A Frequency-domain Compensation Scheme for IQ-Imbalance in OFDM Receivers

 Shu Feng, Wang Mao, Shi Xiajie, Liu J unhao, Sheng Weixing, and Xie Renhong Abstract—A special pilot pattern across two OFDM sy mbols is devised for channel estimation in OFDM systems with IQ imbalance at receiver. Based on t his pilot pattern, a high-efficiency time-domain (TD) least square (LS) channel estimator is proposed to suppress channel noise by a factor of N /( L +1) in comparison with the frequency-domain LS one in [1] where N a nd L+1 are the total number of subcarriers and the length of cyclic prefix, respectively. Following this , a low-complexity frequency-domain (FD) Gaussian elimination (GE) equalizer is proposed to eliminate IQ distorti on by using only 2 N complex multiplications per OFDM symbol. From simulation, the proposed TD-LS/FD-GE scheme using only two pilot OFDM symbols achieves the same bit error rate ( BE R) p erformance under ideal channe l knowledge and no IQ imbalances at low and medium signal-to-noise ratio (SNR) regions whereas these co mpensation schemes including FD-LS/Post-FFT LS, FD-LS/Pre-FFT Corr, and SPP/Pre-FFT Corr in [1] require about twenty OFDM training symbols to reach the same performa nce wh ere A/B denotes comp en sation scheme with A being channel estimator and B being equalizer. Index Terms—IQ imbalance, equalizer, channel estima tion, time domain, frequency domain, least square. I INTRODUCTION Orthogonal frequency division multip lexing (OFDM) has been adopted in several standards such as wireless local area network (IEEE 802.11a, g and n) , wireless metropolitan area network (IEEE 802.16d, e and m), digital audio broadcasting, LTE/LTE-advan ced, digital radio mondi ale and digital video broadcasting. Compared with the he terodyne receiver, the direct convers ion RF receiving architecture is recently reconsidered as a promising solution in OFDM system s to reduce the cost and power consumption of the receiver [1]-[3]. However, the latter is severely dist orted by gain and phase imbalances between the I and Q path s due to imperfections of the anal og components [1]-[3]. This will Manuscript received June 21, 2011. This work was supported in part by the open resear ch fund of National Mobile Communications Research Laboratory (No. 2010D13), Southeast University, China. The authors are with the Departme nt of Communication E ngineering, Nanjing Univer sity of Science and Technology, Nanjing, China. Shu Feng is also with National Mobile Communications Research Laboratory, South east University, China (email: shuf eng@mail.njust.edu.cn, wang mao 123@gmail.com ). A Frequency-domain Compensation Scheme for IQ-Imbalance in OFDM Receivers severely destroy the orthogonality among the OFDM su bcarriers and cause inte rcarrier interference, giving rise to a high bit error rate (BER) fl oor. Therefore, estimation and co mpensation of IQ imbalance in the direct conversion receivers are cr ucial to OFDM receiver perform ance. The schemes of canceling IQ imbalan ce have been investigated by severa l scholars. In [ 1], the authors derive the SNR loss of IQ-imbalance in OFDM rece ivers and propose several frequency domain(FD) and time-domain (TD) m ethods including post-FFT least- squares, adaptive least mean square (LMS) and pre-FFT TD compensation to eliminate IQ distortions . They extend these methods to IQ imbalances at both transmitter and receiver [ 4]. Blind estimation and compensation sch emes in the time dom ain have also been proposed [5]. Joint estimation of IQ imb alance and several other im pairments such as phase noise, frequency offset are inves tigated in [6]-[10]. In [6], a fi nite impulse response (FIR) fi lter f ollowed by an asymmetric phase compensator has been pr oposed to correct both frequency dependent and frequency independent IQ imbalance. In [8], a differential fi lter is employed to estimate the f requency offset and IQ imbalance. A compensation m ethod base d on the subcarrier alloca tion of OFDM signals is proposed in [9]. [11] extends the research of Tx/Rx IQ imbalances to the case of packet-switched systems. In [12] and [13], authors focus on pilot design an d reduced c omplexity compensation in MIMO-OFDM systems with IQ imbalance. Unfortunately, the FD LS channel estimation in [1] doesn’t exploit the TD property of the channel in the presence of IQ imbalance. Thus, it requires more than twenty training OFDM symbols to achieve the same BER performance with ideal channel knowledge and no IQ-imb alance (abbreviated as id eal IQ below). Obviously, this scheme is low on bandwidth ef fi ciency. To overcome this problem , we design an LS channel estimator which f ully exploits the TD prope rty of channel param eters to reduce the impact of channel noise. Hence, it requires only two OFDM sy m bols to reach the BER pe rformance of ideal IQ. Notations: Bold letters denote vectors and m atrices.  T  ,  *  ,and  H  denotes transpose, conjugate, and conjugate transpose operations, respectively. Operation diag( x ) places vector x on diagonal of a diagonal matrix. n I and 0 n× m are the nn  identity and nm  zero matrices, respectively. This paper is organized as follows. Section II describes the system m odel. Pilot p attern, TD-LS channel estimation and Gaussian elimination (GE) equalization are proposed in se ction III. Simulation results are listed in section IV. Sect ion V concludes the paper. II SYSTEM MODEL In OFDM systems with IQ imbalance as shown in Fig.1, the transmitted block of N data symbols over N subcarriers is denoted as      =1 2 T ss s N     s  (1) whose IDFT operation yields H s= F s (2) where      -2 π -1 -1 1 =e x p jn m m,n N N    F (3) =- 1 j   1, 2, , m,n N   Then, the received OFDM symbol before being distorted by IQ im balance is expressed as H y= F Λ Fs + w (4) where   di ag  H  and (- - 1 ) 1 = NL     h HF 0 (5) and       T =[ 1 2 1 ] hh h L  h  is the channel impulse response (CIR). The received OFDM sym bol distorted by IQ imbalance is written as * =+ μν zy y (6) where   = c o s /2 + s i n /2 μθ j αθ and     =c o s / 2 - s i n / 2 να θ j θ , θ and  are phase and a mplitude imbalance between I and Q branches[1].Taking FFT operation on (6) gives     ## =+ + μν zH s H s w diag diag (7) where the operation # is de fi ned as [1]           #* * * * * * T =[ 1 2 /2 +2 / 2 + 1 /2 2 ] X X XN XN XN X X  (8) where            T =[ 1 2 /2 /2+ 1 /2+2 ] X X XN XN XN XN X  (9) From [1], if = XF x , then  # #* == XF x F x (10) Thus, we obtain the following equality     ## #* * * == = XF x F x X (11) For the simplicity of discussion, bloc k fading is assum ed in the following, i.e., channel is assumed to be constant within a frame and variable from frame to frame. III PROPOSED SCHEME COMBINING PILOT DESIGN, EQUALIZATION AND CHANNEL ESTIMATION In the following, a low-complexity Gaussian elimination is adopted to cancel the IQ distortion based on operation #. Then, a particular training pattern us ing two adjacent OFDM sym bols is designed and a TD-LS channel estimation is presented to provide a high-precision estim ation of channel parameters * ν / μ , μ H and * ν H . A. Gaussian Elimination Equalizer Due to the equalities,     # ## diag = di ag Hs H s and     # ## diag = dia g Hs H s , making # operation on (7) yields   # #* * # # =d i a g + d i a g + νμ zH s H s w (12) Assuming * ν κ = μ is known, based on (7) and (12), we construct     #* # -= - d i a g + - κμ κ ν κ zz H s w w (13) (13) no longer has the IQ distortion present in (7) and can be rewritten as     #* # -= d i a g - d i a g + - κμ κ ν κ zz H H s ww (14) which gives the following detector as      -1 *# ˆ =d i a g -d i a g - μκ ν κ sH H z z (15) Equation (15) can be simpli fi ed as        # * - = - ν k κ k k μ k κ k zz s HH  (16) To obtain s from (15) or (16), κ ,  H , and *  H need to be estimated in advance, among which  h and *  h are  -1 1 = N- L μ μ      h HF 0 (17) and  * * -1 1 = N- L ν ν      h HF 0 (18) Similar to [1], the loss in signal-to-noise (SNR) fr om the difference between th e error variance given by (16) and the error variance  2 / 2 w σ k H is  2 22 2 * 1+ Loss i n SN R = 10l og -2 R e + κ μκ ν μ κ ν     (19) where Re ( x ) denotes the real part of x . Fig. 2 shows the theoretical lower bounds concerning the loss in SNR due to IQ imbalances. The 2D surfaces of loss in SNR are based on the resu lts (19) derived in Sectio n III and (31 ) in [1]. These bounds are computed with perfect channel and distortion param eter knowledge available at the receiver, therefore serving as the theoretical lower bou nds on the SNR lo ss due to imbalances. From this figure, the lower bound of the GE equalizer proposed in this letter is better than that of the LS equalizer in [1]. B. Pilot Pattern Design and TD-LS Estim ation of Channel Parameters. Let us devise the frequency-domain pilot vectors of two pilot OFDM symbols at the beginning part of frame as  /2-1 1 = N η η         p 1 s s 0 (20) and  /2 -1 1 = N j η j η         2 p 0 s s (21) where 2 s η =P with s P being the average transmit power for signal constellation and p s is an /2 1 N  dimensional column pilot vector with      H s tr E = - 1 NP pp ss . After the two pilot sy mbol vectors passes through multipath channel, we get the following rece ived training vectors and symbols in frequency domain as follows         2: / 2 = d i a g 2: / 2 + 2: / 2 N μ NN 1p 1 zH s w (22)         # /2 + 2 : = d i a g /2 + 2 : + /2 + 2 : NN ν NN NN 1p 1 zH s w  (23)         # 2: / 2 = d i a g 2: / 2 + 2: / 2 N ν NN 2p 2 zH s w  (24)        / 2+2: = d i a g / 2+2: + / 2+2: NN μ NN NN 2p 2 zH s w (25) and     * 1= 1 + 1 + 1 μη ν η 11 zH H w (26)     * / 2 +1 = / 2 +1 + / 2 +1 + / 2 +1 N μ N ην N η N 11 zH H w (27)     * 1= 1 - j ν 1+ 1 j μη η 22 zH H w (28)     * / 2 +1 = / 2 +1 - / 2 +1 + / 2 +1 Nj μ N η j ν N η N 22 zH H w (29) Then, combining (26)-(29) forms the following equations           -0.5 1 + 0.5 1 = 1 - 0 .5 1 + 0 .5 1 j μη j 21 2 1 zz H w w (30)      * 0.5 1 + 0. 5 1 = 1 + 0.5 1 + 0. 5 1 j νη j 21 2 1 zz H ww (31)           -0.5 / 2 + 1 + 0 .5 / 2 + 1 = / 2 + 1 - 0 .5 / 2 + 1 + 0 .5 / 2 + 1 jN N μ N η jN N 21 2 1 zz H w w (32)   0. 5 / 2 + 1 + 0. 5 /2 + 1 = jN N 21 zz    * /2 +1 + 0 .5 /2 +1 + 0 . 5 /2 +1 ν N η jN N 21 Hw w (33) where  # 1 =/ 2 + 2 : NN p ss  . Stacking (22), (25), (30) and (32) gives a large matrix-vector form of         - 0.5 1 + 0.5 1 2 : / 2 == - 0.5 / 2 + 1 + 0.5 / 2 + 1 / 2 + 2 : j N jN N NN        21 1 a 21 2 zz z z zz z          - 0.5 1 + 0.5 1 2 : / 2 diag + -0.5 / 2 + 1 + 0.5 / 2 + 1 / 2 + 2 : j η N μ η jN N NN               a 21 p 1 21 p 2 w ww s w H ww s w            11 diag diag N- L - ? η μ μ η            p p ap a p s s h =H + w = s F + w 0 s        (34) Thus, the LS estimate of μ h is given as   -1 -1 HH = diag = + diag TD LS μμ   pa p a hP F s z h P F s w   (35) where +1 ( +1 ) ( - -1 ) = LL N L    PI 0 . Then, we have the estimate of μ H as LS (- 1 ) 1 = TD LS N- L μ μ         h HF 0 (36) In the same manner, combining (23), (24), (31), and (33) into a large ma trix-vec tor form yields                 # # 0.5 1 + 0.5 1 0.5 1 + 0.5 1 2 : / 2 2 : / 2 =d i a g + 0.5 / 2 + 1 + 0.5 / 2 + 1 0.5 / / 2 + 2 : b jj η NN ν η jN N j N NN                p 21 21 p 2 2 21 2 p 1 s zz ww s zw zH zz w s z        2+ 1 +0 . 5 / 2+ 1 / 2 + 2 : N NN        b 1 1 w w w       (37) whose # operation forms         * ## # # ~~ ~ ~ # * -1 1 = = diag + = diag + bb N- L ν h ν      bb pp zz H s w s F w 0   (38) which gives the LS estimate of *  h     -1 -1 ~~ ~ ** = diag = + diag HH TD LS νν   # b b pp hP F s z h P F s w  (39) Then, we have the estimate of *  H as  * * -1 1 = TD LS TD LS N- L ν ν          h HF 0 (40) In terms of (35), (36), (39) , and (40), the estimate of  can be formulated as ** +1 ** =1 =1 ** +1 =1 =1 () () == () () NL TD LS TD LS kk LS NL TD LS TD LS kk ν k ν k κ μ k μ k        Hh Hh (41) From (35), (36), (39), and (40), we obtain the estimation m ean square errors of *  H and  H as follows E- - = H TD LS TD LS μμ μμ N              HH HH  ** ** ** E- - +1 = H TD LS TD LS νν νν L β NN γ                HH HH (42) where  is signal-to-noise ratio (SNR) and is de fi ned as     * E/ 2 n kk σ ss [14], and          * * E = E1 / 1 / kk β kk ss ss (43) IV SIMULATION AND DISCUSSION In the following, a typical OFDM system is simulate d to evaluate the perfor mance of the proposed scheme against an ideal IQ OFDM receiver, a receiv er with no compensation schem e, and those aforementioned compensation schem es, FD-LS/Post -FFT LS, FD-LS/Pre-FFT Corr, and SPP/Pre-FFT Corr in [1]. where A/B denotes compensation sche me with A being channe l estim ator and B being equalizer. Simulation parameters were : OFDM sym bol length N=128 , cyclic prefix L+1=16 , signal bandwidth BW=2MHz, digital m o dulation QPSK, carrier frequency f c =2GHz. A typical urban (TU) channel was employed in the simulation as in [15]. Figs. 3 to 4 compare the proposed scheme with the FD-LS channel estim ator plus Post-FFT LS equalizer (FD-LS/Post-FFT) in [1], for different va lues of IQ imbalance param eters where N T denotes the number of consecutive training OFDM sym bols (TOSs) with all subcarriers c arrying pilot symbols at the beginning of each frame as shown in [1]. Here, our sc heme uses two TOSs as shown in (21) and (20). From these figures, it is evident tha t the proposed scheme with only two TOSs achieves the sam e BER performance as ideal IQ at low m edium SNRs whereas the LS schem e in [1] costs about N T =2 T O S s t o realize almost the same BER perform ance. Therefore, the proposed scheme is more effective in the sense of overhead. Figs. 5 and 6 plot the curves of BER versus SNR of the proposed TD-LS/FD-GE scheme, the FD-LS/Post-FFT LS, the FD-LS channel estimator pl us pre-FFT distortion co rrection (FD-LS/Pre-FFT Corr), and the special pilot structure based channel estimator plus p re-FFT distortion correction (SPP/Pre-FFT Corr) in [1] for different values of IQ imbalance parameters where N T =2 . In these two figures, our scheme is obviously better on BE R performance than the FD-LS/Post-FFT, the FD-LS/Pre-FFT Corr, and SPP/Pre-FFT Corr in [1]. The complexity of three channel estimators the proposed TD-LS, FD-LS in [1] and the SPP in [1] are      23 22 41 21 4 / 3 l o g l o g 1 LN LN N N N L       , 2 36 64 / 3 0.5 log TT N NN N N N   , and  2 2 1 0.5 log TT NN N N  complex multiplic ations (CMs) where N F is the total number of non-training OFDM symbols. Clearly, three channel estimation ha s almost com plexity. The computational am ounts of three equalizer FD-GE, Post-FFT LS, and Pre-FFT Corr are 2 30 . 5 l o g FF N NN N N  , 2 32 / 3 2 0. 5 l og FF N NN NN N  , and   2 20 . 5 l o g F N NN N  CMs. The proposed FD-GE and Pre-FFT are slight lower on com plexity than Post -FFT LS. Thus, we conclu de that four schemes including the proposed TD-LS/FD -GE scheme, the FD-LS/Post-FFT LS, FD-LS/Pre-FFT Corr, and SPP/Pre-FFT Corr has almost the same computational am ount. Hence, our scheme is very attractive for mitigating IQ imbalance in practic al OFDM receivers. Additionally, our scheme can be a pplied to the case of frequency-de pendent IQ imbalance parameters  and *  like the FD-LS /Post-FFT LS scheme in [1] (  and *  depends on the subcarriers k (frequency-domain), these time-domain can not solve this problem, in general, when bandwidth <20MHz, they can be viewed as constants [1]). However, these TD compensation schemes based on pre-FFT (TD) distortion correction are not su itable for this case [1] . V CONCLUSIONS In this paper, a compensation scheme com bining a TD-LS channel estimator and a FD GE equalizer is investigated in OFDM systems with IQ-imbalance at receiver. Com pared with the FD-LS/Post-FFT LS, SPP/Pre-FFT Corr, and FD-LS/Pre-FFT Corr schemes in [1], this scheme shows better BER performance. More importantly, it needs only two OFDM training sy mbols to achieve the same BER performance as ideal IQ in the low and medium SNR regions. The proposed TD-LS/FD-GE can function in the case of frequency-de pendent distortion parameters  and *  . However, the schemes based on pre-FFT correction lack this capability. Due to a short training pattern, the proposed scheme can be directly applied to tim e-variant wireless channels. R EFERENCES [1] A. Tarighat, R. Bagheri , and A. H. 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[15] ETSI TR 125 943, Universal Mobile Telecommunications System (UM TS); Deployment (3GPP TR 25.943 Version 5.1. 0 Release 5), J une 2002. 12 (1 ) () s s N                  s  ˆ (1 ) ˆ ˆ () s s N          s     ** ,, / v   HH s y z Fig. 1.Discrete baseband OFDM syst ems with IQ imbalance at receiver 13 Fig. 2 Comparison of loss in SNR 0 0. 5 1 1. 5 2 2. 5 0 5 10 15 20 0 5 10 15 20 P has e I m bala nc e(degree) Gai n Imbal anc e(dB ) Los s in SN R ( dB) P rops oed GE LS equal iz er in [ 1] 14 Fig. 3 Comparison of BER perfo rmance for t h e proposed scheme with two training OFDM symbols and the LS in [1] with different numbers of training symbols in the case of o 2   and 1   dB 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SN R ( dB) BER Q PSK , ph ase i m bal an ce=2 degr ee, gain i m balan ce=1dB I deal I Q N o C ompen sat i on F D -L S /P o s t-F F T LS i n [1 ], N T =2 F D -L S /P o st -F F T L S i n [1] ,N T =4 F D -L S /P o s t-F F T LS i n [1 ], N T =8 F D -L S /P o s t-F F T LS i n [1 ], N T =3 2 Proposed T D -LS/ FD -G E, N T =2 15 Fig. 4 Comparison of BER perfor mance for the proposed scheme with t wo training OFDM symbols and the LS in [1] with different numbers of trainin g symbols in the case of o 20   and 4   dB. 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SN R ( dB) BER Q P SK , ph ase imbal an ce=20 d egr ee, gai n i mbal an ce=4 dB I deal I Q N o C om pen sat i on F D - L S /P o s t-F F T L S i n [1 ], N T =2 F D - L S /P o s t-F F T L S i n [1 ] ,N T =4 F D - L S /P o s t-F F T L S i n [1 ], N T =8 F D - L S /P o s t-F F T L S i n [1 ], N T =3 2 Propos ed T D -LS/ FD -G E , N T =2 16 Fig. 5 Comparison of BER performance for t he proposed scheme and three schemes in [1] for two OFDM training symbols in the case o f o 2   and 1   dB. 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 S NR(dB ) BER QP S K , P has e Imbal a nc e= 2 degree, Gai n Imbal anc e= 1 dB Ideal FD - LS/Po st - FF T LS [1 ] P ropos ed T D-LS / F D-GE FD - LS/Pr e - FFT C o rr [1 ] SPP/Pr e - FF T C o r r [1 ] 17 Fig. 6 Comparison of BER perfor mance for the pr oposed scheme and three schem es in [1] for two OFDM training symbols in the case of o 20   and 4   dB. 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 Q P SK ,Ph ase I mbal an ce=20degr ee, G ai n I mbal a n ce=4dB S NR(d B ) BER I deal FD- LS/ Post - FFT LS [ 1] Proposed T D -LS/ FD -G E F D -L S /P re-F F T C o rr [1 ] SPP/ Pr e- FFT Cor r [ 1]