for the transmitter and receiver in combination. The transmitter includes an I/Q modulator 5510 and a transmit filter 5515. In some embodiments, both the transmit and receive LOs are shared. When the transmitter creates a single tone, the I/Q modulator 5510 introduces transmit I/Q impairments. Then these impairments travel through the transmit signal path, cabling, and the receive signal path before finally reaching the I/Q demodulator. This path between the I/Q modulator output and the I/Q demodulator input corrupt the transmit I/Q impairments measurement taken at the receiver. In addition, the I/Q impairments of the demodulator further corrupt the measurement of the transmitter I/Q impairments taken at the receiver. In alternative embodiments, the receiver may be based on an alternative RF architecture (i.e., other than a direct conversion architecture) such that the I/Q impairments of the receiver are very small, i.e., small enough to neglect.

[0531] Shown in FIG. 55C is an example of how the non-flat amplitude response in the signal path corrupts the I/Q impairments seen at the receiver. Coming out of the I/Q modulator are the actual I/Q impairments. Then the transmit signal path corrupts them, followed by a phase rotation due to the electrical delay of the cable, followed by another corruption by the receive signal path. In addition to the amplitude, the phase response (not shown in FIG. 55C) also creates a different but related problem.

[0532] Upon initial observation, it would appear that the ideal solution would first characterize the magnitude and phase of the signal path between the I/Q modulator and the I/Q demodulator. Then the effects of the signal path could be removed from the receiver-measured I/Q impairments by using the calculation in the section “Altering Gain Imbalance and Phase Skew through a Filter”. However, this is not a reasonable task given the performance requirements for the impairment suppression. In order to achieve image suppression of better than −80 dB, the phase skew would need to be less than 0.01 degrees. Even at lower RF frequencies, this means that the absolute phase must be stable and measureable to better than picosecond accuracy. In addition, the I/Q impairments alter the magnitude and phase of the signal coming out of the modulator as described in the section “Magnitude and Phase Corruption from I/Q Impairments” and expressed in equation (4.9) of FIG. 58A. As a result, the I/Q impairments of the transmitter, the very thing we are trying to measure, would need to be known in order to determine the magnitude and phase response of the signal path.

[0533] A better approach to determining the exact I/Q impairments through a signal path is to iterate on the solution. Given a coarse estimate of the amplitude and phase of the signal path and an estimate of the I/Q impairments, the exact I/Q impairments can be determined through enough iterations. (The iterations may be performed using shared LOs or offset LOs, as described in detail below. In the case of shared LOs, the receiver's I/Q impairments need to be known. In the case of offset LOs, the receiver I/Q impairments do not need to be known, although knowing them helps. In both cases, the transmitter's I/Q impairments do not need to be known beforehand. They will be determined as a result of the iterations.) The total number of iterations will depend largely on the initial estimates and the performance criteria. Listed below is a procedure for determining the transmitter's impairments for both shared and offset LOs. This procedure measures all calibration frequency locations within the instantaneous bandwidth and only iterates over these measurements once all measurements for a given instantaneous bandwidth have been completed. Given in the section on Optimizations is a modified procedure that obtains the same result but generally requires fewer iterations.

[0534] Iterative Method Steps (Overview):

[0535] 1. Tune the RX and TX LOs.

[0536] 2. Measure RX Impairments.

[0537] 3. Measure the mapping between RX and TX.

[0538] 4. Apply estimated impairment correction at TX.

[0539] 5. Generate tone at TX and measure at RX.