FREQUENCY-DOMAIN IQ-IMBALANCE AND CARRIER FREQUENCY OFFSET COMPENSATION FOR OFDM OVER DOUBLY SELECTIVE CHANNELS

Transcription

1 FREQUENCY-DOMAIN IQ-IMBALANCE AND CARRIER FREQUENCY OFFSET COMPENSATION FOR OFDM OVER DOUBLY SELECTIVE CHANNELS Imad Barhumi, and Marc Moonen Deartement Elektrotechniek-ESAT, Katholieke Universiteit Leuven Kasteelark Arenberg 1, 31, Leuven (Heverlee), Belgium hone: , fax: , ABSTRACT In this aer we roose a frequency-domain IQ-imbalance and carrier frequency offset (CFO) comensation and equalization for OFDM transmission over doubly selective channels IQ-imbalance and CFO arise due to imerfections in the receiver and/or transmitter analog front-end, whereas user mobility and CFO give rise to channel time-variation In addition to IQ-imbalance and the channel time-variation, the cyclic refix (CP) length may be shorter than the channel imulse resonse length, which in turn gives rise to inter-block interference (IBI) While IQ-imbalance results in a mirroring effect, the channel time-variation results in inter-carrier interference (ICI) The frequency-domain equalizer is roosed to comensate for the IQ-imbalance taking into account ICI and IBI The frequency-domain equalizer is obtained by transferring a timedomain equalizer to the frequency-domain resulting in the so-called er-tone equalizer (PTEQ) 1 INTRODUCTION Orthogonal frequency division multilexing (OFDM) has been adoted for digital audio and video broadcasting [1] and chosen by the IEEE 8211 standard [2] as well as by the HIPERLAN-2 standard [3] for wireless local area networks (WLAN) This is due to its robustness against multi-ath fading channels and its simle imlementation But OFDM is sensitive to analog front-end imerfections; mainly the amlitude- and hase- imbalances (IQ-imbalance) and the carrier frequency offset (CFO) In OFDM a cyclic refix (CP) with a length equal to or longer than the channel delay sread is required to maintain orthogonality between sub-carriers This is deending on the fact that ideal conditions are satisfied such as: no IQ-imbalance is resent, zero CFO, and the channel is timeinvariant (TI) over the OFDM block eriod In ractice it is difficult to satisfy all of these conditions On the one hand, IQ-imbalance and CFO are resent due to the analog front end imerfections, in articular when low comlexity low cost receivers/transmitter are sought On the other hand, the channel time-variation arises due to user mobility and CFO Different aroaches have been roosed to overcome the analog front-end roblems for OFDM transmission In [4] a training based-technique for CFO estimation is roosed assuming erfect IQ-balance A maximum likelihood (ML) CFO estimation is roosed in [5], also assuming erfect IQ-balance The IQ-imbalance only roblem is treated in [6, 7, 8] assuming zero CFO Joint comensation of IQ-imbalance and CFO is treated in [9, 1] In [9] it is assumed that the CFO is corrected based on erfect knowledge of the IQ-imbalance arameters, and the IQ-imbalance arameters can be estimated correctly in the resence of CFO The assumtion This research work was carried out at the ESAT laboratory of the Katholieke Universiteit Leuven, in the frame of Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office IUAP P5/11 ( Mobile multimedia communication systems and networks ) The scientific resonsibility is assumed by its authors here is valid only for small CFO and small IQ-imbalance arameters, and so these algorithms are unable to achieve the desired accuracy for moderate to large IQ-imbalance arameters and large CFO values In [1], nulled sub-carriers are used to estimate the CFO by maximizing the energy on the designated sub-carrier and its image In an earlier work [11] the authors roosed frequency-domain IQ-imbalance and CFO comensation for OFDM transmission over time-invariant channels There the CP-length was also assumed to be shorter than the channel imulse resonse length However, in the above mentioned works, the channel is assumed to be TI, and the CP length is consistently assumed to be longer than or equal to the channel imulse resonse length In this aer we roose a frequency-domain er-tone equalizer (PTEQ) to equalize the channel and comensate for the IQimbalance The channel is assumed to be time-varying due to user mobility and/or CFO, and the CP-length may be shorter than the channel imulse resonse length The PTEQ is obtained by transferring a time-domain equalizer to the frequency-domain The resulting PTEQ combines adjacent sub-carriers and their mirrors to combat the effect of ICI/IBI and to comensate for IQ-imbalance This aer is organized as follows In Section 2, we introduce the system model In Section 3, the er-tone equalizer is roosed Our simulations are introduced in Section 4 Finally, our conclusions are drawn in Section 5 Notation: We use uer (lower) bold face letters to denote matrices (column vectors) Suerscrits, T, and H reresent conjugate, transose, and Hermitian, resectively We denote the exectation as E { } and the Kronecker roduct as We denote the N N identity matrix as I N, the M N all-zero matrix as M N The kth element of vector x is denoted by [x] k Finally, diag{x} denotes the diagonal matrix with vector x on the diagonal 2 SYSTEM MODEL We consider an OFDM transmission over a time-varying frequencyselective channel We assume a single-inut single-outut (SISO) system, but the results can be easily extended to single-inut multile-outut (SIMO) or multile-inut multile-outut (MIMO) systems At the transmitter the information-bearing symbols are arsed into blocks of N frequency-domain QAM symbols Each block is then transformed to the time-domain by the inverse discrete Fourier transform (IDFT) A cyclic refix (CP) of length ν is added to the head of each block The time-domain blocks are then serially transmitted over the time-varying channel When no IQimbalance is resent, the discrete time-domain baseband equivalent descrition of the received signal at time index n is given by: L y[n] = g[n;l]x[n l]+v[n], l= where g[n; θ] is the discrete time equivalent baseband reresentation of the time-varying frequency-selective channel taking into account

2 the multi-ath hysical channel and the transmitter and receiver ulse shaing filters as well as the effect of CFO (viewed as art of the channel time-variation) L is the channel order L = τ max /T +1 with τ max the channel maximum delay sread v[n] is the discrete time additive white noise (AWN), and x[n] is the discrete timedomain sequence transmitted at a rate of 1/T symbols er second Assuming S k [i] is the QAM symbol transmitted on the kth sub-carrier of the ith OFDM block, x[n] can be written as: x[n] = 1 N 1 N S k [i]e j2π(m ν)k/n, k= where i = n/(n + ν) and m = n i(n + ν) Note that this descrition includes the transmission of a CP of length ν In the resence of IQ-imbalance, namely an amlitudeimbalance of a and hase-imbalance of φ, the baseband equivalent received sequence at time index n is given by: r[n] = αy[n]+βy [n] (1) where the arameters α and β are given by [12]: α = cos(φ)+ jasin(φ) β = acos(φ) j sin(φ) 3 PER-TONE EQUALIZER In general a er-tone equalizer (PTEQ) is obtained by transferring a time-domain equalizer (TEQ) to the frequency-domain For the case of IQ-imbalance, the conventional TEQ is not enough to comensate for IQ-imbalance and reduce or eliminate IBI/ICI For this urose, two TEQs are alied, where one is used to filter the received sequence and the other one is used to filter a conjugated version of the received sequence The urose of the TEQs is to comensate for IQ-imbalance, equalize the time-varying channel and ossibly eliminate IBI In other words, the urose of the TEQs is to shorten the time-varying channel imulse resonse length to fit within the CP-length, eliminate the channel time-variation and finally comensate for the mirroring effect induced by the IQimbalance Assuming the time-varying TEQs w 1 [n;θ] and w 2 [n;θ] are alied to the received sequence in the fashion described above, the outut of the TEQ subject to some decision delay d can be written as: L z[n d] = w 1 [n;l ]r[n l L ]+ w 2 [n;l ]r [n l ], (2) l = l = where L is the order of the time-varying TEQs It was shown in [13, 14] that modeling the TEQ using the basis exansion model (BEM) is an efficient way to tackle the roblem of IBI/ICI for OFDM systems Using the BEM to model w 1 [n;l ] and w 2 [n;l ], the BEM equivalent of w 1 [n;l ] and w 2 [n;l ] can be written as: w a [n;l Q ] = w a,q,l [i]e j2πq n/k, for a = 1,2, (3) q = Q where 2Q + 1 is the number of basis functions, K is the BEM resolution of the time-varying TEQs taken as integer multile of the block size K = PN, where P is an integer 1 w a,q,l is the q th basis of the l th ta of the ath TEQ, which is ket fixed over a window length of N + L, and may change from window to window indeendently Substituting (3) in (2), and by using a block level formulation, we arrive at Q z[i] = D q [i]w H Q 1,q [i]r[i]+ D q [i]w2,q H [i]r [i], (4) q = Q q = Q where z[i] = [z[i(n + ν) + ν],,z[(i + 1)(N + ν) 1]] T, r[i] is the received block in the ith OFDM block after removing the CP and taking into account the time-domain filter san and the decision delay defined as r[i] = [r[i(n + ν) + ν + d L ],,r[(i+1)(n + ν)+d 1]] T, D q [i] is a diagonal matrix with the q th time-varying basis comonents on its diagonal D q [i] = diag{[e j2πq (i(n+ν)+ν+d)/k,,e j2πq ((i+1)(n+ν)+d 1)/K ] T }, and W a,q [i] is an (N + L ) N Toelitz matrix with the first column equal to [w a,q,l [i],,w a,q, [i], 1 (N 1) ] T and the first row equal to [w a,q,l [i], 1 (N 1) ] Since we are only interested in the ith block (without loss of generality), and for the sake of a simle notation the block index i will be droed form now on By means of a 1-ta frequency-domain equalizer, an estimate of the transmitted symbol on the kth sub-carrier in the ith OFDM block can be written as: S k = 1 γ k F (k) z, (5) where γ k is the 1-ta frequency-domain equalizer on the kth subcarrier in the ith OFDM block, and F (k) is the (k+1)st row of the unitary discrete Fourier transform (DFT) matrix F Transferring the TEQ to the frequency-domain, the estimate of the transmitted QAM symbol on the kth sub-carrier in the ith OFDM symbol can be written as: Ŝ k = F (k k ) D R ˆD w (k k ) 1, = k = K + F (k k ) D R ˆD w (k k ) 2,, (6) = k = K where 2K + 1 is the san of adjacent sub-carriers involved in the equalization rocess, ˆD = diag{[1,,e j2π L /K ] T }, w a, (k) = [w (k) a,,,,w(k) a,,l ] T, and R is an N (L + 1) Toelitz matrix with first column [r[i(n + ν) + ν + d],,r[(i + 1)(N + ν) + d 1]] T, and first row [r[i(n + ν)+ν + d],,r[i(n + ν)+ν + d L ]] For more details about the last ste the reader is referred to [13] The estimate in (6) corresonds to the so-called PTEQ in the frequencydomain Note that, we defined the second filter differently than the first filter and at the same time we unified the san of the different PTEQs to serve our urose and simlify the forthcoming analysis The estimate in (6) can now be equivalently written as: Ŝ k = w (k k )H 1, = k = K + w (k k )H 2, = k = K F (k k ) r where the (L + 1) (N + L ) matrix F (k) is given by: F (k) F (k) F = (k), F (k) F (k k ) r, (7) and r = D r where D is the th haseshift matrix in the time-domain given by D = diag{[e j2π (i(n+ν)+ν+d L )/K,,e j2π ((i+1)(n+ν)+d 1)/K ] T } In (7), the estimate of the transmitted symbol on the kth sub-carrier of the ith OFDM block is obtained by erforming a sliding DFT

3 on the received sequence, ie by erforming an N-oint DFT within a sliding-window of size L The oututs of the sliding DFT (L + 1 oututs) are fed to the PTEQ equalizer The imlementation comlexity of the sliding DFT can be significantly reduced by erforming only one DFT and comensate for the other sliding DFTs by means of L difference terms, as exlained in the following roerties [15]: F (k) r = T (k) (k) R 1 1 r L 1, (8a) + N-Point FFT k K k k + K 1,, K 1,,K and F (k) r = T (k) R (N k) r 1 1 L 1, (8b) r[n] N k K N k N k + K 2,, K 2,,K where R (k) is the kth sub-carrier frequency resonse of the received sequence on the th branch defined as R (k) = F (k)[ r [i(n + ν)+ ν + d],,r [(i+1)(n + ν) + d 1] ] T, and T (k) is the circulant shift matrix corresonding to the kth sub-carrier, which is an (L + 1) (L + 1) lower triangular Toelitz matrix given by: 1 T (k) = δ k, (9) δk L δ k 1 with δ k = e j2πk/n The difference terms vector r is given by: Defining v (k)h a, ( Ŝ k = v (k k )H 1,,k [r ] L [r ] N+L r = [r ] 1 [r ] N+1 = w a, (k)h T (k), (7) can be written as: [ R (k k ) r ] [ +v (k k )H 2, R (N k ) r ]) (1) Notice that the difference terms are common to all sub-carriers which allows for a further reduction in comlexity To do so, we first collect the first element of the vectors v (k k ) a,, for k { K,,K } in the vector ũ (k) a,, ie ũ (k) a, = [v (k K ) ) a,,,,v(k+k a,, ]T Second, we sum over the remaining elements that corresond to the difference terms as ū (k) a, = K [v (k k ) ) k = K a,,1,,v(k k a,,l ] T Collecting these two vectors in one vector as a, = [ũ (k)t a,,ū (k)t a, ] T, (1) can now be written as: ( P 1 Ŝ k = H 1, = R (k K ) R (k+k ) r +u(k)h 2, R (N k K ) R (N k+k ) r ) (11) The imlementation of (11) is shown in Figure 1 Define a = [T a,,,u(k)t a,p 1 ]T, (11) can be written in a comact form as: Ŝ k = H 1 A (k) r+h 2 A (N k) r, (12) j2π(p 1)n/K e + N-Point FFT k K k k + K N k K N k N k + K 1,P 1, K 1,P 1,K 2,P 1, K 2,P 1,K Figure 1: PTEQ for OFDM over doubly selective channel with IQimbalance where A (k) = [F (k)t,,f (k)t P 1 ]T, with 1 L F (k K ) F (k) = 1 L F (k+k ) D, Ī L L (N L ) Ī L with Ī L is an anti-diagonal identity matrix of size L L Due to the IQ-imbalance, the kth sub-carrier and its mirror the N kth sub-carrier are combined to obtain an estimate of the transmitted symbol on the kth sub-carrier for k {1,,N/2 1} The same holds for estimating the transmitted symbol on the (N k)th sub-carrier This suggests, that a roer equalizer estimates the transmitted symbol on the kth sub-carrier and the one transmitted on the (N k)th sub-carrier in a joint fashion For this urose we can obtain the following: [ ] Ŝ k H ŜN k = 1 H 2 A (k) r u (N k)t }{{} 2 u (N k)t A (N k) r (13) 1 }{{}}{{} s k U (k)h à (k) At this oint we may introduce a model for the received sequence r Note that due to the time-domain filter san and the decision delay, the received sequence is written to cover three consecutive OFDM blocks; i 1, i and i+1 blocks In the absence of Ŝk[i]

4 IQ-imbalance, the received sequence can therefore be written as: y = [O 1,G,O 2 ](I 3 P) (I 3 F H) s[i 1] s[i] +v[i], (14) }{{} s[i+1] G where y is similarly defined as r, O 1 = (N+L ) (N+2ν+d L L ), O 2 = (N+L ) (N+ν d), and G is an (N + L ) (N + L + L) matrix reresenting the time-varying channel g[n;l] g[n;] G =, g[n ;L] g[n ;] BER QAM, φ=5, a=1, K =5, L=6, ν=6, N=128 with IQ no com with IQ com 16 QAM over TI channels where n = i(n + ν)+ν + d L, and n = (i+1)(n + ν)+d 1 P is the CP insertion matrix given by: ν (N ν) I P = ν, I N and s[i] = [S [i],,s N 1 [i]] T is the vector of QAM symbols transmitted on the ith OFDM block v is the noise vector similarly defined as r Note that, we can also aroximate the channel using the BEM with window size N N indeendent of the BEM resolution of the TEQs In this aer we restrict ourselves to the channel definition given earlier Hence, we can write the received sequence and its conjugate as in (15) shown at the to of next age There Z 1 is an N N matrix defined as: Z 1 = 1 Ī N 1 To obtain the PTEQ coefficients for the kth and (N k)th subcarriers (ie solve for U (k) ), we define the following mean-squared error (MSE) cost function: { } J = E s k U (k)h r 2 r The minimum MSE (MMSE) solution can then be obtained as: U (k) = argmin U (k) J (16) The solution of (16) is obtained by solving J / U (k) = and is equal to: U (k) =(Ã(k) ( HR s H H +Rṽ)Ã(k)H ) 1Ã(k) HR s [e 3N+k+1 e 4N+k+1 ], (17) where R s and Rṽ are the transmitted sequence covariance matrix, and the noise covariance matrix resectively, and e k is a 6N long unity vector with a 1 at the k th osition The roosed PTEQ unifies and extends many reviously roosed PTEQs for OFDM In this context the roosed PTEQ extends and unifies: The PTEQ for OFDM transmission over doubly selective channels with erfect IQ-balance roosed in [16] The PTEQ for OFDM transmission over TI channels with IQimbalance and CFO roosed in [11] There only critically samling is used since CFO was the only source of time-variation SNR (db) Figure 2: BER vs SNR for OFDM over doubly selective channels ν = L With no IQ-imbalance, no CFO, and the channel is TI, the roosed PTEQ boils down to the PTEQ roosed in [17] for xdsl The imlementation comlexity of the roosed PTEQ is P(2K + L + 1) multily-add (MA) oerations er sub-carrier lus O(PN log 2 N) MA oeration for the FFTs The design comlexity of the PTEQ (see (17)) on the other hand, is of O(N 3 ) MA oerations for N 2P(2K + L + 1) 4 SIMULATIONS In this section we resent some of the simulation results of the roosed equalization technique for OFDM transmission over doubly selective channels We consider an OFDM system with N = 128 sub-carriers The doubly selective channel is assumed to be of order L = 6 The channel tas are simulated as iid random variables with uniform ower delay rofile, correlated in time according to Jakes model with correlation function r h (τ) = J (2π f max τ), where J is the zeroth-order Bessel function of the first kind, with maximum Doler sread f max = 1Hz and samling time T = 1µsec The IQ-imbalance arameters are assumed to be known at the receiver with amlitude-imbalance a = 1 and hase-imbalance φ = 5 16-QAM signaling is used in the simulations We measure the erformance in terms of BER vs SNR In the first setu we assume the CP-length ν fits within the channel imulse resonse length The oversamling factor is assumed to be P = 1 The PTEQ is designed to have order L =, and ICI san of K = 5 The simulation results are shown in Figure 2 The IQ-imbalance (if not roerly comensated for) results in a significant degradation of the system erformance For this setu, the IQ-imbalance results in a BER error floor at BER= The PTEQ with IQ-imbalance comensation enhances the erformance significantly, which roughly coincides with that of 16-QAM OFDM transmission over TI channels, esecially for low to moderate SNR values In the second setu, we assume the CP-length is shorter than the channel imulse resonse length The CP-length ν = 3 in this case The PTEQ is designed to have order L = 8, and the ICI san is K = 2 We consider the critically samled case P = 1 as well as the oversamled case with oversamling factor P = 2 As shown in Figure 3, IQ-imbalance degrades the erformance significantly for the critically samled as well as for the oversamled case, where a BER error floor is again observed at BER= The PTEQ with IQ-imbalance comensation enhances the system erformance

5 [ r r ] = [ α G(I 3 P) ( I 3 F H) β G (I 3 P) ( I 3 F H) ] β G(I 3 P) ( I 3 F H) α G (I 3 P) ( I 3 F H) }{{} H s[i 1] s[i] s[i+1] Z 1 s [i 1] Z 1 s [i] Z 1 s [i+1] +ṽ (15) BER QAM, φ=5, a=1, K =2, L=6, ν=3, N=128 P=1 (critically samled) P=2 (oversamled) with IQ no com with IQ com 16 QAM over TI channels SNR (db) Figure 3: BER vs SNR for OFDM over doubly selective channels ν = 3 significantly for both the critically samled and the oversamled cases and aroaches the erformance of that of 16-QAM OFDM transmission over TI channels A slight enhancement is observed for the case of oversamling over the critically samled case, where an SNR gain of 1 db is observed at BER= CONCLUSIONS In this aer we have roosed a frequency-domain er-tone equalizer (PTEQ) for OFDM transmission over doubly selective channels with IQ-imbalance and CFO (viewed as art of the doubly selective channel) The PTEQ is designed to equalize the channel and comensate for the IQ-imbalance The channel is assumed to be time-varying, and the CP-length may be shorter than the channel imulse resonse length The PTEQ is obtained by transferring a time-domain equalizer to the frequency-domain The channel and the IQ-imbalance arameters are assumed to be known at the receiver The resulting PTEQ combines adjacent sub-carriers and their mirrors to combat ICI/IBI and comensate for IQ-imbalance While IQ-imbalance degrades the system erformance significantly, the roosed PTEQ aroaches the erformance of OFDM transmission over TI channels with erfect IQ-balance REFERENCES [1] ETSI Standard EN v141, ETSI Digital Video Broadcasting (DVB); Framing Structure, Channel Coding and Modulation for Digital Terrestrial Television, Aug 21 [2] IEEE 8211 Task Grou a, Part 11, Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Secifications: High-seed Physical Layer in the 5 GHz Band, 1999 [3] ETSI TS V122 (21-2), ETSI Broadband Radio Access Networks (BRAN): HIPERLAN Tye 2: Physical (PHY) Layer, Aug 1999 [4] P H Moose, A Technique for Orthogonal Frequency Division Multilexing Frequency Offset Correction, IEEE Trans Commun, vol 42, , Oct 1994 [5] J H Yu and Y T Su, Pilot-Assisted Maximum-Likelihood Frequency-Offset Estimation for OFDM Systems, IEEE Trans Commun, vol 52, , Nov 24 [6] J Tubbax, B Come, L V der Perre, L Deneire, S Donnay, and M Engels, Comensation of IQ Imbalance in OFDM Systems, in IEEE Int Conf on Communications, (Anchorage, Alaska USA), May, [7] C-L Liu, Imacts of I/Q Imbalance on QPSK-OFDM- QAM Detection, IEEE Trans Consumer Electron, vol 44, , Aug 1998 [8] A Targhiat and A H Sayed, On the Baseband Comensation of IQ Imbalance in OFDM Systems, in IEEE Int Conf on Acoustics, Seech, and Signal Processing, (Montreal, Canada), May 24 [9] J Tubbax, A Fort, L V der Perre, S Donnay, M Engels, M Moonen, and H D Man, Joint Comensation of IQ Imbalance and Frequency Offset in OFDM Systems, in IEEE Global Communications Conference, (San Francisco, CA USA), December, [1] S Fouladifard and H Shafiee, Frequency Offset Estimation in OFDM Systems in Presence of IQ Imbalance, in ICCS, (Singaore), Nov 22 [11] I Barhumi and M Moonen, IQ Comensation for OFDM in the Presence of IBI and Carrier Frequency-Offset, IEEE Trans Signal Processing, Jan 26 (acceted) [12] M Valkama and V Koivunen, Advanced Methods for I/Q Imbalance Comensation in Communication Receivers, IEEE Trans Signal Processing, vol 49, , Oct 21 [13] I Barhumi, G Leus, and M Moonen, Per-Tone Equalization for OFDM over Doubly Selective Channels, in IEEE Int Conf on Communications, (Paris, France), June 24 [14] I Barhumi, G Leus, and M Moonen, Time-Domain and Frequency-Domain Per-Tone Equalization for OFDM over Doubly Selective Channels, Signal Processing, vol 84/11, , 24 Secial Section Signal Processing in Communications [15] E Jacobsen and R Lyons, The Sliding DFT, IEEE Signal Processing Mag, vol 2, 74 8, Mar 23 [16] I Barhumi, G Leus, and M Moonen, Equalization for OFDM over Doubly Selective Channels, IEEE Trans Signal Processing, vol 54, , Ar 26 [17] K van Acker, G Leus, M Moonen, O van de Wiel, and T Pollet, Per-tone Equalization for DMT-based Systems, IEEE Trans Commun, vol 49, Jan 21