CODE-AIDED ADAPTIVE DECORRELATOR FOR IQ IMBALANCE COMPENSATION IN

CODE-AIDED ADAPTIVE DECORRELATOR FOR IQ IMBALANCE COMPENSATION IN ITERATIVE RECEIVERS FOR FLAT FADING CHANNELS Raghunath Cherukuri, Poras T. Balsara Center for Integrated Circuits and Systems University of Texas Dallas Richardson, Texas, 75083 ABSTRACT Long convergence times associated with adaptive algorithms are not practical in burst mode communications for synchronization and channel estimation tasks including IQ imbalance parameters. This motivates one to find techniques to increase the rate of convergence of the learning algorithm. We propose a novel acquisition aid that is also low in complexity to the adaptive decorrelator for IQ imbalance compensation. The acquisition aid is an estimator that is embedded in the iterative detector and passes the estimated IQ imbalance parameters back to the compensation filter controlled by the adaptive decorrelator. The resulting adaptive decorrelator with acquisition aid can converge on the first packet and resumes in the tracking mode and offers steady-state performance superior to that offered by the estimator. The proposed adaptive decorrelator aided by the embedded estimator is blind in the sense that it does not require any training symbols nor tones. The performance is simulated for 64QAM under flat fading channel conditions. INTRODUCTION A practical communication system has to deal with impairments arising from the channel, algorithms, architecture, and/or the circuit, in addition to AWGN. It is necessary to mitigate these impairments to improve the performance of the system. By using TURBO codes [1], a type of capacity achieving channel codes, it is possible to remove the effect of AWGN almost completely. The power of these codes lies in the iterative decoder which exploits belief propagation principles. The technique is generalized as "TURBO principle" [10] and can be used to solve other detection and estimation problems. The redundant bits introduced in channel coding cause bandwidth (BW) expansion and hence loss in spectral efficiency. Similar redundancy can also be introduced at the symbol level without incurring loss in spectral efficiency, and coded modulation is such a strategy. Bit-interleaved-code-modulation (BICM) [2] is proven to be a better choice for fading channels, but, the complexity of the optimal ML detector for a BICM is prohibitive because of the interleaver present between the encoder and the mapper and sub-optimal solutions are thus encouraged. Iterative detection inspired by the TURBO principle is an attractive solution with much lesser complexity [11]. Such an iterative detector is termed as BICM_ID, and it is realized that the iterative detector yields performance gain only when a non-Gray labeling 1-4244-1513-06/07/$25.00 c 2007 IEEE for the constellation symbols are used [11]. In-phase and Q-phase imbalance correction is essential for higher data rates and for higher spectral efficiencies. Mismatch in I and Q paths gives rise to image frequencies which become interference upon down conversion to baseband. Low-IF and Direct conversion receiver (DCR) topologies are more vulnerable since IQ separation happens early in the (Radio Frequency) RF/analog portion. In a DCR, there is no image frequency present, and thus the induced image is the self-image of the signal. In the last several decades, numerous solutions based on calibration techniques have been proposed to improve image rejection ratio (IRR). A major class of these calibration techniques employ trimming or adjusting the physical properties of the circuit such as the geometry of the device or the bias current or voltage of the circuit. These techniques take considerable test time and thus are not preferred. Another major class is based on DSP techniques that employ filtering to improve the IRR. The filter parameters can be designed by employing specialized tones or pilot signals [3] or by using algorithms based on adaptive techniques [4], [5], [6]. The filter parameters can also be designed by using statistical signal processing techniques such as estimation techniques [7]. A cost function of choice during the optimization of these filter designs is the orthogonality of the I and Q branches and do not require any desired response and thus are inherently unsupervised [4], [5], [6], [7]. In this study we address the problem of IQ imbalance correction in an iterative receiver designed to transmit and receive short bursts of packets over a flat fading channel. For a BER of 10−5 , channel coding is needed and the gain from the iterative decoder can be beneficially used for channel recovery including IQ compensation. We are unaware of any previously published work along these lines for IQ compensation. The complexity involved in using techniques based on statistical signal processing can be considerable since they use operations such as matrix inversion and square root operations that are power hungry [7]. Adaptive decorrelator based techniques may not be able to converge with the limited amount of data and in the limited amount of time. In order to keep the turbo iterations to a practical number and to achieve the desired BER in the first packet, there should be an additional IQ imbalance estimator to aid the adaptive decorrelator until convergence. Such an estimator should also be low in complexity to be an attractive solution. In [8] we presented such an estimator, which we called as a TURBO IQ decorrelator, and it was developed as an extension to the work in [9]. We propose a solution for IQ imbalance correction in a BICM_ID as a combination of adaptive decorrelator and a non-data aided batch estimator embedded in the iterative detector that we developed in [8]. In our proposed solution, the TURBO IQ decorrelator acts as an acquisition aid to the adaptive decorrelator and adaptive decorrelator in turn aids the embedded estimator by reducing the bit errors on the first iteration of each packet. This leads to quick convergence and the adaptive decorrelator resumes in the tracking mode, without demanding a training phase. The quick acquisition is accomplished by passing the estimated IQ imbalance parameters back to the adaptive decorrelator and the filter weights are updated with the estimated values. The novelty of our proposed solution, is thus quick convergence as well as tracking of variations with a superior steady state performance. In [14], a technique is proposed for coarse and fine estimation of IQ imbalance parameters for a WLAN OFDM case, where they used special tones for coarse estimation and training symbols for fine estimation and thus falls into the data aided category. Our goal is to propose a non-data aided solution that is independent of any particular standard. The remaining paper is organized as follows: communication signals and system model are reviewed in section 2, IQ compensator is formulated in section 3, simulation results and conclusions are presented in sections 4 and 5 respectively. RECEIVER MODEL In the transmitter, the source data is encoded, interleaved and mapped before given to the RF front-end. In the RF front-end, the complex symbols are up-converted to a carrier frequency by a complex mixer, and after sufficient power gain, are launched by the antenna into the physical channel. When high spectral efficiencies are desired M-ary modulation schemes such as QAM are preferred. When high power efficiencies are desired, capacity achieving codes are required. We assume the channel to be flat fading with AWGN. The channel is also assumed to be time invariant during one block of data. Fig.1 highlights the RF front-end portion of the receiver. The I and Q separation happens in the complex mixer, and has highest contribution for IQ imbalance. The mismatches in the receiver local oscillator circuit that generates quadrature phases contribute to IQ phase imbalance, and mismatch in mixers gives rise to gain imbalance. The I and Q baseband analog signals are further filtered to reject the higher frequency terms resulting from mixing operations. Variations in filter time constants gives rise to fre- MIXER Figure 1: Receiver RF front-end architecture source encoder BI mapper sink decoder BDI Filter demapper BI BI: Block Interleaver BDI: Block Deinterleaver LMS Algorithm IQ estimate Figure 2: Receiver Baseband Architecture quency dependent IQ phase imbalance in wideband systems. The filtered analog baseband signal is sampled in the analog-to-digital converter (ADC) to generate a bit sequence for further processing by the digital baseband processor. We focus on frequency independent IQ imbalance model for a narrowband single carrier system. Fig.2 represents the baseband block diagram. Assuming that the channel is known through estimation, the received symbol is equalized for the multiplicative distortion and then passed to the soft demapper [11]. The demapper gives a most likely soft representation of the corresponding bits of the received complex symbols, upon deinterleaving these bits are decoded by the symbol level decoder. Decoding is done at symbol level and an algorithm of type BCJR [12] can be used to perform this operation. The Soft-Input-Soft-Output (SISO) decoder generates soft representation of the most likely bits at the encoder input. The SISO decoder also generates soft representation of the demapper output bits which are the coded bits, to be exchanged with the demapper. This soft information, termed as extrinsic information, is generated by each block based on information available to itself and also based on the extrinsic information they receive from each other. The generation and transfer of extrinsic information is the key ingredient for the performance gain in using the iterative loops. Convergence of this interaction is said to happen when the incremental growth of extrinsic information is negligible from successive iterations. The loop should be designed to guarantee such a beneficial feedback. Constellation labeling is an important issue in BICM_ID and it is observed that, Gray Code (GC) does not improve the ∗ ∗ G = K1 − K2 .K2 /K1 (8) The filter for IQ imbalance compensation is implemented based on Eq. 5, and the optimal weights are given by Eq. 6. The value of the optimal weight can be computed either adaptively or through K1 and K2 . A proper estimator can be identified to estimate K1 and K2 directly, or can be computed indirectly using Eq. 1 and Eq. 2 by esSignal and IQ Imbalance model timating the values g and φ. In the next section we review an adaptive decorrelator for designing the optimal weight A block of data d is encoded into a block c and after inadaptively [16] and a code-aided estimator [8] to estimate terleaving is mapped to a complex symbol sequence a. the values of g and φ. After pulse shaping this sequence, is upconverted by the transmit local oscillator operating at angular frequency IQ Compensation by Adaptive Decorrelator (AD) wc , to a waveform s(t) which can be given as s(t) = K−1 x(t)ejwc (t) + x∗ (t)e−jwc (t) where x(t) = k=0 ak p(t − kT ), There are several choices available in choosing an adaptive T is the symbol period and p(t) is the unit-energy pulse decorrelator. Two issues that play a dominant role in the shaping function.The received RF signal r(t) is given as selection is a) lower complexity of implementation and b) r(t) = Aejθ s(t)+n(t) where Aejθ is the multiplying chanability to feedback the estimated values of the mismatch nel parameter and n(t) is the AWGN with N (0, σ 2 ). The parameters either (K1 , K2 ) or (g, φ), and c) no residual receiver local oscillator signal l(t) with IQ imbalance can phase and gain imbalances that are delegated to automatic be modeled as l(t) = K1 e−jwc (t) + K2 ejwc (t) and gain control (AGC) or channel estimation blocks down K1 = cosφ − jg sinφ (1) stream. From this perspective we have chosen the adaptive BER performance over successive iterations [11]. In this investigation we used binary labeling which seems to offer comparable performance between offset gain (improvement in extrinsic information through feedback) and implementation loss (IL). Since we have chosen a non data aided (NDA) approach (or blind) for channel recovery, the effect of data on the channel parameters during the estimation process needs to be averaged out. For this purpose, marginal probabilities of the channel symbols are required and these are the exact quantities that are available in the form of extrinsic information. ∗ Wopt = K2 /K1 (6) (7) and y (t) = G.y(t) ˆ where K2 = g cosφ + jsinφ (2) and g and φ are the gain and phase imbalance parameters modeled as shown in Fig. 1. The down converted received signal z(t) after the low-pass filter is given as z(t) = LP {r(t).l(t)} = K1 y(t) + K2 y ∗ (t) (3) where, y(t) is the down-converted signal if there were no IQ imbalance in the receiver local oscillator. After sampling the signal z(t), we have a sequence of complex numbers z[n] which become input to the LMS filter. The output of LMS filter is denoted as the sequence of complex numbers z[n] from which we can recover A, θ, g, φ and the ˆ data d. Channel correction is done by scaling with A and ˆ derotating by θ where ^ denotes estimates. PROPOSED IQ IMBALANCE COMPENSATION From Eq. 3 it can be observed that the desired signal is corrupted by the image signal. This can be considered as a widely linear transform in y(t) [15]. Under this consideration a widely linear inverse transform can be proposed as y (t) = W1 z(t) + W2 z ∗ (t) ˆ (4) Since the goal is to suppress the image, we can rewrite Eq. 4 as y (t) = z(t) − W2 z ∗ (t) ˆ (5) From Eq. 3 and Eq. 5, we can derive the weight W2 as decorrelator [16] that was proven in a real product as our choice. For a slight reduction in performance and slower rates of adaptation [17] will be another choice. We recapitulate the update equations as follows Wopt = WI + jWQ (9) WI (n + 1) = WI (n) + µ(|zI (n)|2 − |zQ (n)|2 ) WQ (n + 1) = WQ (n) + 2µ(zI (n)zQ (n)) (10) (11) IQ Estimation and Compensation using TURBO IQ Decorrelator [8] In this section we develop the embedded estimator via the EM algorithm, but first some observations about adaptive decorrelator are in order for the sake of motivation. The SAD problem is formulated as leakage between two independent channels. A clean copy of Q-branch (I-branch) is needed in order to remove the leakage of Q-branch (Ibranch) from I-branch (Q-branch). Since both branches are corrupted, leakage free I and Q branches are available only at the output of the filter. Upon convergence of the adaptive algorithm, the I and Q signal paths are separated from each other, or the leakages are suppressed and thus the orthogonality is restored. These samples are further processed by the iterative detector for message recovery. The error correcting capability of the iterative detector can be seen as an additional IQ imbalance compensator. When u z z ZI Splitter aQ Estimation, Derotation and Scaling ZI Combiner aQ Ch. Samples Splitter Splitter Z Combiner aI Splitter Estimation, Derotation and Scaling Combiner aI ZQ ZQ Decision Feedback from Detector a v istic and unknown we can take the maximum likelihood approach and formulate the data detection as a joint ML problem as a = arg maxa {arg max˜{ln p(r|a, b)}. The ˆ ˜ b maximum-likelihood (ML) data detection is formulated as a = arg maxa {ln p(r|a)} and the channel estimation ˆ ˜ problem as ˆ = arg max˜{ln p(r|˜ where p(r|˜ = b b)} b) b ˜ a p(r|a, b)p(a)da which in effect is the pdf of the channel estimates by averaging out the unknown data dependency. By decoupling estimation and detection, and invoking EM algorithm [13] it allows to perform iterative channel estimation and data detection. In our case, after some manipulation and simplification [9] the E step and M step can be stated as E step : a Figure 3: IQ phase imbalance conversion to carrier gain and phase error. p(a|r, ˆn−1 )ln p(r|a, ˜ da b b) (12) a leakage free I and Q signals are available as past decisions (upon re-coding and re-mapping) from the previous iteration that can be fed back, the SAD can be modified by forming the cost function with these signals, which results in a decision aided SAD. The non-adaptive and batch estimator equivalent to this would be a decision aided IQ imbalance parameter estimator. In order to facilitate this we need to form the I-branch (Q-branch) which has leakage from Q-branch (I-branch), as another complex number with a clean Q-branch (I-branch), which can be easily derived from the Q-branch (I-branch) of the previous decisions. And thus the rotation and scaling of the I-branch and Q-branch are now seen as common mode with respect to the decisions. This is how the IQ imbalances are viewed as channel gain and phase (common mode) imbalances. The conversion process is explained further in the next paragraph. When there is IQ imbalance in the RF front-end, sequence z reflects this condition. We propose splitting the z sequence into I and Q sequences zI and zQ , and also splitting the a into its corresponding I and Q sequences aI and ˆ ˆ aQ . For estimation purposes, within the estimator block, ˆ we form two new sequences u = zI + aQ , v = zQ + aI . ˆ ˆ Both these sequences do not have IQ imbalance but have gain and phase offset with respect to hard decisions a. The ˆ conversion process is illustrated in Fig.4. Then by using a proper estimator that we develop later in equations (14) and (15) these offsets can be computed and corrected for by scaling and derotation. After the correction is done, the proper I and Q sequences corresponding to z are paired to reconstruct the channel sample sequence which is now equalized for IQ imbalances. Now let us proceed on the development of the estimator. We suggest the reader to refer to reference [9] for a complete treatment on the development of the solution, and we only provide a brief overview. By treating the parameters b = [A, θ, g,φ], and a as determin- M step : arg max˜{Q(˜ ˆn−1 )} b, b (13) b The E step corresponds to data detection assuming the estimated channel ˜ and then formulating the Q function b using the a-posteriori probabilities p(a|r, ˆn ) that are in b fact the extrinsic probabilities that are generated by the SISO detector. For our channel conditions the received K−1 RF signal r(t) can be written as, r(t) = A k=0 ak p(t − jθ kT )e + w(t) and the estimates can be derived as [9] ; ˆ θn = arg{ | ˆ An = K−1 k=0 K−1 ∗ k=0 ηk zk | K−1 k=0 ρk 2 ˆ a A |ak | p(ak = a|r, bn−1 ) ∗ ηk zk } (14) (15) where, ρk = and ηk = ˆn−1 ). Here ηk can be viewed as soft a A ak p(ak = a|r, b symbol estimates from the detector. Refer to [8] for the behavior of the estimator as an IQ compensator. Code-Aided Adaptive Decorrelator We propose a solution for IQ imbalance correction in a BICM_ID as a cascade of adaptive decorrelator and the batch estimator. These two techniques working alone may not be optimal in terms of convergence and steady state performance, but both aiding each other in a synergic way results in a superior solution. There are several possible operating modes using a cascade of adaptive decorrelator and a batch estimator, the cascade can work in the open loop mode or in the closed loop mode. Furthermore, the adaptive decorrelator can work in the data-repeat mode or non data-repeat mode. In the data-repeat mode, the channel samples are recirculated in the adaptive algorithm over and over again during the turbo iterations, which helps in faster convergence. In this work we consider the case of non data-repeat closed loop configuration of the adaptive decorrelator and the batch estimator. Since we have developed both the components we present our z'I + + _ zI hi LMS algorithm Code-aided Estimator BASE BAND hq _ z'Q + + zQ Figure 4: Adaptive Decorrelator with acquisition aid from the Codeaided Estimator. Algorithm 1 Acquisition-aided AD (for a variable number of TURBO Iterations, AD is not embedded in the TURBO loop, steps related to channel estimation are not shown) Wopt = 0; 1. AD iteration; 2. Perform TURBO iteration 3. Is BER achieved? Yes Go to 8 No Go to 4 4. Estimate IQ phase imbalance 5. Is the estimate less than threshold or misadjustment is tolerable? No Go to 7 Yes Go to 2 7. Compute the weights using Eq. 1 and 2 and add to WI and WQ on the penultimate TURBO iteration, Go to 2 8. Process New packet Go to 1. 10 0 proposed approach as an algorithm shown as Algorithm 1 and illustrated by the block diagram in Fig.5. The accuracy of the embedded estimator is dependent on the BER on the first iteration and this in turn dependents on the Eb/No and other impairments present. In the case when the BER is high, the estimator takes several turbo iterations to estimate the imbalances accurately, through successive improvement in BER over the iterations. When enough iterations are allowed (through iteration control on every packet) to achieve the required BER, the best estimate of the IQ imbalance occurs on the penultimate iterations. The weights in the AD filter are updated on the last turbo iteration of the current packet and the AD is ready to continue with the updated values on the new packet. It can be so arranged that these estimated values can be treated as fixed component of the weights and then adaptive decorrelator will continue on the adaptive component of the weight. The step size for the adaptive decorrelator algorithms needs to be properly chosen so that the embedded estimator has better performance. We envision a control block that monitors error vector magnitude (EVM), Eb/No, BER and thus controls the number of iterations, step size and also when to disable the embedded estimator. Through simulations the step size and the number of iterations for the estimator to contribute to the filter Wopt , can be determined. The batch estimator can work with either hard or soft decision feedback, and the difference in performance can be quantified through exhaustive simulations. In terms of complexity, hard decision may be favored since the encoder and mapper on the transmit side can be used for remapping. Another point about complexity is the additional resources needed for implementing Eq. 1, Eq. 2 and Eq. 6 using the estimated IQ imbalance values (which involves evaluation of trigonometric functions and division of complex numbers). 10 −1 Gray 10 −2 Binary Binary, iter3 Binary, iter2 Binary, iter1 Gray BER 10 −3 10 −4 10 −5 10 −6 8 9 10 11 Eb/No 12 13 14 Figure 5: BER vs. Eb/No, Binary Labeling and g = 0 dB, φ = 0;. and a constraint length of 7, a random interleaver of size 5760 bits and a non-Gray mapper for a 64QAM modulation format. A block of 2880 bits are encoded into a block of 5760 bits, and after random interleaving, the bits are mapped to a packet of 960 64QAM symbols. Since the estimator is embedded in the BICM-ID loop, there may be an EM estimator iteration for one of the detector iteration and this will be decided by the control block based on the error signals from the AD block. For the example system, we have a baseline BER performance of 10−5 on the third iteration at Eb/No of 13.5 dB. Fig.5 shows the BER vs. Eb/No curve. As can be seen, there is a mild threshold in the detector behavior. This threshold corresponds to the waterfall region of the BICM_ID BER curve, and for Eb/No values below this region there will be no convergence and hence no successful estimation and convergence. For Eb/No value of 13.5 dB, and for IQ imbalance values of (1.74 dB, 15o ), the EM estimator aided adaptive decorrelator is able to completely correct the imbalances and allow the detector to reach the error floor or the desired BER of 10−5 . The learning curves of the adaptive algorithms are presented in Fig.6. The step size and step size control philosophy is the same for both code-aided and non-aided versions of AD. In the following, we present several cases to illustrate the performance of the estimator. SIMULATION RESULTS Case 1. g = 1.74 dB, φ = 15o ; up to these imbalance The transmitter consists of a rate-1/2 non-systematic con- values there is no IL and target BER 10−5 on the third itvolutional encoder with a generator polynomial (133,171) eration is maintained at Eb/No of 13.5 dB. 0.15 [2] 0.1 Wi [3] Un-Aided Adaptive Decorrelator, Wi Acquisition Aided Adaptive Decorrelator 0.05 0 -0.05 0 [4] 0.5 1 1.5 2 2.5 time 3 3.5 4 4.5 x 10 5 5 [5] 0.15 [6] 0.1 Un-Aided Adaptive Decorrelator, Wq Wq 0.05 Acquisition Aided Adaptive Decorrelator 0 -0.05 0 [7] 0.5 1 1.5 2 2.5 time 3 3.5 4 4.5 x 10 5 5 [8] Figure 6: Learning curves for the W I and W Q of aided and un-aided AD, φ = 15o , g = 1.74 dB [9] [10] Case 2. When there is channel impairment present, g = 0.5 dB, φ = 10o , θ = 5, A = 0.95; for these imbalance [11] values there is no IL and the target BER is achieved on the third iteration at Eb/No of 13.5 dB. [12] CONCLUSIONS We have proposed a code-aided adaptive decorrelator to compensate IQ imbalances in an iterative receiver for bursty wireless channel. This technique is a combination of an adaptive decorrelator (AD) and an estimator embedded in the iterative detector, two techniques that are well understood and implemented in practical systems. The embedded estimator is developed based on the EM algorithm and configured as an acquisition aid to the AD. The proposed technique converges fast without the need of training symbol and offers tracking capability also. Convergence behavior under varying channel conditions needs to be studied further through analysis and simulations. Complexity issues relating to the implementation of the proposed algorithm are also under investigation. And finally the generality of the proposed algorithm is also under study to extend to non iterative receivers and to frequency selective fading channel (OFDM). R EFERENCES [1] Berrou. C., Glavieux, A., and Thitimajshima, P., “Near Shannon limit error-correcting coding and decoding: Turbo-codes”, [13] [14] [15] [16] [17] Proc. ICCC’93, May 1993, pp. 1064-1070. E. Zihavi, “Eight-PSK trellis codes for a Rayleigh channel”, IEEE Trans. Comm., vol. 40, pp. 873-884, May 1992. Giugno, L., Lottici, V., Luise, M., "Efficient compensation of I/Q phase imbalance for digital receivers," ICC 2005, vol. 4, pp. 2462 - 2466, 16-20 May 2005. Van Gerven S., Van Compernolle D., “Signal separation by symmetric adaptive decorrelation: stability, convergence, and uniqueness, IEEE Trans. on Signal Processing, vol.43, pp. 1602-1612, July 1995. M.Valkama, “Adaptive DSP techniques for I/Q imbalance compensation in communication receivers,” half-day tutorial presentation in ISCAS 06, Kos Island, Greece, May 2006. Fred Harris, “Digital filter equalization of analog gain and phase mismatch in I-Q receivers”, 5th IEEE International Conference on Universal Personal Communications, Vol 2, pp. 793 - 796, Sept.-2 Oct. 1996. Marcus Windisch and Gerhard Fettweis, "Standard-independent I/Q imbalance compensation in OFDM Direct-Conversion receivers," Proc. 9th intl. OFDM Workshop (InOWo), pp.15-16, Sept. 2004. Raghunath Cherukuri and Poras T. Balsara, “Iterative(TURBO) IQ imbalance estimation and correction in BICM-ID for flat fading channels,” accepted for publication, VTC2007. N. Noels et al., “Turbo-synchronization: an EM algorithm approach,” in Proc. IEEE ICC Anchorage, May 2003. J. Hagenauer, “The turbo principle: tutorial introduction and state of the art,” in Proc. Int. Symposium on Turbo Codes and Related Topics, pp.1-11, Brest, France, Sep. 1997. S. ten Brink, et al., “Iterative demapping and decoding for multilevel modulation,” in Proc IEEE Globecom’98, pp. 779-584, Sydney, Australia, Nov. 1998. L. R. Bahl, et al., “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Infor. Theory, vol. 20, pp 284-287, Mar. 1974. A. P. Dempster, et al., “Maximum-likelihood from incomplete data via the EM algorithm,” J. Roy. Soc., Ser.B, vol. 39, no.1, pp 1-38, Jan. 1977. Chia-Hung Hsu, Chih-Feng Wu and Chorng-Kuang Wang, “FPGA prototype for WLAN OFDM baseband with STPE of I/Q mismatch self calibration algorithm,” in Proc. Asian Soldstate Circ. Conf., pp 509- 512, Nov. 2005. Lauri Anttila, Mikko Valkama, and Markku Renfors, “Blind Moment estimation techniques for I/Q imbalance compensation in quadrature receivers,” in PIMRC06., pp 1- 5, Sept. 2006. Imtinan Elahi, Khurram Muhammad, and Poras T. Balsara, “I/Q mismatch compensation using adaptive decorrelation in a lowIF receiver in 90-nm CMOS process,” in IEEE J. Solid-State Circuits,., vol. 41, no. 2, pp 395-404, Feb. 2006. Supisa Lerstaveesin, and Bang-Sup Song “A Complex image rejection circuit with sign detection only,” in IEEE J. Solid-State Circuits,., vol. 41, no. 12, pp 2693-2702, Dec. 2006.