Waveform synthesis for imaging and ranging applications

[0003] The present invention is generally related to signal processing. More particularly, the present invention is related to methods and systems providing waveform synthesis for imaging and ranging applications such as RADAR, SONAR, LIDAR, medical imaging, tomography, and communications applications utilizing spread spectrum modulation/demodulation techniques.

[0004] Although the present background describes the functionality and limitations of synthetic aperture radar systems or a particular class of communications, such description is merely provided to exemplify a problem capable of resolution with the present invention. Any discussion herein directed to specific radars or communications protocols should not be taken by those skilled in the art as a limitation on the applicability of the invention described herein.

[0005] Modern high-performance radar systems often generate signals of extraordinarily wide bandwidth. For example, the General Atomics Lynx Synthetic Aperture Radar (SAR) employs a Linear-FM (LFM) chirp waveform and can operate over 3 GHz bandwidth at a 16.7 GHz center frequency. Furthermore, maximum exploitation of these radar signals requires the generated waveforms to be of very high quality, possessing exceptional spectral purity.

[0006] To facilitate high-quality LFM chirp generation, a programmable Digital Waveform Synthesizer (DWS) can often be employed. Use of a GaAs ASIC has been shown to implement well known double accumulator architectures to generate a phase that is quadratic with time, a phase that is generally converted by a memory look-up table to a digital sinusoidal signal and is ultimately converted to an analog signal by a Digital to Analog Converter (DAC). Furthermore, an ability to predistort the phase of the output as a function of instantaneous frequency to correct for unspecified nonlinearities of subsequent components in the signal path has previously been explored. No calibration scheme, however, has been presented for determining correction factors.

[0007] Two principal architectures are presently employed for achieving LFM chirp generation. The first architecture 100 is referred to as single-ended output operation, and is illustrated in FIG. 1 (labeled as prior art). With this architecture a single signal output is generated by the DWS 110 and presented to subsequent components in the signal path. After mixing 115 with a Local Oscillator (LO) 120 signal the nature of a single-ended DWS signal is to generate the desired signal as well as an undesired mirror-image signal, which must be filtered by analog components in the signal path. This filter is often called a sideband filter 130. Consequently, desired and undesired signals are separated by frequency; limiting the usable bandwidth for a generated signal to something less than half the DWS clock frequency. Equivalently, a clock frequency of more than twice the highest output waveform frequency is required. Proper final system bandwidth is achieved through a frequency multiplier 140.

[0008] Multiplexing of multiple parallel chirp generators can allow wide bandwidth single-ended chirps to be generated with commercial silicon Field Programmable Gate Array (FPGA) components. Frequency multiplication can also be employed to widen the bandwidth of a single-ended DWS output signal, but often to the detriment of spectral purity. It is also well known that frequency multiplication raises undesired frequency spurs by 6 dB per doubling with respect to the desired signal level. Frequency spurs, however, are undesired signal perturbations caused by quantization effects and DAC residual nonlinearities. Consequently, minimizing the frequency multiplication factor that can be applied to a DWS output can enhance spectral purity.

[0009] A second architecture that has been used for achieving LFM chirp generation quality can be referred to as balanced or quadrature modulator operation. Such architecture is generally illustrated in FIG. 2 (also labeled as prior art). With this architecture 200, two output signals are generated by the Quadrature DWS (QDWS) 210 and presented to a Single Sideband (SSB) mixer 220 where they are combined 215 to form a single signal to the subsequent signal path. In a perfect system, the two signals generated by the QDWS 210 will differ by a constant 90 degrees of phase, and are termed the In-phase (I) and Quadrature-phase (Q) signals. The signal pair together can be generally referred to in the art as Quadrature signals. In a perfect SSB mixer 220, no mirror-image signal will be generated, obviating the need for a sideband filter. Furthermore, no spectral separation between desired and a nonexistent undesired signal would need to be maintained. Consequently, the QDWS 210 output bandwidth of the desired signal would be able to approach the QDWS clock frequency itself, which is twice the bandwidth of the single-ended DWS system. This, in turn, would require half the frequency multiplication 230 when compared to a single-ended DWS system to achieve a final system bandwidth, and include attendant 6 dB lower spur levels and better spectral purity.

[0010] SSB signal generation techniques, including the employment of quadrature signals, are generally known in the art. Quadrature signals can be generated by a variety of techniques, including Hilbert filters that generate a 90-degree phase shift for all input waveform frequencies, and directly by separate memory look-up tables within the digital signal generation portion of the QDWS. The precision with which quadrature signals can be generated and combined in a SSB mixer, however, is problematic, particularly for high-dynamic-range applications such as imaging radar systems. Imperfections in quadrature signal generation or their combination within a SSB mixer results in the non-cancellation of the undesired mirror-image sideband signal. Such imperfections can result in a relative phase error or an amplitude imbalance. Additionally, the LO 120 may undesirably leak through the mixer and be present in the mixer output in addition to the desired signal. Any of these errors reduce the spectral purity of the resulting SSB mixer output signal, and degrades a SAR image with ghosts and other artifacts. Consequently, quadrature modulators in high-performance radar systems require some form of error cancellation or other mitigation scheme.

[0011] In the field of communications, a quadrature modulator for wireless CDMA systems has been described wherein amplitude and phase of the quadrature component signals are predistorted to provide perfect quadrature signals to the SSB mixer. Furthermore, DC biases are added to the quadrature output signals to mitigate LO leakage. The corrections, however, are derived for a single QDWS output frequency and do not allow for frequency dependent errors. While this may be reasonable for applications such as wireless communications, it is inadequate for high-performance SAR systems. Furthermore, no attempt is made to compensate imbalances in the SSB mixer itself, and the stated procedure precludes this.

[0012] An iterative procedure to adjust for LO leakage, and phase and gain imbalance in a quadrature modulator based on an envelope detection-of a transmitted signal has also been described by the prior art. Other prior art techniques include: use of a similar technique to compensate an SSB mixer-which degrades a resultant output signal and adaptive techniques for achieving quadrature signal balance using a test tone. The adaptation procedures currently described in the prior art is not adequate for wideband LFM chirps. Furthermore, verified wideband frequency-dependent errors are not addressed by the prior art.

[0013] A quadrature modulator that allows frequency-dependent phase and amplitude corrections to be made to the output of the QDWS has been proposed. All corrections are made to analog signals after the DACs. These corrections, however, neglect problematic frequency dependent errors in the SSB mixer. Furthermore, the nature of the errors to be corrected is presumed to be predetermined—that is, no calibration procedure is discussed.

[0014] A quadrature modulator has been described that can be constructed that implements frequency dependent corrections before and at the DACs. The SSB mixer, however, is not addressed, nor is correction due to imbalance from any components or sources addressed.

[0015] Thus far, quadrature modulator prior art generally is inadequate for high-performance radar systems for one or more of the following reasons:

[0016] 1) The balance corrections are relatively narrowband, and do not facilitate frequency dependent phase and amplitude corrections.

[0017] 2) Adaptive schemes are not fast enough for LFM wideband chirp waveforms:

[0018] 3) Imbalances within the SSB mixer itself are not addressed.

[0019] 4) LO leakage (e.g., especially any frequency dependence) is not adequately addressed.

[0020] 5) Where calibration schemes should be required, no calibration schemes are proposed.

[0021] The prior art leaves the skilled in the art with limited choices between narrow-band quadrature modulators that can compensate for imbalances and LO leakage using iterative techniques and wideband quadrature modulators that do not compensate for frequency dependent errors in the SSB mixer. Accordingly, there is a need for an improved wideband quadrature modulator and SSB mixer combination, which produces suitable signals for use in high fidelity radar waveform generation. Additionally, there is also a need for an operational methodology that is insensitive to the imbalances and leakage in a quadrature modulator and associated SSB mixer.

[0022] In the field of quadrature demodulators, which are used in radar receivers, U.S. Pat. No. 6,469,661, issued Oct. 22, 2002 to two of the present inventors teaches how errors due to I and Q channel imbalances in a receiver's demodulator can be mitigated by processing radar signals with pulse-to-pulse, rolling phase shifts. With this technique no calibration is required to achieve a balanced demodulator output signal. The use of quadrature modulators in radar transmitters, however, has not been described. The present inventors have discovered how use of a Quadrature DWS is viable for high-performance radar waveform synthesis. A principal advantage of the present invention is that fewer frequency multipliers are needed to generate radar waveforms of sufficient bandwidth; thereby offering the potential of cleaner waveforms (e.g., signals processing lower frequency-spurs).

[0023] Problems that generally need to be resolved with QDWS signals are those related to the generation of unwanted signal components due to energies such as LO feed-through found in the non-ideal SSB mixer and quadrature signal imbalance, as well as problem typically caused within the SSB mixer. For example, the inclusion of unwanted energy into a radar's output (e.g. SAR images) creates problems associated with the QDWS and SSB components that result in signal waveform degradation, and hence radar performance. Similar problems are encountered with other systems engaged in or requiring signal/waveform synthesis, such as: SONAR, LIDAR, medical imaging, tomography, and communications.

[0024] It is a feature of the present invention to provide an improved Quadrature Digital Waveformn Synthesizer that can further provide systems with frequency dependent corrections for quadrature imbalance and Local Oscillator (LO) feed-through.

[0025] It is another feature of the present invention to provide operational procedures to system that can enable filtration of quadrature imbalance effects from energies, such as LO feed-through energy and imbalance energy, and their effects without prior calibration or equalization.

[0026] It is yet another feature of the present invention to provide calibration procedures that can also be implemented into systems for signal synthesis.

[0027] Techniques that are herein disclosed by the present inventors to mitigate the negative effects of signal components brought on by unwanted energies such as LO feed-through and quadrature signal imbalance include the benefits of:

[0028] 1) Compensating or equalizing energy imbalance by adjusting inputs (DC offsets, phase, and amplitude) to the SSB mixer so that problematic signal components aren't generated, and

[0029] 2) Operating in a manner (e.g. by employing rolling phase shifts) that separates problem signals (i.e., energy) from a desired signal in Doppler to allow filtering to suppress problem energy.

[0030] In accordance with methods of the present invention, waveform generation in a system can be adjusted/corrected, for example, in a synthetic aperture radar system (SAR), where a rolling phase shift is applied to the SAR's QDWS signal where it is demodulated in a receiver; the LO feed-through energy and/or imbalance energy are then separated from a desired signal in Doppler, the separated energy is filtered from the receiver leaving the desired signal; and the separated energy in the receiver is measured to determine the degree of imbalance that is represented by it. The degree of imbalance in the system can be used to determine calibration values that can then be provided as compensation for frequency dependent errors in components, such as a SAR's QDWS and SSB mixer, affecting quadrature signal quality.

[0031] Error correction circuits can be built into the QDWS that compensate wideband frequency dependent phase and amplitude imbalances originating in the QDWS, SSB mixer, and other non-ideal components in the I and Q channels of the quadrature modulator. This can be done by predistorting the relative phase and amplitudes of the QDWS quadrature signal components by programmable frequency dependent digital perturbations into the QDWS prior to analog signal generation. LO leakage in the SSB mixer is thereafter compensated by applying DC offsets to the SSB inputs.

[0032] Frequency dependent phase and amplitude corrections and LO leakage compensation can be determined by a calibration process that involves applying pulse-to-pulse rolling phase shifts to the LFM signal within the QDWS. These phase shifts can be removed in the receiver by conjugate phase shifts. The application of these phase shifts can separate desired signals from differential-mode imbalance energy in a Doppler spectrum (the spectrum taken across the pulses). Separation allows for the unique identification, of the nature of the quadrature imbalance and LO leakage as a function of frequency for calibration purposes.

[0033] Alternatively, the present inventors teach that the calibration process can be dispensed with, where Doppler filtering can be applied to the received data to simply remove the imbalance, such as LO leakage energy, from the received signal. Calibration can thereby be rendered unnecessary, rendering any quadrature signal imbalances, or LO leakage, impotent.

[0034] The accompanying drawings, which are incorporated in and form part of the specification, illustrate embodiments of the present invention and, together with the description, serve to explain relevant principles of the invention.

[0035] FIG. 1 provides a prior art illustration of a single-ended DWS with frequency conversion, sideband filter, and frequency multiplication components.

[0036] FIG. 2 provides a prior art illustration of a quadrature modulator employing a QDWS, SSB mixer, and frequency multiplier.

[0037] FIG. 3 illustrates a non-ideal SSB Mixer model.

[0038] FIG. 4 illustrates a simplified radar-processing model.

[0039] FIG. 5 illustrates a simplified non-ideal SSB Mixer model.

[0040] FIG. 6 illustrates a simplified non-ideal SSB Mixer model with LO feed-through compensation.

[0041] FIG. 7 illustrates a Simplified non-ideal SSB Mixer model with LO feed-through compensation and quadrature signal equalization.

[0042] FIG. 8 illustrates a LFM chirp generation using QECDWS. The PEC RAM contains phase corrections common to both I and Q channels. The QEC phase RAM corrects phase differences, and the QEC magnitude RAM corrects amplitude differences.

[0043] FIG. 9 illustrates a Frequency representation of a conventional technique for generating QDWS output.

[0044] FIG. 10 illustrates the generation of QDWS output with frequency bias.

[0045] FIG. 11 illustrates the generation of QDWS output with frequency bias.

[0046] FIG. 12 illustrates filtering effects of non-ideal DAC.

[0047] FIG. 13 illustrates Doppler separation of LO feed-through signal from desired signal using π rad./pulse modulation of the QDWS phase.

[0048] FIG. 14 illustrates a method for generating π rad./pulse modulation of the QDWS phase.

[0049] FIG. 15 illustrates Doppler separation of LO feed-through signal and I/Q imbalance signal from desired signal using rolling phase shift of the QDWS phase.

[0050] FIG. 16 illustrates an example energy map (image) or point scatterer in radar with LO feed-through and I/Q imbalance.

[0051] FIG. 17 illustrates insertion of a rolling phase shift into the QDWS output.

[0052] FIG. 18 illustrates a method for generating π/2 rad./pulse modulation of the QDWS phase.

[0053] FIG. 19 illustrates a simulation of gradient search method for nulling LO feed-through signal, SNR is for desired signal after compression.

[0054] FIG. 20 illustrates the relationship of residual video LFM chirp due to LO feed-through to video bandwidth and relative echo delay time.

[0055] FIG. 21 illustrates the relationship of residual video LFM chirp due to quadrature channel imbalance to video bandwidth and relative echo delay time.

[0056] FIGS. 22A-22D illustrate simulations of LO feed-through migration and I/Q imbalance mitigation using gradient search parameter estimation. Images displayed are periodic snapshots of iterative cycles.

[0057] Although the following provides a detailed description of the functionality and limitations that can be found in synthetic aperture radar (SAR) systems or may refer to a particular class of communications, such description is merely provided to exemplify a signaling problem commonly found in imaging, ranging and communication equipment that is capable of resolution with the present invention. Any discussion herein directed to specific radars or communications and their associated protocols should not be taken by those skilled in the art as a limitation on the applicability of the invention described herein.

[0058] The following detailed description is comprised of three main sections:

[0059] A. A model for errors and their impact on radar data, including errors due to the SSB mixer, effects of multipliers, and radar processing.

[0060] B. Mitigation techniques for non-ideal performance, including equalization techniques and operational techniques that do not require calibration.

[0061] C. Parameter estimation for equalization, including procedures for calibrating the QDWS.

[0062] FIG. 3 illustrates an architecture 300 for the non-ideal mixer. The effects of non-ideal QDWS, SSB mixer, and related components can be modeled in the following development:

[0063] Module H_I(f)=the application of the perturbations offered to the I channel 310,

[0064] Module H_Q(f)=the application of the perturbations offered to the Q channel 320,

[0065] Module H′_O(f)=the application of the perturbations offered to the LO signal 330, and

[0066] Module H_L(f)=the transfer function of the LO leakage signal to the output 340.

[0067] When the Local Oscillator is a single-frequency sinusoid with frequency ω₀=2πf₀, the transfer functions H_O(f) and H_L(f) represent constant perturbations to amplitude and phase. Consequently, for the single-frequency LO, intermediate signals can be identified as:

[0068] i₀=cos(ω₀t),

[0069] q₀=(1+k_q0)sin(ω₀t+θ_q0) and

[0070] g₀=k_g0cos(ω₀t+θ_g0).

[0071] For ideal quadrature input signals, k_q0=k_g0=0, and θ_q0=θ_g0=0 are desired.

[0072] Correspondingly, the transfer functions can be identified by H_I(f) and H_Q(f) as descriptions of frequency-dependent amplitude and phase perturbations to their respective quadrature signal inputs. These quadrature signals would have been originated by the QDWS.

[0073] Consider now the inputs to be defined as single frequency sinusoids:

[0074] x_QDWS,I(t)=cos(ω_Tt), and

[0075] x_QDWS,Q(t)=sin(ω_Tt).

[0076] In a radar waveform, these signals are normally finite in duration. The envelope in the following development, however, can be ignored to simplify the discussion. Nevertheless, an underlying presumption is generally that the envelope does in fact exist, and will need to be reinserted at some point.

[0077] The corresponding intermediate signals would then be identified as

[0078] i₁=(1+k_i1)cos(ω_Tt+θ_i1and

[0079] q₁=(1+k_q1)sin(ω_Tt+θ_q1).

[0080] Again, if the applied quadrature signals were ideal, k_i1=k_q1=0 and θ_i1=θ_q1=0 would be desirable. If not equal to zero, the next best condition can be that the respective parameters are equal to each other, to at least maintain balance. In any case, in a practical system we expect these perturbations to be frequency dependent.

[0081] Multiplying out the intermediate signals yields: $\begin{array}{c}{x}_{\mathrm{Mixer}}\left(t\right)=i_0{i}_1-q_0{q}_1+g_0\\ =\left(\begin{array}{c}\left(1+k_{\mathrm{i1}}\right)\cos \left({\omega }_{0}t\right)\cos \left({\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)-\\ \left(1+k_{\mathrm{q0}}\right)\left(1+k_{\mathrm{q1}}\right)\sin \left({\omega }_{0}t+{\theta }_{\mathrm{q0}}\right)\sin \left({\omega }_{T}t+{\theta }_{\mathrm{q1}}\right)+\\ k_{\mathrm{g0}}\cos \left({\omega }_{0}t+{\theta }_{\mathrm{g0}}\right)\end{array}\right)\end{array}.$

[0082] It can be noted that as far as the output is concerned, a scaling error k_q0is indistinguishable from a scaling error k_q1. That is, for every perturbation in one, a corresponding perturbation exists in the other. Consequently, no generality can be lost by assuming k_q0=0. This yields: $x_{\mathrm{Mixer}}\left(t\right)=\left(\begin{array}{c}\left(1+k_{\mathrm{i1}}\right)\cos \left({\omega }_{0}t\right)\cos \left({\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)-\\ \left(1+k_{\mathrm{q1}}\right)\sin \left({\omega }_{0}t+{\theta }_{\mathrm{q0}}\right)\sin \left({\omega }_{T}t+{\theta }_{\mathrm{q1}}\right)+\\ k_{\mathrm{g0}}\cos \left({\omega }_{0}t+{\theta }_{\mathrm{g0}}\right)\end{array}\right).$

[0083] The phase perturbations θ_q0, however, is not indistinguishable from any other term. Consequently, θ_q0=0 cannot be assumed and must be carried along.

[0084] For small imbalances and small perturbations, the mixer output can be expanded and approximated as: $x_{\mathrm{Mixer}}\left(t\right)=\left(\begin{array}{c}\left[\left(1+k_{\mathrm{i1}}\right)+\frac{\left(k_{\mathrm{q1}}-k_{\mathrm{i1}}\right)}{2}\right]\cos \left({\omega }_{0}t+{\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)-\\ \left[\frac{{\theta }_{\mathrm{q1}}+{\theta }_{\mathrm{q0}}-{\theta }_{\mathrm{i1}}}{2}\right]\sin \left({\omega }_{0}t+{\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)-\\ \left[\frac{k_{\mathrm{q1}}-k_{\mathrm{i1}}}{2}\right]\cos \left({\omega }_{0}t-{\omega }_{T}t-{\theta }_{\mathrm{i1}}\right)-\\ \left[\frac{{\theta }_{\mathrm{q1}}-{\theta }_{\mathrm{q0}}-{\theta }_{\mathrm{i1}}}{2}\right]\sin \left({\omega }_{0}t-{\omega }_{T}t-{\theta }_{\mathrm{i1}}\right)+\\ k_{\mathrm{g0}}\cos \left({\omega }_{0}t+{\theta }_{\mathrm{g0}}\right)\end{array}\right).$

[0085] This signal is considered real valued. Consequently, spectral symmetry allows us to work with just the positive frequencies. Positive frequencies can be extracted and manipulated to the form: $x_{\mathrm{Mixer}}\left(t\right)=\frac{1}{2}\exp j\left({\omega }_{0}t\right)\left(\begin{array}{c}\left[1+k_{\mathrm{i1}}+\frac{\Delta k}{2}+j\left(\frac{\Delta \theta }{2}+{\theta }_{\mathrm{q0}}\right)\right]\exp j\left({\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)-\\ \left[\frac{\Delta k}{2}-j\frac{\Delta \theta }{2}\right]\exp -j\left({\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)+k_{\mathrm{g0}}\exp j{\theta }_{\mathrm{g0}}\end{array}\right),$

[0086] where the imbalances have been defined as (k_q1−k_i1)=Δk, and (θ_q1−θ_i1)=Δθ. In this form, it should be noted that the carrier is modulated by three terms, each with distinctly different characteristics.

[0087] If, ideally, no imbalances exist, this would reduce to elements of the first modulation term, that is: $x_{\mathrm{Mixer},\mathrm{No}\mathrm{Imbalances}}\left(t\right)=\frac{1}{2}\left[1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}\right]\exp j\left({\omega }_{0}t+{\omega }_{T}t+{\theta }_{\mathrm{i1}}\right).$

[0088] No perturbations of any kind would further reduce this to: $x_{\mathrm{Mixer},\mathrm{Ideal}}\left(t\right)=\frac{1}{2}\exp j\left({\omega }_{0}t+{\omega }_{T}t\right).$

[0089] The first modulation term in x_Mixer(t) is desired, and for small imbalances is very nearly ideal, in the relative sense.

[0090] The second modulation term is due to imbalances, and contains the conjugate, or “image” of the desired modulation.

[0091] The third term is due to local oscillator feed-through, which manifests itself as a DC modulation of the carrier, albeit with some phase shift.

[0092] The effects of imbalance to the magnitude and phase of the desired term are typically inconsequential. Consequently, the mixer output can be further simplified to: $x_{\mathrm{Mixer}}\left(t\right)=\frac{1}{2}\exp j\left({\omega }_{0}t\right)\left(\begin{array}{c}\left[1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}\right]\exp j\left({\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)-\\ \left[\frac{\Delta k}{2}-j\frac{\Delta \theta }{2}\right]\exp -j\left({\omega }_{T}t+{\theta }_{\mathrm{i1}}\right)+\\ k_{\mathrm{g0}}\exp j{\theta }_{\mathrm{g0}}\end{array}\right).$

[0093] Furthermore, if ω_Tis allowed to vary with time itself, as with a LFM chirp, ω_Tt can be replaced with the more general phase function Φ(t), but the fact that Δk and Δθ are frequency dependent should be kept in mind. The mixer output can then be described as: $x_{\mathrm{Mixer}}\left(t\right)=\left(\begin{array}{c}{A}_0\exp j\left({\omega }_{0}t+\Phi \left(t\right)+{\theta }_{\mathrm{i1}}+{\varphi }_0\right)\times \\ \left(\begin{array}{c}1+A_1\exp -j\left(2\Phi \left(t\right)+2{\theta }_{\mathrm{i1}}+{\varphi }_1\right)+\\ A_L\exp -j\left(\Phi \left(t\right)+{\theta }_{\mathrm{i1}}-{\theta }_{\mathrm{g0}}-{\varphi }_L\right)\end{array}\right)\end{array}\right)$

[0094] where frequency dependent terms can be defined as: $A_0\exp j{\varphi }_0=\left(\frac{1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}}{2}\right)$ $A_1\exp -j{\varphi }_1=-\frac{1}{2}\left(\frac{\Delta k-j\mathrm{\Delta \theta }}{1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}}\right),\mathrm{and}$ $A_L\exp j{\varphi }_L=\left(\frac{k_{\mathrm{g0}}}{1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}}\right).$

[0095] The frequency multiplier output can initially take this expression to the power M, and then applies appropriate filtering. Furthermore, the non-linear compression nature of the multiplication stages typically mitigates to some extent any absolute amplitude fluctuations. For small imbalances, the net result is then reasonably modeled as: $x_M\left(t\right)\approx \left(\begin{array}{c}\exp j\left(M{\omega }_{0}t+M\Phi \left(t\right)+M{\theta }_{\mathrm{i1}}+M{\varphi }_0\right)\times \\ \left(\begin{array}{c}1+{\mathrm{MA}}_1\exp -j\left(2\Phi \left(t\right)+2{\theta }_{\mathrm{i1}}+{\varphi }_1\right)+\\ {\mathrm{MA}}_L\exp -j\left(\Phi \left(t\right)+{\theta }_{\mathrm{i1}}-{\theta }_{\mathrm{g0}}-{\varphi }_L\right)\end{array}\right)\end{array}\right).$

[0096] The increase in the level of the undesired terms by a factor M is now evident. As this happens to all pre-multiplier spurs, it rationalizes a desire to minimize the multiplication factor in an overall system design.

[0097] The Radar Receiver Video Signal and Correlator Output

[0098] For purposes of discussion, a simplified radar system 400 is illustrated in FIG. 4. Output from the multiplier 401 can be presumed to be a radar transmitted (TX) signal. The received (RX) echo signal is delayable 410 by an amount t_s. The receiver can mix 415 echo with a receiver local oscillator signal 420 in the form:

[0099] x_L(t)=x_M*(t−t_m)

[0100] to form a video signal 421. The superscript * denotes complex conjugate.

[0101] The video signal can be written in a reduced form (ignoring some negligible terms) as: $x_V\left(t\right)=\left[\exp j\left(M{\omega }_0\left(t_m-t_s\right)+M\left(\Phi \left(t-t_s\right)-\Phi \left(t-t_m\right)\right)+M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)\right)\times \left(1+{\mathrm{MA}}_{l,m}\exp j\left(2\Phi \left(t-t_m\right)+2{\theta }_{i1,m}+{\varphi }_{l,m}\right)+{\mathrm{MA}}_{l,s}\exp -j\left(2\Phi \left(t-t_s\right)+2{\theta }_{i1,s}+{\varphi }_{l,s}\right)+{\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{i1,m}-{\theta }_{g0,m}-{\varphi }_{L,m}\right)+{\mathrm{MA}}_{L,s}\exp -j\left(\Phi \left(t-t_s\right)+{\theta }_{i1,s}-{\theta }_{g0,s}-{\varphi }_{L,s}\right)\right)\right].$

[0102] It should be noted that a number of terms are in fact frequency dependent—that is, dependent on the instantaneous frequency of either Φ(t−t_s) or Φ(t−t_m), which may in fact be different from each other at any instant in time. Consequently additional subscripts have been added herein to distinguish these terms as being influenced by which instantaneous frequency.

[0103] Strictly speaking, because pulses are being dealt with, unless t_m=t_s, there will typically be some non-overlap in the envelopes of the two signals. This is generally deemed normal for “stretch processing” and can be compensated by lengthening the envelopes (pulse widths) slightly beyond the interval over which data is typically collected from the video signal. Nevertheless, development can proceed as if the envelopes do in fact overlap sufficiently. The envelope of a video signal itself is adequately represented by: $\mathrm{rect}\left(\frac{t-t_m}{T}\right)\mathrm{rect}\left(\frac{t-t_s}{T}\right)\approx \mathrm{rect}\left(\frac{t-t_m}{T}\right)$

[0104] where T is the pulse width.

[0105] The phase term M(φ_0,s−φ_0,m) represents undesired phase variations in the desired video component, due to non-ideal signal paths. The result in SAR images, for example, is range-sidelobes, even where I and Q channels are completely balanced and LO feed-through entirely suppressed. This is precisely the term that more traditional DWS Phase-Error Correction is targeted to fix.

[0106] The integral of the video signal over time is the correlator output for the radar and describes the Impulse Response (IPR) of the radar. A correlation peak occurs when t_m=t_s, for which the video signal is: ${x_V\left(t\right)|}_{t_m=t_s}=\left(1+2{\mathrm{MA}}_{i,s}\cos \left(2\Phi \left(t-t_s\right)+2{\theta }_{i1,s}+{\varphi }_{l,s}\right)+2{\mathrm{MA}}_{L,s}\cos \left(\Phi \left(t-t_s\right)+{\theta }_{i1,s}-{\theta }_{g0,s}-{\varphi }_{L,s}\right)\right).$

[0107] Several observations can be noted:

[0108] The “1” describes the properly focused signal that would be sought.

[0109] The second term is the residual due to imbalance in the I and Q channels of the QDWS and SSB mixer. This signal retains time dependence, with frequency components of 2Φ(t)—that is, its energy is spread across twice the bandwidth of Φ(t).

[0110] The third term is the residual due to LO feed-through in the SSB mixer. This signal also retains time dependence, with frequency components of Φ(t). That is, its energy is spread across just the bandwidth of Φ(t).

[0111] The second and third terms are undesired and constitute ‘multiplicative noise’ in that their energy level is proportional to the desired component's response energy level. The total multiplicative noise generated relative to the signal is then:

[0112] MNR_DWS=2M²(A_I²+A_L²)

[0113] where some sort of average value of A_Iand A_Lare used.

[0114] While this expression indicates the total relative undesired energy available, it fails to account for any diminishment due to video pass-band filtering, and may in some circumstances be too pessimistic. The video signal will normally have a bandwidth much reduced from that of x_M(t). It can be noted, however, that undesired terms are functions whose phase are themselves time-dependent—that is, whose phase depends on delayed versions of Φ(t). Typical phase functions Φ(t) for radar are such that exp jΦ(t) has a spectral bandwidth (as with a LFM chirp), that when appropriately multiplied, allows a desired range resolution. Consequently, if we denote B_Φ as the bandwidth of exp jΦ(t), then energy due to I and Q channel imbalance is spread over a bandwidth 2B_Φ, and energy due to LO feed-through is spread over a bandwidth B_Φ.

[0115] If the video signal is filtered (as it typically is) to some bandwidth B_V, then some of the offending undesired energy may be filtered, and rendered unavailable to the correlator output. Consequently, fractional bandwidths can be defined as: ${\beta }_l=\frac{2B_V}{2B_{\Phi }}=\frac{M{B}_V}{B_M},\mathrm{and}$ ${\beta }_L=\frac{2B_V}{B_{\Phi }}=\frac{2M{B}_V}{B_M}.$

[0116] but limited to β_I≦1 and β_L≦1, and with B_Mthe bandwidth of the TX signal x_M(t) as previously defined. With these definitions, the Multiplicative Noise Ratio (MNR) from a single return echo is limited to

[0117] MNR_DWS=2M²(β_IA₁²+β_LA_L²).

[0118] As an example, given M=8, B_V=30 MHz, and B_M=1800 MHz for 0.1-m resolution, would result in is β_I=0.133, and β_L=0.267, offering 8.75 dB and 5.74 dB reductions in the MNR due to their respective sources. The reduction in MNR, however, may not always be as much as one might think. For example, given M=6, B_V=30 MHz, and B_M=180 MHz for 1-m resolution, would result in β_I=β_L=1 with no MNR reduction due to the reduced video filter. Note that higher multiplication factors M, and coarser resolutions render the video filter less effective in reducing MNR due to imbalance and LO feed-through.

[0119] An additional problem can be found when the bandwidth of the undesired signals extends beyond the video filter bandwidth, which also implies that radar echoes that would ordinarily be filtered by the video filter might still throw energy into the video pass band due to LO feed-through and I/Q imbalance.

[0120] In any case, for a versatile high-performance radar system, the video filter cannot be relied upon to adequately reduce the effects of imbalance and LO feed-through. Consequently, the imbalance and LO feed-through (or at least the effects thereof) generally need to be mitigated by other techniques.

[0121] Techniques for dealing with problematic QDWS and SSB mixer associated I/Q channel imbalance and LO feed-through can generally fall into three broad classes:

[0122] 1) Compensating or equalizing the imbalance and LO feed-through so that problematic signal components aren't generated,

[0123] 2) Separating the problematic signal components from the desired signal in a manner that allows filtering to work, and

[0124] 3) Dealing with the effects of the problematic signal components in the receiver.

[0125] Of course, in light of the present description, one should appreciate that various ideas can be mixed and matched from all three classes into an overall strategy to maximize system performance.

[0126] Equalization Techniques

[0127] The first mitigation technique that will now be described involves equalizing the quadrature channels. One goal of this technique is to cause the SSB mixer output to be as nearly ideal as possible. This can be achieved by pre-distorting the SSB input signal in a complementary manner from its own distortion, such that the net result is an ideal signal at the mixer output.

[0128] There are generally two distinct problem signals. The first is due to LO feed-through, and the second is due to I/Q channel imbalance. Each can be treated in order. To facilitate this development the architecture of an SSB mixer 520 itself can be desired to, as is illustrated in the system 500 of FIG. 5, where one can appreciate the mathematical model: $x_{\mathrm{Mixer}}\left(t\right)=\left(A_0\exp j\left({\omega }_{0}t+\Phi \left(t\right)+{\theta }_{i1}+{\varphi }_0\right)\times \left(1+A_l\exp -j\left(2\Phi \left(t\right)+2{\theta }_{i1}+{\varphi }_i\right)+A_L\exp -j\left(\Phi \left(t\right)+{\theta }_{i1}-{\theta }_{g0}-{\varphi }_L\right)\right)\right).$

[0129] Also, it should be noted that the effects of k_q0have been absorbed into k_q1, and hence A_Iand φ_I. The goal here is to make the quantities A_Iand A_Las small as possible, ideally zero.

[0130] Compensating LO Feed-Through

[0131] The LO feed-through energy may be mitigated by forcing A_L=0 by appropriately modifying or selecting properties of x_QDWS,I(t) and x_QDWS,Q(t). Note that the LO feed-through signal can manifest itself as an equivalent DC modulation of the LO signal. By forcing input signals x_QDWS,I(t) and x_QDWS,Q(t) to be DC signals, with respective values:

[0132] x_QDWS,I(t)=V_L,I, and

[0133] x_QDWS,Q(t)=V_L,Q,

[0134] intermediate signals can be identified as:

[0135] i₁=V_L,IH_I(0), and

[0136] q₁=V_L,QH_Q(0).

[0137] For the DC signal input, the SSB mixer 520 output can then be described by: $x_{\mathrm{Mixer}}\left(t\right)=(\lfloor V_{L,l}{H}_l\left(0\right)+k_{g0}\cos \left({\theta }_{g0}\right)-V_{L,Q}{H}_Q\left(0\right)\sin \left({\theta }_{q0}\right)\rfloor \cos \left({\omega }_{0}t\right)-\left[V_{L,Q}{H}_Q\left(0\right)\cos \left({\theta }_{q0}\right)+k_{g0}\sin \left({\theta }_{g0}\right)\right]\sin \left({\omega }_{0}t\right)).$

[0138] Positive frequencies can be described by: $x_{\mathrm{Mixer}}\left(t\right)=\frac{1}{2}(\lfloor V_{L,l}{H}_l\left(0\right)+k_{g0}\cos \left({\theta }_{g0}\right)-V_{L,Q}{H}_Q\left(0\right)\sin \left({\theta }_{q0}\right)\rfloor +j\left[V_{L,Q}{H}_Q\left(0\right)\cos \left({\theta }_{q0}\right)+k_{g0}\sin \left({\theta }_{g0}\right)\right])\exp j\left({\omega }_{0}t\right).$

[0139] The mixer output could be zeroed, thereby mitigating the effects of LO feed-through, by setting: $V_{L,Q}=-\left(\frac{k_{g0}\sin \left({\theta }_{g0}\right)}{H_Q\left(0\right)\cos \left({\theta }_{q0}\right)}\right),\mathrm{and}$ $V_{L,l}=-\left(\frac{k_{g0}\cos \left({\theta }_{g0}\right)-V_{L,Q}{H}_Q\left(0\right)\sin \left({\theta }_{q0}\right)}{H_l\left(0\right)}\right).$

[0140] Clearly, as long as these DC values are properly input to the SSB mixer 520, the LO feed-through can be nullified, that is A_L=0, even in the presence of other spectral components. FIG. 6 illustrates a system 600 wherein DC biases 610 can be added to the SSB mixer 620 input to accomplish compensation. It can be noted also that θ^q0≠0 can cause the appropriate value for V_L,Ito depend on the proper value for V_L,Q. If the LO feed-through amplitude and phase are dependent on the frequency of the input signal x_QDWS(t), then the DC compensation values V_L,Iand V_L,Qneed to be dependent on the instantaneous frequency generated by a QDWS.

[0141] Balancing I/Q Channels with Error Correction

[0142] The quadrature channels can be balanced by forcing A_I=0 by appropriately modifying or selecting x_QDWS,I(t) and x_QDWS,Q(t). In this aspect of the invention the imbalance can be found in the transfer functions H_I(f) and H_Q(f), and LO phase imbalance term θ_q0.

[0143] To compensate for the undesired transfer functions H_I(V) and H_Q(f), and phase imbalance θ_q0, equalization filters 710 can be inserted between the DWS output and the SSB mixer 720, as illustrated for the system 700 shown in FIG. 7. Alternately the behavior of the equalization filters 710 can be incorporated into the QDWS itself by suitably perturbing the quadrature signals.

[0144] A model for the input signals can be returned to, namely:

[0145] x_QDWS,I(t)=cos(Φ(t)), and

[0146] x_QDWS,Q(t)=sin(Φ(t))

[0147] and the limiting envelope rect(t/T) can continue to be ignored. In the absence of the equalization filters 720 (and ignoring the DC offset inputs), we recall the intermediate signals to be:

[0148] i₁=(1+k_i1)cos(Φ(t)+θ_i1), and

[0149] q₁=(1+k_q1)sin(Φ(t)+θ_q1)

[0150] where perturbation parameters k_i1, θ_i1, k_q1, and θ_q1embody the effects of H_I(f) and H_Q(f), and depend on (may vary with) the instantaneous frequency ω_T=dΦ(t)/dt.

[0151] The effects of the equalization filters 710 can now be added, and it should be noted that these transfer functions are also just descriptions of amplitude and phase variations as a function of frequency. Consequently the effects of the equalization filters 710 can be mathematically modeled as:

[0152] i₁=(1+k_i1,Heq)(1+k_i1)cos(Φ(t)+θ_i1+θ_i1,Heq), and

[0153] q₁=(1+k_q1,Heq)(1+k_q1)sin(Φ(t)+θ_q1+θ_q1,Heq)

[0154] where k_i1,Heq, θ_i1,Heq, k_q1,Heqand θ_q1,Heqdescribe the effects of the equalization filters 710 at frequency ω_T.

[0155] For the subsequent development equivalent parameters can now simply be defined as:

[0156] (1+k_i1,eq)=(1+k_i1,Heq)(1+k_i1),

[0157] θ_i1,eq=θ_i1+θ_i1,Heq,

[0158] (1+k_q1,eq)=(1+k_q1,Heq)(1+k_q1), and

[0159] θ_q1,eq=θ_q1+θ_q1,Heq.

[0160] The mixer 720 output without any LO feed-through, can also be described by: $x_{\mathrm{Mixer}}\left(t\right)=\left(\begin{array}{c}{A}_0\exp j\left({\omega }_{0}t+\Phi \left(t\right)+{\theta }_{\mathrm{i1},\mathrm{eq}}+{\varphi }_0\right)\times \\ \left(\begin{array}{c}1+\\ A_1\exp -j\left(2\Phi \left(t\right)+2{\theta }_{\mathrm{i1},\mathrm{eq}}+{\varphi }_1\right)\end{array}\right)\end{array}\right).$

[0161] with the grouped parameters now defined by: $A_0\exp j{\varphi }_0=\left(\frac{1+k_{\mathrm{i1},\mathrm{eq}}+j{\theta }_{\mathrm{q0}}}{2}\right),A_1\exp -j{\varphi }_1=-\frac{1}{2}\left(\frac{\Delta k-j\mathrm{\Delta \theta }}{1+k_{\mathrm{i1},\mathrm{e1}}+j{\theta }_{\mathrm{q0}}}\right),$

[0162] where now Δk=(k_q1,eq−k_i1,eq), and Δθ=(θ_q1,eq−θ_i1,eq−θ_q0).

[0163] To force A_I=0, Δk=Δθ=0 is required and further requires:

[0164] $k_{\mathrm{q1},\mathrm{Heq}}=\left(1+k_{\mathrm{i1},\mathrm{Heq}}\right)\frac{\left(1+k_{\mathrm{i1}}\right)}{\left(1+k_{\mathrm{q1}}\right)}-1,\mathrm{and}$  θ_q1,Heq=θ_i1−θ_q1+θ_i1,Heq+θ_q0.

[0165] k_i1,Heq=θ_i1,Heq=0 can be arbitrarily set and still maintain complete balance, or complete equalization can be forced by setting: $k_{\mathrm{i1},\mathrm{Heq}}=\frac{1}{\left(1+k_{\mathrm{i1}}\right)}-1,\mathrm{and}$

[0166] This would force k_q1,eq=k_i1,eq=0, and θ_i1,eq=(θ_q1,eq−θ_q0)=0.

[0167] Equalization can therefore be accomplished for LFM chirp generation by implementing the required amplitude and phase corrections of the equalization filters 710 directly into the QDWS itself. This scheme would produce x_QDWS,I(t) and x_QDWS,Q(t) with the error corrections already “built in”, and can be referred to as Quadrature Error Corrected DWS (QECDWS). A block diagram detailing how this might be incorporated is illustrated for the system architecture 800 shown in FIG. 8.

[0168] In the system architecture 800, the frequency accumulator 810 and phase accumulator 820 constitute the double accumulator architecture for LFM chirp generation. The PEC RAM 830 applies frequency dependent common-mode phase error corrections of the type previously discussed. The QEC Phase RAM 840 applies frequency dependent corrections θ_q1,Heq=θ_i1−θ_q1+θ_i1,Heq+θ_q0. The QEC magnitude

[0169] RAM 850 applies frequency dependent corrections $k_{\mathrm{q1},\mathrm{Heq}}=\left(1+k_{\mathrm{i1},\mathrm{Heq}}\right)\frac{\left(1+k_{\mathrm{i1}}\right)}{\left(1+k_{\mathrm{q1}}\right)}-1.$

[0170] By applying both LO feed-through compensation and quadrature channel balancing simultaneously, the mixer output can be described by the more desirable model:

[0171] X_{Mixer,No,Imbalance}(t)=A₀exp j(ω₀t+Φ(t)+θ_i1,eq+φ₀).

[0172] Consequently, the imbalances due to a non-ideal SSB mixer have been compensated by corrections within the QDWS.

[0173] A procedure for determining the exact required V_L,I, V_L,Q, H_eq,I(f), and H_eq,Q(f) based on observable and measurable parameters can still be considered problematic. A solution to such an issue can be facilitated by procedures and techniques that allow independent measurement of errors due to LO feed-through and quadrature channel imbalance.

[0174] Separating Problem Signals from the Desired Signal

[0175] Techniques are herein presented to separate the desired signal from the undesired problem signals in a manner that allows spectral filtering to select signal components of interest. As in the previous section, one can begin with techniques for mitigating LO feed-through and follow with I/Q imbalance.

[0176] Separating LO Feed-Through in Frequency

[0177] A technique is now presented herein that separates desired signal components from undesired LO feed-through signals.

[0178] Conventional techniques tend to treat QDWS output signals as occupying a significant majority of the spectrum between −f_s/2 and f_s/2. This is illustrated graphically in Step A of FIG. 9. The baseband signal is then filtered and applied to the SSB mixer as shown at Step B. The problem now is that the LO feed-through is equivalent to a DC signal added to QDWS output. As such, it exists in the center of the output signal spectrum as shown in Step C.

[0179] Notch-filtering the LO feed-through at the output of the SSB mixer would filter a portion of the desired signal, thereby degrading the desired signal, and ultimately the radar data collected by the receiver. This process and result is generally undesirable. A frequency bias, however, can be added to the desired signal during its generation by forcing its spectrum to lie between 0 and f_s, as is illustrated at Step A in FIG. 10. This would allow frequency separation between the desired spectrum and the LO feed-through equivalent, thereby offering an opportunity to filter the problematic signal without adversely affecting the desired signal. A principal difficulty with this approach, however, is that the reconstruction filter as proposed in FIG. 10 is one-sided, that is, it exists for positive frequencies only as shown at Step C, and consequently is unrealizable. This results in a slightly more elaborate scheme being required.

[0180] FIG. 11 depicts an improved scheme for separating LO feed-through from the desired signal. In this scheme, all filters are real-valued. A signal is generated with its spectrum between 0 and f_s. It is initially filtered by a realizable reconstruction filter 1110 and then modulated upwards in frequency to some intermediate carrier frequency. The extra sideband can be filtered using a sideband filter 1120, leaving the desired sideband. This is considered by the inventors to be a more complicated procedure.

[0181] A remaining concern is whether the sideband filter 1120 can suitably reject the LO feed-through and lower-sideband duplicate chirp. Nevertheless, spectral separation of the desired signal from the LO feed-through can be achieved.

[0182] Another problem with this approach stems from the model presumed for the QDWS output spectrum. As is well known, sampled signals have their spectrum perfectly replicated around integer multiples of the sampling frequency f_s, over all frequencies. Perfect replication over all frequencies, however, presumes that a train of ideal impulses each with area proportionate to the instantaneous signal value describes the samples. Mathematically this is described by: $x_{\mathrm{QDWS}}\left(t\right)=\sum _n{x}_{\mathrm{QDWS},\mathrm{ideal}}\left(t\right)\delta \left(t-n\text{/}{f}_s\right)$

[0183] where x_QDWS,ideal(t) is the desired signal to be reconstructed, and δ(t) is a unit-area impulse function.

[0184] Real Digital to Analog Converter (DAC) components, however, do not generate ideal impulses, but rather generate voltage values that are held until the next sample is clocked in. This constitutes an implicit filtering operation. Mathematically this is described more accurately by the convolution: $x_{\mathrm{QDWS}}\left(t\right)=\left(\sum _n{x}_{\mathrm{QDWS},\mathrm{ideal}}\left(t\right)\delta \left(t-n\text{/}{f}_s\right)\right)*\mathrm{rect}\left(f_{s}t\right).$

[0185] This, of course, implies that the spectrum is weighted (multiplied) by a sinc function of the form: $\mathrm{Spectral}\mathrm{weighting}=\frac{1}{f_s}\mathrm{sinc}\left(\frac{f}{f_s}\right)$

[0186] where: $\mathrm{sinc}\left(f\right)=\frac{\sin \left(\pi f\right)}{\pi f}.$

[0187] FIG. 12 illustrates how the sinc weighting might affect QDWS outputs. Note that for f=0.9 f_sthe signal attenuation by the implicit filtering is almost 20 dB. The remedy for this would be to equalize what amounts to an effective sinc function filter shape that is applied to the QDWS output. In a manner similar to the QECDWS scheme herein described, amplitude pre-distortions vs. frequency might be made during waveform synthesis such that the desired portion of the reconstructed output of the QDWS exhibits lesser (and preferably no) rolloff with frequency. Some rolloff can in fact be tolerated in LFM chirp waveforms, as downstream components, especially in the multiplier circuits, which will tend to flatten the response anyway.

[0188] Separating LO Feed-Through in Doppler

[0189] The detailed description up to now has presumed that one wants to correct the signal for each and every pulse, independent of other pulses. Radar systems, however, typically process multiple pulses to arrive at a final signal-processing product. For example, a typical high-performance Synthetic Aperture Radar (SAR) system might process several tens of thousands of pulses to form a single image. Combining returns from multiple pulses in a coherent fashion is termed Doppler processing. An adequate presumption for subsequent discussion is that a Doppler frequency shift manifests itself primarily as a pulse-to-pulse phase shift of a signal. The Doppler characteristics of a radar echo can be manipulated by artificially applying pulse-to-pulse phase shifts in some desirable manner. The LO feed-through can be separated from the desired signal in its Doppler characteristics.

[0190] Consider a receiver video signal model exhibiting only LO feed-through and no I/Q imbalance, that is: $x_V\left(t\right)=\left[\begin{array}{c}\exp j(M{\omega }_0\left(t_m-t_s\right)+M(\Phi \left(t-t_s\right)-\\ \Phi \left(t-t_m\right))+M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right))\times \\ \left(\begin{array}{c}1+\\ {\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\right)+\\ {\mathrm{MA}}_{L,s}\exp -j\left(\Phi \left(t-t_s\right)+{\theta }_{\mathrm{i1},s}-{\theta }_{\mathrm{g0},s}-{\varphi }_{L,s}\right)\end{array}\right)\end{array}\right].$

[0191] A pulse-to-pulse phase shift of π radians can now be added per pulse, such that Φ(t)→(Φ(t)+nπ) is replaced where n is the pulse number index. This effectively alters the reference phase between values of 0 and π on a pulse-to-pulse basis. By noting that exp jnπ=exp−jnπ the video signal then becomes: $x_V\left(t\right)=\left[\begin{array}{c}\exp j(M{\omega }_0\left(t_m-t_s\right)+M(\Phi \left(t-t_s\right)-\\ \Phi \left(t-t_m\right))+M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right))\times \\ \left(1+\exp j\left(n\pi \right)\left(\begin{array}{c}{\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\right)+\\ {\mathrm{MA}}_{L,s}\exp -j\left(\Phi \left(t-t_s\right)+{\theta }_{\mathrm{i1},s}-{\theta }_{\mathrm{g0},s}-{\varphi }_{L,s}\right)\end{array}\right)\right)\end{array}\right].$

[0192] The result is that the LO feed-through term is modulated by exp jnπ, but the desired energy is not. That is, the problematic LO feed-through signal has been separated in Doppler from the desired signal.

[0193] FIG. 13 illustrates how, if the desired signal 1310 is Doppler band-limited and centered around a zero Doppler frequency, and if the radar Pulse Repetition Frequency (PRF) 1320 is defined as f_p, then the problematic LO feed-through signal will also be Doppler band-limited but will instead be centered around f_p/2. Doppler filtering can then quite effectively mitigate the unwanted energy.

[0194] The PRF should be high enough so that the LO feed-through Doppler spectrum 1330 does not overlap that portion of the desired signal Doppler spectrum of interest. If the Doppler spectrum is limited to a bandwidth B_Doppler, then it can be guaranteed by insisting f_p≧2B_Doppler.

[0195] Referring to FIG. 14, the requisite 0/π phase modulation can be accomplished by inserting the appropriate 0 or π radian phase shift into the output of the QDWS 1410, prior to the DACs 1420, while the signal is still just data. It should be noted that a π radian phase shift is just multiplication by −1. Consequently, exp j(nπ)=(−1)ⁿ.

[0196] An important point that should not be lost is that a particular index n value is for a transmitted (TX) pulse and demodulating receiver (RX) pulse pair. Any pulse formed to assist RX demodulation needs to remain coherent with (i.e. the same pulse-to-pulse phase relationship to) the corresponding TX signal. That is, n increments on TX only, and remains unchanged for any RX pulses generated.

[0197] Separating I/Q Imbalance Energy

[0198] A separating technique can be used to also mitigate I/Q imbalance. Simple 0/π modulation, however, is inadequate for this task. Nevertheless, the principle that appropriate phase shifts in the QDWS on a pulse- to-pulse basis can separate problem signals from the desired signal in Doppler is still sound.

[0199] Consider again the video signal, but now including the effects of I/Q imbalance as well as LO feed-through: $x_V\left(t\right)=\left[\begin{array}{c}\exp j\left(M{\omega }_0\left(t_m-t_s\right)+M\left(\Phi \left(t-t_s\right)-\Phi \left(t-t_m\right)\right)+M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)\right)\\ x\left(\begin{array}{c}1+\\ {\mathrm{MA}}_{I,m}\exp j\left(2\Phi \left(t-t_m\right)+2{\theta }_{\mathrm{i1},m}+{\varphi }_{I,m}\right)+\\ {\mathrm{MA}}_{I,s}\exp -j\left(2\Phi \left(t-t_s\right)+2{\theta }_{\mathrm{i1},s}+{\varphi }_{I,s}\right)+\\ {\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\right)+\\ {\mathrm{MA}}_{L,s}\exp -j\left(\Phi \left(t-t_s\right)+{\theta }_{\mathrm{i1},s}-{\theta }_{\mathrm{g0},s}-{\varphi }_{L,s}\right)\end{array}\right)\end{array}\right].$

[0200] Now add a pulse-to-pulse phase shift of some fixed amount, such that Φ(t)→(Φ(t)+nΔφ) is replaced where Δφ is the pulse-to-pulse phase shift, and n is the pulse index number (incrementing by one with each new transmitted pulse). Consequently, x_QDWS(t) contains a rolling phase shift that rolls at a rate Δφ radians per pulse. The video signal then becomes: $x_V\left(t\right)=\left[\begin{array}{c}\exp j\left(M{\omega }_0\left(t_m-t_s\right)+M\left(\Phi \left(t-t_s\right)-\Phi \left(t-t_m\right)\right)+M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)\right)\\ x\left(\begin{array}{c}1+\\ \exp j\left(2n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{I,m}\exp j\left(2\Phi \left(t-t_m\right)+2{\theta }_{\mathrm{i1},m}+{\varphi }_{I,m}\right)\right]+\\ \exp -j\left(2n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{I,s}\exp -j\left(2\Phi \left(t-t_s\right)+2{\theta }_{\mathrm{i1},s}+{\varphi }_{I,s}\right)\right]+\\ \exp j\left(n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\right)\right]+\\ \exp -j\left(n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{L,s}\exp -j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{il},s}-{\theta }_{\mathrm{g0},s}-{\varphi }_{L,s}\right)\right]\end{array}\right)\end{array}\right].$

[0201] A notional video Doppler spectrum is illustrated in FIG. 15. As with the 0/π modulation described earlier, problem terms are modulated in Doppler, whereas the desired signal 1510 remains unaffected. Note also that the LO feed-through spectrum 1530 and the I/Q imbalance spectrum 1520 are separated from each other.

[0202] A single point scatterer would form a range-Doppler energy distribution (i.e. SAR image) as illustrated in FIG. 16. Note how the LO feed-through energy and I/Q imbalance energy in a focussed SAR image manifest themselves as a chirp spectrum appropriately offset in Doppler from the desired impulse 1610 at the image center. While the present inventors propose that this separation can be exploited as is, they also have noted that this separation can assist in making independent measurements of the various undesired signal components to facilitate compensation as described earlier. Nevertheless, if the desired signal is really a set of signals that span a Doppler bandwidth of B_Doppler, then the LO feed-through and I/Q imbalance signals will each also contain Doppler bandwidths of B_Doppler, but centered at their respective Doppler offsets. If no contamination of the entire Doppler bandwidth associated with the desired signal is insisted on, then a minimum imposed value on the pulse-to-pulse phase shift will be: $\mathrm{\Delta \phi }\ge 2\pi \left(\frac{B_{\mathrm{Doppler}}}{f_p}\right).$

[0203] Furthermore, to keep the Doppler bandwidths of aliased undesired signals from contaminating any part of the desired signal's Doppler spectrum, require: $\mathrm{\Delta \phi }\le \pi \left(1-\frac{B_{\mathrm{Doppler}}}{f_p}\right).$

[0204] These combine to form the restriction f_p≧3B_Doppler.

[0205] If, however, one is ultimately interested in merely a lesser portion of the desired signal's available Doppler spectrum, then some encroachment can be tolerated, and the restriction f_p>3B_Doppler, can be relaxed.

[0206] FIG. 17 illustrates a system 1700 wherein Doppler modulation can be implemented within a QDWS (ignoring any amplitude corrections). A particularly interesting configuration is for Δφ=π/2, which requires f_p≧4B_Doppler. The following table describes the QDWS outputs for this case (neglecting amplitude modulations).

[0207] QDWS outputs for Δφ=π/2: $\begin{array}{ccc}\left(n\Delta \phi \right)& x_{\mathrm{QDWS},I}\left(t\right)& x_{\mathrm{QDWS},Q}\left(t\right)\\ 0& \cos \Phi \left(t\right)& \sin \Phi \left(t\right)\\ \pi \text{/}2& -\sin \Phi \left(t\right)& \cos \Phi \left(t\right)\\ \pi & -\cos \Phi \left(t\right)& -\sin \Phi \left(t\right)\\ 3\pi \text{/}2& \sin \Phi \left(t\right)& -\cos \Phi \left(t\right)\end{array}$

[0208] As indicated, the phase shifts can be implemented as merely transpositions and negations of the I and Q channel outputs. This might be accomplished with suitable switch combinations as illustrated for the QDWS 1810 illustrated in FIG. 18.

[0209] In any case, employing rolling phase shifts on a pulse-to-pulse basis allows Doppler separation in the receiver of the desired signal from those undesired and due to LO feed-through and quadrature imbalance. Doppler filtering can then mitigate the effects of LO feed-through and quadrature imbalance, without the need for any calibration in the transmitter.

[0210] Parameter Estimation for Equalization

[0211] Some occasions might nevertheless exist where Doppler filtering is not an available option, perhaps because an adequate radar PRF cannot be employed. In this situation, balancing and equalization as previously described are still available to facilitate clean signals. No calibration scheme, however, has yet been identified to accomplish balance and equalization.

[0212] The present inventors provide estimation, with some precision, of the parametric values, of LO feed-through and I/Q imbalance (or their equivalent) using measurements of the video signal. Values so derived could then be used to compensate the LO feed-through and equalize the I/Q imbalance in a manner previously discussed. A system can be placed in a calibration or test mode to facilitate these measurements and techniques, and that many pulses can be integrated to achieve good Signal to Noise ratios. A test signal can also be employed with a delay-line to avoid any Doppler spreading of the echo signal.

[0213] To unambiguously measure a parameter, it must first be isolated from other obscuring or otherwise confusing factors. To separate the desired signal from the LO feed-through, and these from the I/Q imbalance energy, Doppler separation can be employed as previously discussed. That is, QDWS outputs employing a rolling pulse-to-pulse phase shift can be used. The video signal model then becomes: $x_V\left(t\right)=\left[\begin{array}{c}\exp j\left({\mathrm{M\omega }}_0\left(t_m-t_s\right)+M\left(\Phi \left(t-t_s\right)-\Phi \left(t-t_m\right)\right)+M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)\right)\\ x\left(\begin{array}{c}1+\\ \exp j\left(2n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{I,m}\exp j\left(2\Phi \left(t-t_m\right)+2{\theta }_{\mathrm{i1},m}+{\varphi }_{I,m}\right)\right]+\\ \exp -j\left(2n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{I,s}\exp -j\left(2\Phi \left(t-t_s\right)+2{\theta }_{\mathrm{i1},s}+{\varphi }_{I,s}\right)\right]+\\ \exp j\left(n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\right)\right]+\\ \exp -j\left(n\mathrm{\Delta \phi }\right)\left[{\mathrm{MA}}_{L,s}\exp -j\left(\Phi \left(t-t_s\right)+{\theta }_{\mathrm{i1},s}-{\theta }_{\mathrm{g0},s}-{\varphi }_{L,s}\right)\right]\end{array}\right)\end{array}\right].$

[0214] A pulse-to-pulse phase shift can be chosen to allow clear separation and identification of the LO feed-through from the I/Q imbalance energy. A reasonable value might be Δφ=π/4. This can cause LO feed-through energy to be centered at Doppler frequencies ±f_p/8, and I/Q imbalance energy to be centered at Doppler frequencies ±f_p/4, thereby separating them from the desired signal as well as any aliased energy.

[0215] Recall that the following was earlier defined: $A_I\exp -j{\varphi }_I=-\frac{1}{2}\left(\frac{\Delta k-j\Delta \theta }{1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}}\right),\mathrm{and}$ $A_L\exp j{\varphi }_L=\left(\frac{k_{\mathrm{g0}}}{1+k_{\mathrm{i1}}+j{\theta }_{\mathrm{q0}}}\right).$

[0216] where Δk, Δθ, and k_i1are frequency dependent. That is, if Φ(t)=ω_Tt, then these depend on ω_T. Note that θ_q0in the present model is a phase shift of a constant LO frequency signal, and consequently is not normally expected to depend on ω_T, but yet might.

[0217] LO Feed-Through Parameter Estimation

[0218] With appropriate Doppler processing, a Doppler component exp j(nΔφ) can be extracted uniquely due to the LO feed-through signal, and can be identified as: ${x_v\left(t\right)|}_{\mathrm{\Delta \phi }}=\left[\begin{array}{c}\exp j(M{\omega }_0\left(t_m-t_s\right)+M\left(\Phi \left(t-t_s\right)-\Phi \left(t-t_m\right)\right)+\\ M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right))\times {\mathrm{MA}}_{L,m}\exp j\left(\Phi \left(t-t_m\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\right)\end{array}\right].$

[0219] A convenient QDWS modulation signal can be chosen (e.g., a sinusoid) where:

[0220] Φ(t)=ω_Tt=2πf_Tt,

[0221] where f_Tis less than the video cutoff frequency, that is |f_T|<B_V.

[0222] Then a video signal can be expanded and modeled by: ${x_v\left(t\right)|}_{\mathrm{\Delta \phi }}={\mathrm{MA}}_{L,m}\exp j\left(\begin{array}{c}2\pi f_T\left(t-t_m\right)+M\left({\omega }_0+2\pi f_T\right)\left(t_m-t_s\right)+\\ M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\end{array}\right).$

[0223] Performing a Fourier Transform over the envelope finite interval −T/2≦(t−t_m)≦T/2 yields: ${X_V\left(f\right)|}_{\Delta \phi }={\mathrm{TMA}}_{L,m}\exp j\left(\begin{array}{c}M\left({\omega }_0+2\pi f_T\right)\left(t_m-t_s\right)+\\ M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\end{array}\right)\sin c\left(\frac{f-f_T}{T}\right).$

[0224] In particular, at the spectral peak, we identify: ${X_V\left(f_T\right)|}_{\Delta \phi }={\mathrm{TMA}}_{L,m}\exp j\left(\begin{array}{c}M\left({\omega }_0+2\pi f_T\right)\left(t_m-t_s\right)+\\ M\left({\varphi }_{0,s}-{\varphi }_{0,m}\right)+{\theta }_{\mathrm{i1},m}-{\theta }_{\mathrm{g0},m}-{\varphi }_{L,m}\end{array}\right)$

[0225] which has magnitude: ${\left|X_V\left(f_T\right)\right|}_{\mathrm{\Delta \phi }}|={\mathrm{TMA}}_{L,m}=\frac{{\mathrm{TMk}}_{\mathrm{g0}}}{\sqrt{{\left(1+k_{\mathrm{i1}}\right)}^2+{\left({\theta }_{\mathrm{q0}}\right)}^2}}.$

[0226] The known parameters are T, M, ω₀, f_T, t_m, t_s, and of course X_V(f_T). The unknown parameters are k_g0, k_i1, θ_i1, θ_q0, and θ_g0. The best we can do here is ascertain the quantities k_g0/√{square root over ((1+k_i1)²+(θ_q0)²)}{square root over ((1+k_i1)²+(θ_q0)²)} and └M(φ_0,s−φ_0,m)+θ_i1m−θ_g0,m−φ_L,m┘ which still leaves some ambiguity in the specific parameters whose value is desired to be known. Identifying precisely k_g0, and θ_g0is problematic given that they are perturbed by unknown values of k_i1and θ_i1, among others.

[0227] Rather than identify these parameters specifically, it should be recognized given the frequency that the aim of calibration is to ascertain the DC offsets to minimize the offending components in the received signal.

[0228] Equivalent SSB Mixer DC Offset Estimation

[0229] Reexamining the earlier discussion about compensating LO feed-through with a DC offset into the SSB mixer, it should be recognized that the mixer output with no QDWS input signal is given by (ignoring I/Q imbalance and LO phase shift term θ_q0for the moment): $x_{\mathrm{Mixer}}\left(t\right)=\frac{1}{2}\left(k_{\mathrm{g0}}\cos \left({\theta }_{\mathrm{g0}}\right)+j{k}_{\mathrm{g0}}\sin \left({\theta }_{\mathrm{g0}}\right)\right)\exp j\left({\omega }_{0}t\right)$

[0230] with magnitude k_g0/2. Even small values for θ_q0won't change this much.

[0231] On the other hand, with DC offsets applied to the SSB mixer input, as previously discussed, the output is described by: $x_{\mathrm{Mixer}}\left(t\right)=\frac{1}{2}\left(\begin{array}{c}\lfloor V_{L,I}{H}_I\left(0\right)+k_{\mathrm{g0}}\cos \left({\theta }_{\mathrm{g0}}\right)-V_{L,Q}{H}_Q\left(0\right)\sin \left({\theta }_{\mathrm{q0}}\right)\rfloor +\\ j\left[V_{L,Q}{H}_Q\left(0\right)\cos \left({\theta }_{\mathrm{q0}}\right)+k_{\mathrm{g0}}\sin \left({\theta }_{\mathrm{g0}}\right)\right]\end{array}\right)\exp j\left({\omega }_{0}t\right).$

[0232] By now defining: $k_{\mathrm{g0},\mathrm{eq}}=\sqrt{\begin{array}{c}{\left(V_{L,I}{H}_I\left(0\right)+k_{\mathrm{g0}}\cos \left({\theta }_{\mathrm{g0}}\right)-V_{L,Q}{H}_Q\left(0\right)\sin \left({\theta }_{\mathrm{q0}}\right)\right)}^2+\\ {\left(V_{L,Q}{H}_Q\left(0\right)\cos \left({\theta }_{\mathrm{q0}}\right)+k_{\mathrm{g0}}\sin \left({\theta }_{\mathrm{g0}}\right)\right)}^2\end{array}},\mathrm{and}$ $A_{L,m,\mathrm{eq}}=\frac{k_{\mathrm{g0},\mathrm{eq}}}{\sqrt{{\left(1+k_{\mathrm{i1}}\right)}^2+{\left({\theta }_{\mathrm{q0}}\right)}^2}}$

[0233] the measured video signal spectral peak, when a DC offset is applied to the SSB mixer input, becomes: ${\left|X_V\left(f_T\right)\right|}_{\mathrm{\Delta \phi }}|={\mathrm{TMA}}_{L,m,\mathrm{eq}}=\frac{{\mathrm{TMk}}_{\mathrm{g0},\mathrm{eq}}}{\sqrt{{\left(1+k_{\mathrm{i1}}\right)}^2+{\left({\theta }_{\mathrm{q0}}\right)}^2}}.$

[0234] The task at hand is now to find V_L,Iand V_L,Qto force as nearly as possible k_g0,eq=0, and thereby X_V(f_T)|_Δφ=0. Clearly, these DC offsets also encompass the effects of other relevant factors influencing LO feed-through, such as H_I(0) and H_Q(0). In any case, the essential task is to find appropriate parameters V_L,Iand V_L,Qto nullify LO feed-through.

[0235] Of interest should be the fact that if θ_q0=0, then V_L,Iand V_L,Qcan be independently minimized to achieve an overall solution. That is, the value for V_L,Ithat minimizes k_g0,eqdoes so regardless of V_L,Q, and likewise the value for V_L,Qthat minimizes k_g0,eqdoes so regardless of V_L,I. However, generally θ_q0≠0, so their values do influence each other somewhat, although probably not very much since θ_q0is typically expected to be small.

[0236] Several possible techniques to find optimal V_L,Iand V_L,Qare presented herein.

[0237] Technique 1

[0238] The first procedure amounts to guessing about new solutions and keeping the better solution between the new and the old.

[0239] 1) Guess an initial value for V_L,Iand V_L,Q, and calculate |X_V(f_T)|_Δφ|.

[0240] 2) Guess another value for V_L,Iand recalculate |X_V(f_T)|_Δφ|.

[0241] 3) Remember the value for V_L,Ithat gave the minimum |X^V(f_T)|_Δφ|.