Gain and phase imbalance compensation for OFDM systems

[0001] This application is related to the commonly owned, concurrently filed U.S. patent application Ser. No. 10/259,108, entitled “Cubic Spline Predistortion, Algorithm and Training, for a Wireless LAN System” by the same inventor. The related application is hereby incorporated by reference.

[0002] The present invention includes methods and devices to compensate for gain and phase imbalance for OFDM and other multi-carrier symbol transmission systems. More particularly, methods and devices for determining compensation parameters are provided. This invention may be applied to a variety of standards utilizing OFDM technology, including IEEE 802.11a, Hiperlan/2 and MMAC.

[0003] Orthogonal frequency division multiplexing (OFDM) is a spectrally efficient modulation scheme with application in both wired and wireless communications. It is applied in existing wireless systems including IEEE 802.11a and 802.11g and is proposed for several next-generation wireless systems including IEEE 802.16 and 4th generation cellular. Because OFDM has high spectral efficiency, it is more susceptible to radio impairments. One such impairment, and the topic of this study, is gain and phase imbalance (also known as IQ offset).

[0004] Gain and phase imbalance are introduced at both the transmit and receive radios due to the typical variations in mixer components. The D/A and A/D converters may also introduce some gain imbalance; however, this imbalance can be treated as combined with that of the mixer. Gain imbalance occurs whenever the gains in the in-phase and quadrature paths are not identical and phase imbalance occurs whenever the phase separation of the in-phase and quadrature paths is not exactly 90 degrees. Typical gain and phase imbalances of analog mixers are on the order of 1 dB and 5 degrees, respectively. The effect of imbalance is a loss of orthogonality between the in-phase and quadrature paths. This loss of orthogonality can seriously degrade the link performance of an OFDM system, particularly when larger constellations are transmitted.

[0005] A major challenge in OFDM transceiver design is ensuring gain and phase balance at the transmitter and the receiver. Standards for OFDM transmission typically lack specific requirements for gain and phase balance, relying instead on performance standards that necessitate good balance. Typically, designers focus on improving the mixer section of the system, to avoid imbalance. This comes at the cost of a relatively expensive mixer section.

[0006] Accordingly, an opportunity arises to develop gain and imbalance compensation method and apparatus that improves system performance.

[0007] The present invention includes methods and devices to compensate for gain and phase imbalance for OFDM and other multi-carrier symbol transmission systems. More particularly, methods and devices for determining compensation parameters are provided. This invention may be applied to a variety of standards utilizing OFDM technology, including IEEE 802.11a, Hiperlan/2 and MMAC. Further description of the invention and its embodiments are found in the figures, specification and claims that follow.

[0012] The following detailed description is made with reference to the figures. Preferred embodiments are described to illustrate the present invention, not to limit its scope, which is defined by the claims. Those of ordinary skill in the art will recognize a variety of equivalent variations on the description that follows.

[0013] This describes an OFDM symbol transmission and receiver model, particularly an 802.11a system model, and analyzes the effects of gain and phase imbalance. A method is proposed for measuring the imbalance at a receiver assuming OFDM modulation. Once the imbalance is measured, it is compensated for. Optionally, compensation can be performed at both the transmitter and receiver. Transmitter based compensation pre-compensates for imbalance introduced at the transmit mixer. Two receiver based compensation algorithms are defined: one corrects for imbalance introduced at the receiver and the other compensates for imbalance introduced at the far-end transmitter. Although the primary goal is improving link performance, imbalance compensation will increase production yields by reducing the transceiver's sensitivity to mixer imbalance.

[0014] An alternate solution (besides specifying a high performance, and thus more expensive mixer) is to perform mixing in the digital domain. The disadvantage of digital mixing is the high sampling rate D/A's and A/D's that would be required (potentially equaling the costs of more expensive mixers). Furthermore, even if a digital mixer is used at the receiver, there is no guarantee that the far-end transmit mixer will be digital (or even a well balanced analog mixer).

[0015] Many symbols are used in this application, so their meanings are set out here for reference.

• • [0016] A/D Analog to Digital Converter
  • [0017] D/A Digital to Analog Converter
  • [0018] dB Decibel
  • [0019] FFT Fast Fourier Transform
  • [0020] IEEE Institute of Electrical and Electronic Engineers
  • [0021] IFFT Inverse Fast Fourier Transform
  • [0022] IQ In-phase/Quadrature
  • [0023] ISI Inter-Symbol Interference
  • [0024] LAN Local Area Network
  • [0025] LNA Low Noise Amplifier
  • [0026] OFDM Orthogonal Frequency Division Multiplexing
  • [0027] PA Power Amplifier
  • [0028] RX Receiver
  • [0029] TX Transmitter

[0030] FIG. 1 illustrates a generic OFDM transceiver model used in this study. It is not comprehensive, but rather highlights those system components required to analyze the effects of imbalance and the corresponding compensation schemes. Imbalance compensation is shown in both the transmit and receive chains for both frequency domain and time domain approaches. In practice, the transmitter typically will have either time or frequency domain TX imbalance compensation and the receiver will have frequency domain TX imbalance compensation and either time or frequency domain RX imbalance compensation. OFDM transceiver standards typically do not mention or mandate imbalance compensation. Typical performance requirements make it desirable to use either well-balanced mixers or imbalance compensation to ensure compliance.

[0031] The FIG. 1 model incorporates all the functionality of the 802.11a transmitter and receiver at the OFDM symbol level and below. It does not incorporate the 802.11a framing, scrambling, or convolutional encoding/interleaving, which were not necessary for this evaluation. The radio model consists of digital-to-analog converters (D/A's) 109, 111, a mixer 113, 115, 117 and a PA 117. A predistorter may be inserted between the OFDM transmitter and radio, as part of a time domain compensator 101 or a frequency domain compensator 107 to pre-compensate for non-linearities introduced by the PA. The predistorter is not mandated (or mentioned) in the 802.11a standard. IEEE Std 802.11a Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High-speed Physical Layer in the 5 GHz Band, 1999. Reconstruction filters between the D/A's and the PA are assumed to have negligible effects and thus are ignored in the analysis. Those of skill in the art will recognize that additional steps may be implemented to address ripple and delay introduced by reconstruction filters. The time domain 101 and frequency domain 107 compensators also may be adapted to compensate for gain and phase imbalances.

[0032] OFDM symbols typically include 48 data carriers and 4 pilot carriers. The data carriers are modulated by a set of data symbols whose values for the purpose of this evaluation are independent, uniformly distributed random variables over the range of 0, 1, . . . , 2^Nbpsc−1 where N_bpsc represents the number of bits encoded onto each subcarrier. N_bpsc may equal to 1, 2, 4 and 6 bits corresponding to gray-coded BSPK, QPSK, 16-QAM and 64-QAM constellations, respectively. The QAM Mapper maps each data symbol to its corresponding complex value. The pilot carriers are encoded as BPSK symbols with pseudo-randomly alternating phases in accordance with 802.11a.

[0033] The data and pilot carriers are combined into d⁽¹⁾_n, n=0, . . . , 63 complex points, and translated to the time domain by an N=64-point complex IFFT 103. Each block of N samples output from the IFFT forms a single OFDM symbol, which is then cyclic extended (105) with N_cp=16 samples and windowed 105. Although 802.11a specifies a raised cosine window of arbitrary length w_T (where w_T represents the number of samples overlap between adjacent OFDM symbols), its use is optional. For this analysis, windowing with w_T=1 was used to meet the PSD requirement of 802.11a. Oversampling of the transmit signal can be performed either in the time-domain using standard interpolation techniques or in the frequency domain by performing an osr×N-point IFFT on a zero padded input, where osr represents the oversampling rate. For this analysis, oversampling is performed in the frequency domain and N_cp and w_T are correspondingly scaled by osr.

[0034] The two D/A's convert the in-phase 107 and quadrature 109 components of the digital baseband signal 113, 115 to the analog domain for input to the mixer. The mixer 117 is modeled as a direct-to-RF mixer, for this analysis. A power amplifier 119 boosts the signal for transmission through a channel 121, which typically injects noise 123.

[0035] On the receiver side, the model components generally correspond to the transmitter components. A low noise amplifier (LNA) 149 is used to boost the received signal for processing. The in phase 143 and 145 quadrature components of the signal are reconstructed. A pair of A/D's convert the components from the analog domain to the digital domain. A receiver imbalance compensator 137 may be applied to correct for gain and phase imbalance, in the time domain. A sync component 135 processes the signal before a Fast Fourier Transform 133 is applied to convert the signal from the time to frequency domain. An equalizer 132 uses one or more reference carriers to equalize the received signal. Additional compensation may be applied in the frequency domain 131.

[0036] In this analysis, the following simplifying assumptions are made: The duration of the channel impulse response is shorter than the duration of the cyclic extension minus the duration of the windowing overlap. This allows the cyclic extension and any windowing operations to be omitted from this analysis. And, the carrier frequency offset between frequencies of the transmitter and receiver is assumed to be negligible. Those of skill in the art will recognize that the equations presented here may be generalized to include such offsets.

[0037] End-to-end frequency-domain models describing transmitter and receiver imbalance are derived here for both no channel and multipath channel cases. These frequency domain models relate the received complex constellation points to the transmitted points. From this, methods of measuring and compensating for imbalance are developed.

[0038] This simplified model considers the impact of transmitter and receiver imbalance when there is no channel. This exercise is useful for characterizing the impact of imbalance (without other impairments), as well as for determining how to compensate for TX imbalance at the transmitter since TX compensation at the transmitter is independent of the channel.

[0039] Consider the set of N complex constellation points a_k=a^R_k+ja^I_k, k=0, 1, . . . , N−1 which are transmitted as a single OFDM symbol. These are converted from the frequency domain to the time domain using an N-point IFFT as follows

[0040] $\begin{array}{c}x\left(n\right)=\frac{1}{N}\sum _{k=0}^{N-1}\left(a_k^R+j{a}_k^I\right)\left[\cos \left(2\pi \mathrm{nk}/N\right)+j\sin \left(2\pi \mathrm{nk}/N\right)\right]\\ =\frac{1}{N}\sum _{k=0}^{N-1}\left[a_k^R\cos \left(2\pi \mathrm{nk}/N\right)-a_k^I\sin \left(2\pi \mathrm{nk}/N\right)\right]+\\ j\frac{1}{N}\sum _{k=0}^{N-1}\left[a_k^R\sin \left(2\pi \mathrm{nk}/N\right)+a_k^I\cos \left(2\pi \mathrm{nk}/N\right)\right]\end{array}$
for n=0, 1, . . . , N−1. The IFFT output, x(n), is passed through the transmitter's TX I/Q Imbalance Compensator to obtain y(n). For the development of this model, the imbalance compensator is disabled such that y(n)=x(n). The in-phase and quadrature components of y(n) are each passed through a D/A converter and then mixed to some carrier frequency f_c to obtain the real valued RF signal
z(t)=y_I(t)cos(2πf_ct)−y_Q(t)G^tx_msin(2πf_ct+θ^tx_m)
where G^tx_m and θ^tx_m represent the gain and phase imbalance of the transmit mixer, D/A's and amplifiers referenced to the in-phase component. Assuming no channel impairments, the receiver mixes the received RF signal back to complex baseband (filtering out the image frequencies and scaling by a factor of 2)

[0041] $\begin{array}{c}{\hat{y}}_I\left(t\right)=2{\left\{\left[y_I\left(t\right)\cos \left(2\pi f_{c}t\right)-y_Q\left(t\right)G_m^{\mathrm{tx}}\sin \left(2\pi f_{c}t+{\theta }_m^{\mathrm{tx}}\right)\right]\cos \left(2\pi f_{c}t\right)\right\}}_{\mathrm{LPF}}\\ =y_I\left(t\right)-y_Q\left(t\right)G_m^{\mathrm{tx}}\sin \left({\theta }_m^{\mathrm{tx}}\right)\\ {\hat{y}}_Q\left(t\right)=2\{-\left[y_I\left(t\right)\cos \left(2\pi f_{c}t\right)-y_Q\left(t\right)G_m^{\mathrm{tx}}\sin \left(2\pi f_{c}t+{\theta }_m^{\mathrm{tx}}\right)\right]G_m^{\mathrm{rx}}\sin (2\pi f_{c}t+\\ {{\theta }_m^{\mathrm{rx}})\}}_{\mathrm{LPF}}\\ =G_m^{\mathrm{rx}}\left[y_Q\left(t\right)G_m^{\mathrm{tx}}\cos \left({\theta }_m^{\mathrm{tx}}-0_m^{\mathrm{rx}}\right)-y_I\left(t\right)\sin \left({\theta }_m^{\mathrm{rx}}\right)\right]\end{array}$
where G^rx_m and θ^rx_m represent the combined gain and phase imbalance of the receive mixer, A/D's and amplifiers, again referenced to the in-phase component. Converting to the digital domain and rewriting the received baseband signal in complex notation
ŷ(n)=y_I(n)[1−jG^rx_msin(θ^rx_m)]+jy_Q(n)[G^rx_mG^tx_mcos(θ^tx_m−θ^rx_m)+jG^tx_msin(θ^tx_m)].

[0042] Defined two time-invariant intermediary values
α=1−jG^rx_msin(θ^rx_m)
β=G^rx_mG^tx_mcos(θ^tx_m−θ^rx_m)+jG^tx_msin(θ^tx_m)
such that
ŷ(n)=αy_I(n)+jβy_Q(n).
Ignoring the effects of the receiver's time-domain I/Q imbalance compensator, {circumflex over (x)}(n)=ŷ(n). The reference to whether this compensates for transmitter or receiver imbalance is intentionally omitted because, for the case of no channel, both transmitter and receiver imbalance can be compensated for in the time domain. The received set of constellation points is decoded using the FFT as follows

[0043] $\begin{array}{c}{\hat{a}}_k=\sum _{n=0}^{N-1}\left[\alpha y_I\left(n\right)+j\beta y_Q\left(n\right)\right]\left[\cos \left(2\pi \mathrm{nk}/N\right)-j\sin \left(2\pi \mathrm{nk}/N\right)\right]\\ =\frac{1}{N}\sum _{n=0}^{N-1}\left[\begin{array}{c}\alpha \sum _{k^{\prime }=0}^{N-1}\left[a_{k^{\prime }}^R\cos \left(2\pi {\mathrm{nk}}^{\prime }/N\right)-a_{k^{\prime }}^I\sin \left(2\pi {\mathrm{nk}}^{\prime }/N\right)\right]+\\ j\beta \sum _{k^{\prime }=0}^{N-1}\left[a_{k^{\prime }}^R\sin \left(2\pi {\mathrm{nk}}^{\prime }/N\right)+a_{k^{\prime }}^I\cos \left(2\pi {\mathrm{nk}}^{\prime }/N\right)\right]\end{array}\right]\left[\cos \left(2\pi \mathrm{nk}/N\right)-j\sin \left(2\pi \mathrm{nk}/N\right)\right]\\ =\frac{1}{N}\sum _{n=0}^{N-1}\sum _{k^{\prime }=0}^{N-1}\left[\begin{array}{c}{a}_{k^{\prime }}^R\left[\alpha \cos \left(2\pi {\mathrm{nk}}^{\prime }/N\right)+j\beta \sin \left(2\pi {\mathrm{nk}}^{\prime }/N\right)\right]+\\ j{a}_{k^{\prime }}^I\left[\beta \cos \left(2\pi {\mathrm{nk}}^{\prime }/N\right)+j\alpha \sin \left(2\pi {\mathrm{nk}}^{\prime }/N\right)\right]\end{array}\right]\left[\cos \left(2\pi \mathrm{nk}/N\right)-j\sin \left(2\pi \mathrm{nk}/N\right)\right]\end{array}$
for k=0, 1, . . . , N−1. This can be simplified into the following form (where, for convenience,
A=2πnk′/N
and
B=2πnk/N)

[0044] $\begin{array}{c}{\hat{a}}_k=\frac{1}{2N}\sum _{n=0}^{N-1}\sum _{k^{\prime }=0}^{N-1}\left[\begin{array}{c}\left(\alpha a_{k^{\prime }}^R+j\beta a_{k^{\prime }}^I-\beta a_{k^{\prime }}^R-j\alpha a_{k^{\prime }}^I\right)\cos \left(A+B\right)+\\ \left(\alpha a_{k^{\prime }}^R+j\beta a_{k^{\prime }}^I+\beta a_{k^{\prime }}^R+j\alpha a_{k^{\prime }}^I\right)\cos \left(A-B\right)+\\ \left(j\beta a_{k^{\prime }}^R-\alpha a_{k^{\prime }}^I-j\alpha a_{k^{\prime }}^R+\beta a_{k^{\prime }}^I\right)\sin \left(A+B\right)+\\ \left(j\beta a_{k^{\prime }}^R-\alpha a_{k^{\prime }}^I+j\alpha a_{k^{\prime }}^R-\beta a_{k^{\prime }}^I\right)\sin \left(A-B\right)\end{array}\right]\\ =\frac{1}{2N}\sum _{n=0}^{N-1}\sum _{k^{\prime }=0}^{N-1}\left[\begin{array}{c}{a}_{k^{\prime }}^{*}\left(\alpha -\beta \right)\cos \left(A+B\right)+a_{k^{\prime }}\left(\alpha +\beta \right)\cos \left(A-B\right)-\\ j{a}_{k^{\prime }}^{*}\left(\alpha -\beta \right)\sin \left(A+B\right)+j{a}_{k^{\prime }}\left(\alpha +\beta \right)\sin \left(A-B\right)\end{array}\right]\end{array}$
for k=0, 1, . . . , N−1 where * indicates complex conjugation. Reversing the order of summation and substituting back in the values for A and B

[0045] ${\hat{a}}_k=\frac{\alpha -\beta }{2N}\sum _{k^{\prime }=0}^{N-1}{a}_{k^{\prime }}^{*}\sum _{n=0}^{N-1}\left\{\cos \left[2\pi n\left(k^{\prime }+k\right)/N\right]-j\sin \left[2\pi n\left(k^{\prime }+k\right)/N\right]\right\}+\frac{\alpha +\beta }{2N}\sum _{k^{\prime }=0}^{N-1}{a}_{k^{\prime }}\sum _{n=0}^{N-1}\left\{\cos \left[2\pi n\left(k^{\prime }-k\right)/N\right]+j\sin \left[2\pi n\left(k^{\prime }-k\right)/N\right]\right\}$
for k=0, 1, . . . , N−1. This can be simplified using the following sets of relations:

[0046] $\sum _{n^{\prime }=0}^{N-1}\cos \left(2\pi \mathrm{nm}/N\right)=\{\begin{array}{ccc}N& m=\mathrm{iN},& i=\mathrm{integer}\\ 0& \mathrm{otherwise}& \end{array}$
and

[0047] $\sum _{n=0}^{N-1}\sin \left(2\mathrm{\pi nm}/N\right)=0,m=\mathrm{integer}.$

[0048] Thus in the equation for â_k above, the sine terms disappear, the first cosine term is equal to N whenever k′=k=0 or k′+k=N and the second cosine term is equal to N whenever k′=k:

[0049] ${\hat{a}}_k=\{\begin{array}{cc}\alpha a_0& k=0\\ \frac{\alpha +\beta }{2}{a}_k+\frac{\alpha -\beta }{2}{a}_{N-k}^{*}& k=1,2,\dots ,N-1\end{array}$
Thus, imbalance exhibits itself as deterministic interference from exactly one other sub-carrier. Here, this interference is labeled the IQ imbalance image (or just image) and the (N−k)-th subcarrier as the image carrier. When the imbalance is zero, α=β=1 and â_k=a_k, as expected. This signal represents the FFT output and still requires equalization to compensate for channel distortion. The following two sections consider two types of adaptive equalizers: the first trains over a single OFDM symbol, possible repeated while the latter trains over multiple different OFDM symbols such that the effects of imbalance are averaged.

[0050] The typical mixer imbalance is 1 dB gain imbalance and 5% (or roughly 5°) phase imbalance. FIG. 2 illustrates the eye diagram of a 64 QAM system with this typical imbalance applied only at the transmitter and no other degradations. The dark circles at the center of gray diamonds are the transmitted points and the dashed lines are the decision boundaries. Although there are no decision errors, it is apparent that in a noisy environment, performance is significantly degraded. FIG. 3 shows a similar figure but with both the transmitter and receiver suffering 1 dB gain imbalance and 5° phase offset. The transmitted points again appear as dark circles. Instead of gray diamonds, there are now gray diamonds of diamonds that overlap the decision boundaries. This figure illustrates 320 symbol errors with a 30% error rate. This is an unacceptable error rate.

[0051] If the equalizer is trained on one OFDM symbol, or the same OFDM symbol repeated one or more times, then the effects of imbalance will not be averaged over time and the equalizer taps will be (assuming no noise)

[0052] ${\hat{w}}_k=\frac{a_k}{{\hat{a}}_k}=\frac{2}{\alpha \left(1+{\rho }_k\right)+\beta \left(1-{\rho }_k\right)}$
where
ρ_k=a^*_N−k/a_k
is computed based on the training symbol and is thus known a priori and remains constant over the duration of the OFDM packet. The error vector due to the imbalance image for this non-averaging equalizer is then

[0053] ${\overrightarrow{N}}_{\alpha \beta }={\hat{a}}_k{\hat{w}}_k-a_k=\frac{\left(\alpha -\beta \right)\left(a_{N-k}^{*}-a_k{\rho }_k\right)}{\alpha \left(1+{\rho }_k\right)+\beta \left(1-{\rho }_k\right)}.$

[0054] Using 802.11a as an example when ρ_k=±1, the signal to distortion ratio is

[0055] ${{\gamma }_{\mathrm{Imb}}}_{{\rho }_k=\pm 1}=\frac{1}{2}{\frac{\alpha +\beta }{\alpha -\beta }+{\rho }_k}^2\approx \frac{1}{2}{\frac{\alpha +\beta }{\alpha -\beta }}^2$
where the approximation is valid because the denominator approaches zero and the numerator approaches two when the imbalance is small rendering the ratio term much larger than ±1.

[0056] If only TX imbalance or only RX imbalance is present, then the signal-to-distortion ratio is

[0057] ${{{\gamma }_{\mathrm{Imb}}}_{\mathrm{TX}\mathrm{only}}\approx \frac{1}{2}\times \frac{1+2G_m^{\mathrm{tx}}\cos {\theta }_m^{\mathrm{tx}}+{\left(G_m^{\mathrm{tx}}\right)}^2}{1-2G_m^{\mathrm{tx}}\cos {\theta }_m^{\mathrm{tx}}+{\left(G_m^{\mathrm{tx}}\right)}^2}{\gamma }_{\mathrm{Imb}}}_{\mathrm{RX}\mathrm{only}}\approx \frac{1}{2}\times \frac{1+2G_m^{\mathrm{rx}}\cos {\theta }_m^{\mathrm{rx}}+{\left(G_m^{\mathrm{rx}}\right)}^2}{1-2G_m^{\mathrm{rx}}\cos {\theta }_m^{\mathrm{rx}}+{\left(G_m^{\mathrm{rx}}\right)}^2}$
Therefore, given identical amounts of imbalance, the transmitter and receiver introduce identical amounts of signal distortion. However, when combined, the imbalances may ‘align’ favorably or unfavorably. For example, given G^tx_m=G^rx_m and θ^tx_m=θ^rx_m, the combined signal-to-distortion ratio is about 6 dB worse than at just the transmitter or receiver. Alternatively, when
G^tx_m=1/G^rx_m
and θ^tx_m=−θ^rx_m, the combined signal-to-distortion ratio is 10-15 dB better than at just the transmitter or receiver for typical values of imbalance and improves considerably as the imbalance decreases.

[0058] Given that typically a_k and a^*_N−k are uncorrelated and the average a^*_N−k is zero, the noise due to imbalance will also be uncorrelated with zero mean. This implies then that the taps of an adaptive frequency domain equalizer, given sufficient time and no other impairments, will converge to
ŵ_k=2/(α+β)
Comparing these equalizer taps to those of the non-averaging equalizer, it can be observed that the effect of perfect equalizer averaging is to force ρ_k to zero.

[0059] Thus, in the case of a perfectly converged, adaptive equalizer, the interference due to the imbalance image can be expressed as an additive error vector:

[0060] ${\overrightarrow{N}}_{\alpha \beta }=\frac{\alpha -\beta }{\alpha +\beta }{a}_{N-k}^{*}\mathrm{for}k=1,2,\dots ,N-1$
The average signal-to-distortion ratio due to imbalance is then (assuming E[a²_k]=E[a²_N−k])

[0061] ${\gamma }_{\mathrm{Imb}}=\frac{E\left[{a_k}^2\right]}{E\left[{{\overrightarrow{N}}_{\alpha \beta }}^2\right]}={\left(\frac{\alpha +\beta }{\alpha -\beta }\right)}^2=\frac{{\left[1+G_m^{\mathrm{tx}}{G}_m^{\mathrm{rx}}\cos \left({\theta }_m^{\mathrm{tx}}-{\theta }_m^{\mathrm{rx}}\right)\right]}^2+{\left(G_m^{\mathrm{tx}}\sin {\theta }_m^{\mathrm{tx}}-G_m^{\mathrm{rx}}\sin {\theta }_m^{\mathrm{rx}}\right)}^2}{{\left[1-G_m^{\mathrm{tx}}{G}_m^{\mathrm{rx}}\cos \left({\theta }_m^{\mathrm{tx}}-{\theta }_m^{\mathrm{rx}}\right)\right]}^2+{\left(G_m^{\mathrm{tx}}\sin {\theta }_m^{\mathrm{tx}}+G_m^{\mathrm{rx}}\sin {\theta }_m^{\mathrm{rx}}\right)}^2}$

[0062] ${{{\gamma }_{\mathrm{Imb}}}_{\mathrm{TX}\mathrm{only}}=\frac{1+2G_m^{\mathrm{tx}}\cos {\theta }_m^{\mathrm{tx}}+{\left(G_m^{\mathrm{tx}}\right)}^2}{1-2G_m^{\mathrm{tx}}\cos {\theta }_m^{\mathrm{tx}}+{\left(G_m^{\mathrm{tx}}\right)}^2}{\gamma }_{\mathrm{Imb}}}_{RX\mathrm{only}}=\frac{1+2G_m^{\mathrm{rx}}\cos {\theta }_m^{\mathrm{rx}}+{\left(G_m^{\mathrm{rx}}\right)}^2}{1-2G_m^{\mathrm{rx}}\cos {\theta }_m^{\mathrm{rx}}+{\left(G_m^{\mathrm{rx}}\right)}^2}$
This result is identical to the result when averaging is not performed, with the exception of the ½ term. Thus, the signal-to-distortion ratio in the case where the equalizer averages the effects of imbalance is 3 dB better than when the equalizer trains on a single OFDM symbol and thus does not average the effects of imbalance. Given that the distribution of a_k is known for each constellation type, the distribution of the noise vectors can also be determined as a function of the imbalance; however, this is not as important as determining the average noise.

[0063] The exact number of OFDM symbols over which averaging must be performed in order to average out imbalance depends on the OFDM symbols themselves. Two OFDM symbols are sufficient if, for example, the second symbol is the sign inverted equivalent of the first (i.e. a_k(m)=−a_k(m+1)).

[0064] Note that the frequency domain equalizer is not capable of compensating for imbalance, although allowing it to average the imbalance effects can reduce the distortion by up to 3 dB.

[0065] FIG. 4 illustrates the signal-to-distortion ratio for either gain or phase imbalance applied, individually, at either the transmitter or receiver. The left hand vertical axis 401 represents gain imbalance only, in dB. The right hand vertical axis 402 represents phase imbalance only, in degrees. The horizontal axis 403 represents a signal-to-distortion ratio in dB. Two curves are plotted for gain 411 and phase 412 imbalance. The results are the same regardless of whether the imbalance is at the transmitter or receiver. The distortion will be worse if both phase and gain imbalance are present or if imbalance is introduced at both the transmitter and receiver.

[0066] The effect of multipath fading in conjunction with imbalance is now investigated. It is assumed that the delay spread of the channel (including TX/RX filters) is less than the duration of the cyclic prefix (800 nsec for 802.11a) and that any symbol alignment error is sufficiently small such that inter-symbol interference (ISI) is negligible. This model does not incorporate PA non-linearities.

[0067] Recall from above that the transmitted RF signal, incorporating transmitter imbalance, is
z(t)=y_I(t)cos(2πf_ct)−y_Q(t)G^tx_msin(2πf_ct+θ^tx_m)
The effect of the multipath channel is that the receiver sees N_p copies of the transmitted signal, each with its own gain G_p, delay t_p and phase θ_p, such that

[0068] $\hat{z}\left(t\right)=\sum _{p=0}^{N_p-1}{G}_p\left[y_I\left(t-t_p\right)\cos \left(2\pi f_c\left(t-t_p\right)+{\theta }_p\right)-y_Q\left(t-t_p\right)G_m^{\mathrm{tx}}\sin \left(2\pi f_c\left(t-t_p\right)+{\theta }_p+{\theta }_m^{\mathrm{tx}}\right)\right]$
At the receiver, a carrier with a frequency offset of f₀ with respect to the transmitter's carrier frequency mixes this signal down to baseband. Because mixing is a linear process, each of the paths can be considered independently as

[0069] $\begin{array}{c}{\hat{y}}_{I,p}\left(t\right)={\left\{2G_p\left[\begin{array}{c}{y}_I\left(t-t_p\right)\cos \left(2\pi f_c\left(t-t_p\right)+{\theta }_p\right)-\\ y_Q\left(t-t_p\right)G_m^{\mathrm{tx}}\sin (2\pi f_c\left(t-t_p\right)+\\ {\theta }_p+{\theta }_m^{\mathrm{tx}})\end{array}\right]\cos \left(2\pi \left(f_c+f_o\right)t\right)\right\}}_{\mathrm{LPF}}\\ =G_p[y_I\left(t-t_p\right)\cos \left(2\pi f_{o}t+2\pi f_c{t}_p-{\theta }_p\right)+\\ y_Q\left(t-t_p\right)G_m^{\mathrm{tx}}\sin \left(2\pi f_{o}t+2\pi f_c{t}_p-{\theta }_p-{\theta }_m^{\mathrm{tx}}\right)]\end{array}$

[0070] $\begin{array}{c}{\hat{y}}_{Q,p}\left(t\right)={\left\{-2G_p{G}_m^{\mathrm{rx}}\left[\begin{array}{c}{y}_I\left(t-t_p\right)\cos (2\pi f_c(t-\\ t_p)+{\theta }_p)-y_Q(t-\\ t_p)G_m^{\mathrm{tx}}\sin (2\pi f_c(t-\\ t_p)+{\theta }_p+{\theta }_m^{\mathrm{tx}})\end{array}\right]\sin \left(2\pi \left(f_c+f_o\right)t+{\theta }_m^{\mathrm{rx}}\right)\right\}}_{\mathrm{LPF}}\\ =G_p\left[\begin{array}{c}{y}_Q\left(t-t_p\right)G_m^{\mathrm{tx}}{G}_m^{\mathrm{rx}}\cos \left(2\pi f_{o}t+2\pi f_c{t}_p-{\theta }_p-{\theta }_m^{\mathrm{tx}}+{\theta }_m^{\mathrm{rx}}\right)-\\ y_I\left(t-t_p\right)G_m^{\mathrm{rx}}\sin \left(2\pi f_{o}t+2\pi f_c{t}_p-{\theta }_p+{\theta }_m^{\mathrm{rx}}\right)\end{array}\right]\end{array}$
These signals are converted to the digital domain where
n=it×f_s−mN_s,
f_s is the A/D's sample rate, m is the OFDM symbol number,
N_s=(N+N_cp)
is the number of samples per OFDM symbol and N_cp is the cyclic prefix length. Thus, the received baseband signal from the p-th path is

[0071] ${\hat{y}}_p\left(n+{\mathrm{mN}}_s\right)=G_p\left\{\begin{array}{c}{y}_I\left(n+{\mathrm{mN}}_s-f_s{t}_p\right)\left[\begin{array}{c}\cos \left(2\pi \left(n+{\mathrm{mN}}_s\right)f_o/f_s+2\pi f_c{t}_p-{\theta }_p\right)-\\ j{G}_m^{\mathrm{rx}}\sin (2\pi \left(n+{\mathrm{mN}}_s\right)f_o/f_s+\\ 2\pi f_c{t}_p-{\theta }_p+{\theta }_m^{\mathrm{rx}})\end{array}\right]+\\ j{y}_Q\left(n+{\mathrm{mN}}_s-f_s{t}_p\right)\left[\begin{array}{c}{G}_m^{\mathrm{tx}}{G}_m^{\mathrm{rx}}\cos (2\pi \left(n+{\mathrm{mN}}_s\right)f_o/f_s+2\pi f_c{t}_p-\\ {\theta }_p-{\theta }_m^{\mathrm{tx}}+{\theta }_m^{\mathrm{rx}})-j{G}_m^{\mathrm{tx}}\sin (2\pi (n+\\ {\mathrm{mN}}_s)f_o/f_s+2\pi f_c{t}_p-{\theta }_p-{\theta }_m^{\mathrm{tx}})\end{array}\right]\end{array}\right\}$

[0072] Define four time-invariant intermediary values that are functions of the imbalance only (where α and β are the same as defined above):
α=1−jG^rx_msin(θ^rx_m) β=G^tx_mG^rx_mcos(θ^tx_m−θ^rx_m)+jG^tx_msin(θ^tx_m)
α₀=−jG^rx_mcos(θ^rx_m) β₀=G^tx_mG^rx_msin(θ^tx_m−θ^rx_m)−jG^tx_mcos (θ^tx_m)
Rewriting the received baseband signal in terms of these intermediary values and assuming that ISI is not an issue (i.e. that
f_s×max(t_p)≦N_cp
and ignoring TX/RX filtering effects), the baseband model incorporating imbalance, multipath and frequency offset is obtained:

[0073] ${\hat{y}}_m\left(n\right)=\sum _{p=0}^{N_p-1}{G}_p\left\{\begin{array}{c}{y}_{m,I}\left(n-f_s{t}_p\right)\left[\begin{array}{c}\alpha \cos \left(2\pi \left(n+{\mathrm{mN}}_s\right)f_o/f_s+2\pi f_c{t}_p-{\theta }_p\right)+\\ {\alpha }_0\sin (2\pi \left(n+{\mathrm{mN}}_s\right)f_o/f_s+\\ 2\pi f_c{t}_p-{\theta }_p)\end{array}\right]+\\ j{y}_{m,Q}\left(n-f_s{t}_p\right)\left[\begin{array}{c}\beta \cos (2\pi \left(n+{\mathrm{mN}}_s\right)f_o/f_s+2\pi f_c{t}_p-\\ {\theta }_p)+{\beta }_0\sin (2\pi (n+\\ {\mathrm{mN}}_s)f_o/f_s+2\pi f_c{t}_p-{\theta }_p)\end{array}\right]\end{array}\right\}$
The received set of constellation points for the m-th symbols is obtained by performing the FFT on ŷ_m,p(n). Exploiting the linearity of the FFT, superposition can be applied (where the subscript m has been dropped for convenience)

[0074] ${\hat{a}}_k=\left\{\sum _{p=0}^{N_p-1}{\hat{y}}_p\left(n\right)\right\}=\sum _{p=0}^{N_p-1}\left\{{\hat{y}}_p\left(n\right)\right\}$
where {•} is the Fourier transform. Now, rearranging ŷ_p(n) such that

[0075] $\hat{y}\left(n-f_s{t}_p\right)=\sum _{p=0}^{N_p-1}{G}_p\left\{\begin{array}{c}\cos (2\pi {\mathrm{nf}}_o/f_s+2\pi {\mathrm{mN}}_s{f}_o/f_s+2\pi \left(f_c+f_o\right)t_p-\\ {\theta }_p)\lfloor \alpha y_I\left(n\right)+j\beta y_Q\left(n\right)\rfloor +\sin (2\pi {\mathrm{nf}}_o/f_s+2\pi {\mathrm{mN}}_s{f}_o/f_s+\\ 2\pi \left(f_c+f_o\right)t_p-{\theta }_p)\left[{\alpha }_0{y}_I\left(n\right)+j{\beta }_0{y}_Q\left(n\right)\right]\end{array}\right\}$
the delay property of the Fourier transform, where n₀ is non-integer,
f(n−n₀)exp(−j2πkn₀/N)F(k) for k≦N/2
and
f(n−n₀)exp(−j2πn₀(k−N)/N)F(k) for k>N/2,
can be applied

[0076] ${\hat{a}}_k=\{\begin{array}{cc}\sum _{p=0}^{N_p-1}\exp \left(-j2\pi f_s{t}_{p}k/N\right)\left\{{\hat{y}}_p\left(n\right)\right\}& k\le N/2\\ \sum _{p=0}^{N_p-1}\exp \left(-j2\pi f_s{t}_p\left(k-N\right)/N\right)\left\{{\hat{y}}_p\left(n\right)\right\}& k>N/2\end{array}$
Recognizing that {ŷ_p(n)} is the sum of two products, the Fourier transform's modulation property can be used—if y(n)=h(n)x(n), then

[0077] $Y\left(k\right)=H\left(k\right)*X\left(k\right)=N^{-1}\sum _{n=0}^{N-1}H\left(n\right)X\left(k-n\right)$
where * represents circular convolution. Now define
θ=2πmN_sf₀/f_s+2π(f₀+f_c)t_p−θ_p
which is constant over each symbol, and using the definition of the discrete Fourier transform,

[0078] $\left\{\cos \left(2\pi {\mathrm{nf}}_o/f_s+\theta \right)\right\}=\sum _{n=0}^{N-1}\cos \left(2\pi {\mathrm{nf}}_o/f_s+\theta \right)e^{-j2\pi \mathrm{nk}/N}$

[0079] $\left\{\sin \left(2\pi {\mathrm{nf}}_o/f_s+\theta \right)\right\}=\sum _{n=0}^{N-1}\sin \left(2\pi {\mathrm{nf}}_o/f_s+\theta \right)e^{-j2\pi \mathrm{nk}/N}$
The Fourier transform of αy_I(n)+jβy_Q(n) was derived above and the Fourier transform of α₀y_I(n)+jβ₀y_Q(n) is a trivial variation of this. Thus, (assuming α₀ is zero)

[0080] $\left\{{\hat{y}}_p\left(n\right)\right\}=\frac{G_p}{2N}\sum _{n=1}^{N-1}\left\{\begin{array}{c}[\left(\alpha +\beta \right)a_n+\\ \left(\alpha -\beta \right)a_{N-n}^{*}]\sum _{n^{\prime }=0}^{N-1}\cos \left(2\pi n^{\prime }{f}_o/f_s+\theta \right)e^{-j2\pi n^{\prime }\left(k-n\right)/N}+\\ [\left({\alpha }_0+{\beta }_0\right)a_n+\\ \left({\alpha }_0-{\beta }_0\right)a_{N-n}^{*}]\sum _{n^{\prime }=0}^{N-1}\sin \left(2\pi n^{\prime }{f}_o/f_s+\theta \right)e^{-j2\pi n^{\prime }\left(k-n\right)/N}\end{array}\right\}$
Finally, substituting back into the original equation, the final result is obtained (for
k≦N/2;
see the delay property above for the modification required for the
k>N/2
case)

[0081] ${\hat{a}}_k=\frac{1}{2N}\sum _{p=0}^{N_p-1}{G}_p{e}^{-j2\pi f_s{t}_{p}k/N}\sum _{n=1}^{N-1}\left\{\begin{array}{c}[\left(\alpha +\beta \right)a_n+\\ \left(\alpha -\beta \right)a_{N-n}^{*}]\sum _{n^{\prime }=0}^{N-1}\cos \left(2\pi n^{\prime }{f}_o/f_s+\theta \right)e^{-j2\pi n^{\prime }\left(k-n\right)/N}+\\ [\left({\alpha }_0+{\beta }_0\right)a_n+\\ \left({\alpha }_0-{\beta }_0\right)a_{N-n}^{*}]\sum _{n^{\prime }=0}^{N-1}\sin \left(2\pi n^{\prime }{f}_o/f_s+\theta \right)e^{-j2\pi n^{\prime }\left(k-n\right)/N}\end{array}\right\}$
Again, it is assumed that the frequency offset is negligible and thus can be assumed to be zero. In this case, the above equation simplifies to

[0082] ${\hat{a}}_k=\{\begin{array}{c}\frac{1}{2}\sum _{p=0}^{N_p-1}{G}_p\exp \left(-\mathrm{j2\pi }{f}_s{t}_{p}k/N\right)\left\{\begin{array}{c}\cos \left(\theta \right)\left[\left(\alpha +\beta \right)a_k+\left(\alpha -\beta \right)a_{N-k}^{*}\right]+\\ \sin \left(\theta \right)\left[\left({\alpha }_0+{\beta }_0\right)a_k+\left({\alpha }_0-{\beta }_0\right)a_{N-k}^{*}\right]\end{array}\right\}0\le k<N/2\\ \frac{1}{2}\sum _{p=0}^{N_p-1}{G}_p\exp \left(-\mathrm{j2\pi }{f}_s{t}_p\left(k-N\right)/N\right)\left\{\begin{array}{c}\cos \left(\theta \right)\left[\left(\alpha +\beta \right)a_k+\left(\alpha -\beta \right)a_{N-k}^{*}\right]+\\ \sin \left(\theta \right)\left[\left({\alpha }_0+{\beta }_0\right)a_k+\left({\alpha }_0-{\beta }_0\right)a_{N-k}^{*}\right]\end{array}\right\}N/2\le k<N\end{array}$

[0083] To simplify the notation going forward and to assist in separating the channel effects from the imbalance effects, the following functions are defined (not that these are functions of the multipath channel only and are assumed constant over the duration of the packet)

[0084] $A_{\cos }=\{\begin{array}{c}\sum _{p=0}^{N_p-1}{G}_p\exp \left(-\mathrm{j2\pi }{f}_s{t}_{p}k/N\right)\cos \left(\theta \right)0\le k<N/2\\ \sum _{p=0}^{N_p-1}{G}_p\exp \left(-\mathrm{j2\pi }{f}_s{t}_p\left(k-N\right)/N\right)\cos \left(\theta \right)N/2\le k<N\end{array}$

[0085] $A_{\sin }=\{\begin{array}{c}\sum _{p=0}^{N_p-1}{G}_p\exp \left(-\mathrm{j2\pi }{f}_s{t}_{p}k/N\right)\sin \left(\theta \right)0\le k<N/2\\ \sum _{p=0}^{N_p-1}{G}_p\exp \left(-\mathrm{j2\pi }{f}_s{t}_p\left(k-N\right)/N\right)\sin \left(\theta \right)N/2\le k<N\end{array}$
Thus, rewriting the above expression,

[0086] ${\hat{a}}_k=\frac{1}{2}\left[\left(\alpha +\beta \right)A_{\cos }+\left({\alpha }_0+{\beta }_0\right)A_{\sin }\right]a_k+\frac{1}{2}\left[\left(\alpha -\beta \right)A_{\cos }+\left({\alpha }_0-{\beta }_0\right)A_{\sin }\right]a_{N-k}^{*}$

[0087] Again, assuming that a_k and a^*_N−k are uncorrelated, randomly distributed and of zero mean, a frequency domain equalizer allowed to average over a sufficient number of OFDM symbols will converge to
ŵ_k=2/[(α+β)A_cos+(α₀+β₀)A_sin].
The error vector in this case is
{right arrow over (N)}_αβ=â_kŵ_k−a_k=ε_ka^*_N−k
where

[0088] ${\varepsilon }_k=\frac{\left(\alpha -\beta \right)A_{\cos }+\left({\alpha }_0-{\beta }_0\right)A_{\sin }}{\left(\alpha +\beta \right)A_{\cos }+\left({\alpha }_0+{\beta }_0\right)A_{\sin }}$
Notice that the error vector will be zero only if a^*_N−k=0. The parameter, ε_k, is of particular importance as it will be used extensively. It is a constant value, independent of the transmitted data, which incorporates all the channel and imbalance parameters (i.e. separates the channel and imbalance effects from the data effects).

[0089] When the frequency domain equalizer is trained on a single or repeated OFDM reference symbol (such as for 802.11a systems), the equalizer coefficients will be (assuming no noise)

[0090] ${\hat{w}}_k=\frac{2}{\left[\left(\alpha +\beta \right)A_{\cos }+\left({\alpha }_0+{\beta }_0\right)A_{\sin }\right]+{\rho }_k\left[\left(\alpha -\beta \right)A_{\cos }+\left({\alpha }_0-{\beta }_0\right)A_{\sin }\right]}$
where
ρ_k=a^*_N−k/a_k
is determined from the reference signal used for training the equalizer. In this case, the error vector is

[0091] ${\overrightarrow{N}}_{\mathrm{\alpha \beta }}={\hat{a}}_k{\hat{w}}_k-a_k=\frac{{\varepsilon }_k\left(a_{N-k}^{*}-a_k{\rho }_k\right)}{1+{\rho }_k{\varepsilon }_k}$

[0092] Notice that the error vector is zero whenever a_kρ_k=a^*_N−k. Using the same approach as for the no channel case, it can be shown that averaging the imbalance effects over multiple OFDM symbols results in up to 3 dB reduction in the average distortion power.

[0093] Several algorithms for compensating imbalance have been developed. These are focused in the following areas of pre-compensating TX imbalance at the transmitter, compensating for TX and RX imbalance at the receiver—no channel, compensating for TX and RX imbalance at the receiver—fading channel. Two variations for the fading channel case have been considered: (a) compensating for TX imbalance at the receiver, ignoring RX imbalance, and (b) compensating for RX imbalance at the receiver, ignoring TX imbalance. These two variations are of interest because at the receiver, the RX imbalance will be the same (or very slowly changing) over all packets whereas the TX imbalance may change from packet to packet (particularly for a base station or wireless access point).

[0094] Even when imbalance compensation is applied, some residual imbalance effects may still exist particularly if the previously imbalance measurement is not accurate or the imbalance varies over time. Here, iterative imbalance compensator coefficients are computed such that the coefficients can be updated to compensate for any such residual imbalance. Sufficiently averaging the imbalance measurements over time could negate the need to iteratively update the coefficients. However, this would require that either the imbalance compensation not be applied during this measurement period or that the measurement be performed prior to any compensation. Algorithms for estimating the imbalance are presented below.

[0095] The transmit I/Q Imbalance Compensator is implemented at the transmitter and pre-compensates for I/Q imbalance introduced by the transmit mixer (see FIG. 1).

[0096] TX imbalance compensation can be applied in either the time or frequency domain. Both are relatively simple algorithms to implement; however, if not all samples originate in the frequency domain, then the time-domain approach is preferable. For example, in 802.11a, to reduce the transmitter's wake-up latency, the short preamble is stored in memory as time-domain samples, In this case, a time-domain TX imbalance compensator is preferred unless the preamble samples are compensated before they are committed to memory.

[0097] Compensating for TX I/Q imbalance in the time-domain at the transmitter requires that y(n) be computed as a function of x(n) such that ŷ(n)=x(n), where all but the transmitter imbalance impairments are zero. Using equations derived above, this implies that
ŷ_I(n)=y_I(n)−y_Q(n)G^tx_msinθ^tx_m≡x_I(n)
ŷ_Q(n)=y_Q(n)G^tx_mcosθ^tx_m≡x_Q(n)
Solving these equations for y_I(n) and y_Q(n) as a function of x_I(n) and x_Q(n), the functionality of the time-domain transmit I/Q imbalance compensator is determined:
y_I(n)=x_I(n)+x_Q(n) tan (θ^tx_m)
y_Q(n)=x_Q(n)[G^tx_mcos(θ^tx_m)]⁻¹
This assumes that G^tx_m and θ^tx_m are perfectly known and time-invariant. However, in practice, typically only estimates of G^tx_m and θ^tx_m are known. Fortunately, one can iteratively update these values given new estimates. For example,
G^tx_m(i+1)=G^tx_m(i)Ĝ^tx _m
and
θ^tx_m(i+1)=θ^tx_m(i)+{circumflex over (θ)}^tx_m
where Ĝ^tx_m(i) and {circumflex over (θ)}^tx_m(i) are the i-th estimates of the residual gain and phase imbalance and
G^tx_m(0)=1
and
θ^tx_m(0)=0.