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The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08) REDUCED-RANK LS CHANNEL ESTIMATION FOR MIMO-OFDM SYSTEMS Emiliano Dall’Anese, Antonio Assalini and Silvano Pupolin University of Padua - Department of Information Engineering (DEI) Via Gradenigo 6/B, 35131, Padova, Italy e–mail: {edallane, assa, pupolin}@dei.unipd.it A BSTRACT nique generalizes to the MIMO case known results obtained for SISO-OFDM systems [2, 4]. In this paper reduced-rank least–squares (rr-LS) channel es- The proposed rr-LS scheme is also compared with LMMSE timation for MIMO-OFDM systems is presented. Conven- [3] estimation from the point of view of both estimation perfor- tional LS solution is ﬁrstly introduced and a low complexity mance and computational complexity. In particular, we show per-subcarrier based version derived. Then, estimation mean that for not too long channels, rr-LS and LMMSE perform square error is shown to improve when a reduced-rank ap- close to each other but with the rr-LS having signiﬁcantly proach is adopted. In that case, MIMO channel estimation is lower complexity. reﬁned in time domain by setting to the real value the length of the estimated channel. It results that rr-LS represents a conve- The rest of this paper is organized as follows. In Sect. II nient choice performing close to the linear MMSE solution. we introduce the MIMO-OFDM system and the adopted channel model. Sect. III describes LS estimation including I I NTRODUCTION per-subcarrier and reduced-rank approaches. Sect. IV reports the training sequence structure. Simulation results are pre- Orthogonal frequency division multiplexing (OFDM) is an ef- sented in Sect. V while Sect. VI concludes this paper. ﬁcient multicarrier transmission system for high data rate com- munication over frequency-selective fading channels. To im- Notation: E[·] represents expectation. δ(·) is the Dirac pulse. prove the capacity of wireless systems, multiple antennas can The superscripts (·)T , (·)∗ , (·)H , and (·)−1 represent transpose, be used at both transmitter and receiver sides jointly with conjugate, transpose conjugate (Hermitian), and matrix inver- OFDM. Such a transmission scheme is referred to as multiple- sion, respectively. The superscript (·)+ represents the Moore- input multiple-output (MIMO) OFDM system [1]. Penrose inverse matrix (pseudoinverse matrix), i.e., A+ = The achievable performance with a coherent detection of the (AH A)−1 AH , A ∈ CM ×N , where the columns of A are lin- transmitted information symbols may be compromised by im- early independents. x ∼ CN(mx , Rx ) denotes a random vec- perfect knowledge of the channel state information (CSI) at the tor x with complex circularly-symmetric Gaussian distributed receiver. Therefore, accurate channel estimation techniques re- entries, mean mx and correlation matrix Rx . A = diag(a) sult of basic importance in order to ensure reliable data recov- is a diagonal matrix with the elements of a ∈ CM ×1 on its ery over frequency-selective fading channels. However, since main diagonal. A = bdiag(a1 , . . . , aN ) ∈ CN M ×N P is a in MIMO-OFDM systems there is a higher number of propa- block diagonal matrix with the matrices ai ∈ CM ×P as ele- gation links to be characterized, channel estimation becomes ments. · 2 is the squared Euclidean norm. ⊗ denotes the more complex compared with the single-input single-output Kronecker product. vec[·] is the matrix linear transformation (SISO) case. Hence, it is relevant to achieve a good trade-off known as matrix vectorization which stacks the columns of a between estimator complexity and global system performance. matrix A ∈ CM ×N on a M N × 1 vector. Channel estimation is often based on least-squares (LS) [1, 2] and linear minimum mean square error (LMMSE) [2, 3] techniques. Incidentally, while LS estimator handles the II S YSTEM AND C HANNEL MODEL channel coefﬁcients as deterministic but unknown constants, We consider a point-to-point MIMO-OFDM wireless system MMSE-based solutions require perfect knowledge of the sec- with Nt transmit and Nr receive antennas as depicted in Fig. 1. ond order statistical description of the channel, i.e., power In the following we introduce the system and channel models. delay proﬁle (PDP), spatial correlations, etc. . Nevertheless, MIMO channel statistics depends on many random compo- II.A Transmitter nents which joint estimation may result not sufﬁciently accu- (t) (t) (t) (t) rate and then compromise MMSE estimation capabilities. Let Ak [Ak,0 , Ak,1 , . . . , Ak,M −1 ]T be the information data In this paper, we study a simple and efﬁcient channel estima- vector modulated at time instant kT in correspondence to the tor which presents good performance but lower computational tth transmit antenna, t = 1, 2, . . . , Nt , with M the number complexity than LMMSE. In particular, we ﬁrstly introduce the of OFDM subcarriers. We assume data symbols with average well-known (full-rank) frequency domain LS estimator [1, 2]. 2 energy equal to σA E[|Ak,m |2 ], ∀ m. The modulated vector Successively, in order to save computational complexity, we is extended with a cyclic preﬁx (CP) of Lcp < M samples. implement an LS solution working on a per-subcarrier basis. Therefore, let B be the transmission bandwidth and Tc 1/B Thereinafter, we derive a reduced-rank (rr) LS estimator that the channel sampling time, then the modulation rate is equal to signiﬁcantly outperforms the full-rank LS approach. That tech- 1/T with T (M + Lcp )Tc . The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08) Now, we deﬁne (1,1) (1,N ) Hk,m ... Hk,m t . .. . H k,m = . . . (2) . . (Nr ,1) (Nr ,N ) Hk,m ... Hk,m t as the Nr × Nt matrix with elements given by the sam- ples of the channel frequency response (CFR) of the MIMO channel in correspondence to the mth OFDM tone, m = 0, . . . , M − 1. Moreover, we also introduce the fol- Figure 1: MIMO-OFDM transmission chain. lowing Nr Nt × 1 column vectors H k,m and hk,l obtained by stacking the columns of H k,m and the columns of hk,l , II.B Channel Model respectively, i.e., H k,m = vec[H k,m ], and hk,l = vec[hk,l ]. Finally, let us deﬁne H k = [H T , . . . , H T −1 ]T and k,0 k,M The adopted channel model has Lc distinct Tc -spaced taps. In particular, at time instant kT , the lth component of the MIMO hk = [hT , . . . , hT c −1 ]T , then the coefﬁcients of the MIMO k,0 k,L channel is characterized by: i) an Nr × Nt Gaussian random CFR and channel impulse response (CIR) are related as D matrix Gk,l , ii) a random angle-of-departure (AoD) θk,l , iii) a H k = (F Lc ⊗ I Nr Nt ) hk , (3) A random angle-of-arrival (AoA) θk,l and iv) a propagation delay τk,l . The Lc paths are assumed to be mutually uncorrelated [5]. hk = F +c L ⊗ I Nr Nt H k , (4) Performance of MIMO systems strongly depends on the where [F Lc ]m,l = exp(−j2πlm/M ), m = 0, 1, . . . M − 1, spatial correlation that arises when the separation among ei- l = 0, 1, . . . Lc − 1, is a sub-block of the discrete Fourier trans- ther transmit or receive antennas is insufﬁcient. In this paper, form (DFT) matrix F = [F ]m,l , m, l = 0, 1, . . . M − 1. F +c L we assume that both transmit and receive sides are equipped represents the Lc × M Moore-Penrose inverse matrix of F Lc . with uniform antenna arrays having omnidirectional elements spaced of dt and dr meters, respectively. Therefore, with the II.C Receiver model given in [6], the spatial correlation matrix for the lth tap, At the receiver, we assume perfect time and frequency syn- Rl ∈ CNr Nt ×Nr Nt , is given by the Kronecker product of the chronization. After CP removal and OFDM demodulation, the spatial correlation matrices at the receiver RR,l ∈ CNr ×Nr and resulting signal at the rth receive antenna is given by the super- at the transmitter RT,l ∈ CNt ×Nt as Rl = RR,l ⊗ RT,l . In imposition of Nt components. We can write the received data particular, for small angle spreads the spatial correlation coef- sample, at time kT for the mth OFDM tone, as ﬁcients can be approximated as ([5]) Nt (r) (r,t) (t) (r) 2 Yk,m = Hk,m Ak,m + nm , (5) −j2π∆t (m−n) cos θk,l − 1 (2π∆t (m−n)δt sin θk,l ) D D [RT,l ]m,n ≈ e e 2 , t=1 2(r) 2 where nm represents thermal noise with nm ∼ CN(0, σn ) for [RR,l ]m,n ≈ e A −j2π∆r (m−n) cos θk,l e−1 2 ( A 2π∆r (m−n)δr sin θk,l) , any m and r. Here, we deﬁne the average signal-to-noise ratio 2 2 (SNR) per subcarrier as SN R = σA /σn and we collect the where δt and δr are the angle spreads, while ∆t = dt /λc and (1) (2) (Nr ) samples (5) as Y k,m = [Yk,m , Yk,m , . . . , Yk,m ]T ∆r = dr /λc are the antenna spacing normalized to the wave- length λc corresponding to the transmission carrier frequency. Y k,m = H k,m Ak,m + nk,m , (6) Therefore, the lth path, l = 0, . . . , Lc − 1, of the MIMO (1) (2) (N ) where Ak,m = [Ak,m , Ak,m , . . . , Ak,m ]T . t channel at time kT can be represented by the matrix hk,l ∈ CNr ×Nt deﬁned as III C HANNEL E STIMATION 1/2 1/2 H In this section, we ﬁrst introduce the well-known full-rank fre- hk,l = σhl RR,l Gk,l RT,l , (1) quency domain LS estimator [1, 2]. Then, in order to cut the where the Nr × Nt matrix Gk,l has independent circularly- required computational complexity, we give an LS formula- symmetric zero-mean complex Gaussian distributed entries tion deﬁned on a per-subcarrier basis. Thereafter, we derive a 2 with unitary variance. The value of the average power σhl is “reduced-rank” (rr) LS estimator which is analytically shown obtained by properly sampling the power delay proﬁle (PDP), to signiﬁcantly outperform the full-rank LS approach. More- Lc −1 2 which is normalized as l=0 σhl = 1 in order to ﬁx the chan- over, for completeness, performance comparison also includes nel energy despite of the number of present scatterer clusters. LMMSE estimation [3]. The number of channel taps is given by Lc = ⌊Bτrms ⌋ ≤ Lcp , In this paper, performances are measured through the esti- with τrms the channel root-mean square time delay spread. mation mean square error (MSE) deﬁned as We also assume that the channel is static over at least Nt 1 2 consecutive OFDM symbols. M SE E Hk − Hk , (7) Nt Nr M The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08) where H k is the CFR vector given in (3) while its estimation The last sign of equality holds being data and noise mutually is denoted by H k . In (7) expectation is taken over time. The independent and, as addressed in Sect. IV, Ak is designed following proposition states as the MSE can equivalently be such as training sequences are orthogonal and then AH Ak is k evaluated by considering either the CFR or the CIR [3, 7]. diagonal AH Ak = diag(αk )I Nr Nt M , k (11) Proposition 1 : Let M be the number of subcarriers of the where the elements of αk ∈ CNr Nt M ×1 can be found from the OFDM modulators and let hk and H k be the estimated coefﬁ- speciﬁc realization of Ak . cients of the MIMO CIR and CFR, respectively, then III.B Least-Squares per-subcarrier estimator (LS-PS) 1 2 2 E Hk − Hk =E hk − hk . (8) With the objective of reducing computational complexity of LS M estimator, in the following, we describe a criterion that works Proof: Let ∆H k = H k − H k and ∆hk = hk − hk be the on a per-subcarrier (PS) basis. In principle, complexity is re- Nr Nt M ×1 and Nr Nt Lc ×1 vectors representing the CFR and duced by dividing the channel coefﬁcient vector H k into a set CIR estimation errors, respectively. From the basic relation of blocks of smaller size and then by performing CFR estima- ∆H k = (F Lc ⊗ I Nr Nt )∆hk , it follows that tion independently on each sub-block. Therefore, let Ak,m = [Ak−Nt +1,m , . . . , Ak,m ] denote the E ∆H H ∆H k = sequence of symbols transmitted on the mth subcarrier at time k instants (k − Nt + 1)T, . . . , kT . We can write from (6) H = E ∆hH (F Lc ⊗ I Nr Nt ) (F Lc ⊗ I Nr Nt ) ∆hk k Yk,m = H k,m Ak,m + Nk,m , (12) = E ∆hH k F Hc F Lc L ⊗ I Nr Nt ∆hk where Ym,k = [Y k,m , . . . , Y k−Nt +1,m ] is the Nr × Nt matrix containing the received samples on the mth subcar- = E ∆hH M I Nr Nt Lc ∆hk = M E ∆hH ∆hk . k k rier at time instants (k − Nt + 1)T, . . . , kT , and Nk,m = [nk,m , . . . , nk−Nt +1,m ]. In this case the minimization of the III.A Least Squares estimator (LS) LS−P S 2 In this section, we introduce the channel estimation criterion functional Yk,m − H k,m Ak,m | gives known as frequency domain LS estimator [1, 2]. LS−P S Let Y k,m = [Y T , Y T k,m T k−1,m , . . . , Y k−Nt +1,m ] T be the H k,m = Yk,m A+ . k,m (13) signal samples received at time instants (k − Nt + 1)T, . . . , kT The complete set of M Nr Nt channel fading coefﬁcients is in correspondence to the mth OFDM tone for all re- LS−P S T T T T ceive antennas and let Yk = [Y T Y T , . . . , Y T −1 ]T be k,1 k,2 k,M Hk = H k,0 , H k,1 , . . . , H k,M −1 , where H k,m = the Nr Nt M × 1 column vector holding all received sam- LS−P S ples. Moreover, we also deﬁne Ak,m = [Ak,m ⊗ vec[H k,m ], m = 0, . . . , M − 1. We note (13) leads to the same MSE as for the LS estimator I Nr , Ak−1,m ⊗ I Nr , . . . , Ak−Nt +1,m ⊗ I Nr ]T , where Ak,m (9). In particular, an error component Nk,m A+ is present on k,m is given in Sect. II.C. Finally, we introduce the follow- a generic mth subcarrier. ing Nt Nr M × Nt Nr M block diagonal matrix Ak = bdiag Ak,0 , Ak,1 , . . . , Ak,M −1 . Note that the number of III.C Reduced-rank LS estimator (rr-LS) consecutive OFDM symbols used for channel estimation is With a PS approach the LS estimator presents a signiﬁcantly ﬁxed to Nt . lower complexity, but the MSE is still high. Therefore, to With the above notation the received symbol column vec- achieve a better estimation accuracy we extend the reduced- tor reads Yk = Ak H k + nk . Therefore, frequency domain LS rank strategy for single-antenna systems [2] to the MIMO case. LS estimation H k is obtained by minimizing the functional Let us suppose the receiver knows the length of the CIR of LS 2 Yk − Ak H k yielding [7] the MIMO channel. Then, “reduced-rank” estimation of the CIR, hrr−LS ∈ CNr Nt Lc ×1 , is obtained as k LS 2 H k = A+ Yk , k (9) rr−LS hk = arg min H k LS−P S − (F Lc ⊗ I Nr Nt ) h . (14) h where A+ k = (AH Ak )−1 AH k k is the pseudo-inverse of Ak . The solution is LS technique presents an high MSE. In fact, directly from LS−P S (7) and by (9), we ﬁnd that LS estimation is affected by a rr−LS hk = F +c ⊗ I Nr Nt H k L . (15) noise component A+ nk which leads to the following estima- k + tion MSE per subcarrier Note that (F Lc ⊗ I Nr Nt ) = F +c ⊗ I Nr Nt . L rr−LS 1 2 Therefore, the estimated CFR H k becomes M SELS = E A+ nkk Nr Nt M rr−LS LS−P S 2 Hk = (F Lc ⊗ I Nr Nt ) F +c ⊗ I Nr Nt H k L . σn 1 (16) = E . (10) LS−P S Nt |Ak,m |2 = F Lc F +c ⊗ I Nr Nt H k L . The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08) Table 1: Complex operations required by the considered estimation techniques. Estimator (×) (+) inv(·) LS 3 3 2 2 2Nr Nt M + Nr Nt M 2 2 Nt Nr M (2Nt Nr − Nt Nr − 1) inv(Nr Nt M ) LS-PS 3 2 M (2Nt + Nt Nr ) 3 2 2 M (2Nt + Nt Nr − 2Nt − Nt Nr ) M · inv(Nt ) rr-LS 3 2 M (2Nt + Nt Nr ) + M (M Lc + 2L2 + Nr Nt ) c 3 2 M (2Nt + Nt Nr ) + M (M Lc + 2L2 + Nr Nt ) c M · inv(Nt ) + inv(Lc ) −(M 2 + L2 + Lc M + Nt Nr ) c LMMSE 3 3 M Nt Nr (1 + Lc + Lc M ) 2N 2) + N 3N 3M 2L (Nt Nr − 1)(1 + Lc )(M Nt r 2 · inv(Nr Nt Lc ) t r c 2 2 2 2 +Nt Nr M 2 − M (Nt Nr Lc + Nt Nr ) We emphasize that the entries of the pseudo-inverse matrix by counting the number of operations on complex numbers re- F +c can be stored for some values of Lc of interest. L quired to realize the different criteria. Nevertheless, we note that the CIR length supposed at the In particular, (+) and (×) denote respectively the number receiver may be incorrect. In [4] the robustness of the reduced- of sums and products, while inv(C) represents inversion of a rank estimator to a wrong setting of the CIR length was inves- non-singular C × C matrix. In the analysis, we neglect the cost tigated for the single antenna case. of transpose-conjugate (Hermitian) operation and Kronecker We now compute the MSE of rr-LS estimator. For the products involving identity matrices. sake of simplicity, we suppose CFR estimation is performed In general, matrix inversion has a heavy impact on total com- by using (9). Therefore, from (9) and (15), we have that plexity. However, we note that under (11) inv(Nr Nt M ) and hk S = hk + F + ⊗ I Nr Nt A+ nk , and then the MSE is rr−P L k inv(Nt ) in Tab. 1 for (9) and (13) become straightforward. given by The analysis shows that LS-PS estimator requires a lower number of complex operations than full-rank LS. The imple- 1 2 M SErr−LS = E F + ⊗ I Nr Nt A+ nk L k mentation of rr-LS needs additional complexity than for LS-PS Nr Nt as emphasized by (16). Tab. 1 also shows the higher complex- 2 . σn 1 Lc ity of LMMSE with respect to rr-LS. = E 2 M . (17) Nt |Ak,m | IV O RTHOGONAL T RAINING S EQUENCES The last sign of equality holds under hypothesis (11). In this section we introduce the space-time structure of the III.D Remarks training symbols transmitted on each subcarrier for channel • In OFDM-based systems the channel length is much estimation purposes. In particular, sequences simultaneously shorter than M . In fact, usually Lc ≤ Lcp . From (10) transmitted from the Nt antennas are orthogonal in both space and (17), we note rr-LS signiﬁcantly outperforms full- and time and are Nt OFDM symbols long [1, 8, 9]. rank LS. In details, we obtain We focus on symbols transmitted on a given subcarrier m. Let us consider the Nt × Nt symbol matrix Ak,m = M SELS,dB − M SErr−LS,dB = 10 log10 (M/Lc ) , [Ak−Nt +1,m , . . . , Ak,m ] introduced in Sect. III.B. The inner (18) product of any two distinct rows (space dimension) or columns hence, the gain in estimation accuracy increases with the (time dimension) of Ak,m is zero if they are designed as in decreasing of the channel length Lc . [10], where orthogonal space-time block codes were proposed. In particular, with a BPSK mapping orthogonal sequences can • The estimation MSE of both estimators does not depend be easily generated from an Hadamard matrix for any value of on the second order statistics of the channel, e.g., PDP and Nt [8, 9]. The use of orthogonal sequences is also convenient spatial correlation. In fact, LS technique treats the chan- because it simpliﬁes the computation of the pseudo-inverse ma- nel coefﬁcients as deterministic but unknown constants. trices in (9) and (13). Other MIMO channel estimation criteria, such as LMMSE [3], exploit the spatial correlations to improve channel es- V N UMERICAL R ESULTS timation accuracy. However, the performance of such es- timators may severely be compromised if channel statistic We consider a MIMO-OFDM system with M = 64 and is not perfectly available. Lcp = 16. Channel estimation is performed in the frame preamble. Training sequences are construed by Hadamard or- • LS and rr-LS can be used for both pilot-aided and thogonal bases of length M [9]. decision-directed channel estimation. However, in both The normalized antenna spacing is ﬁxed to ∆t = ∆r = 1/2. cases pilot and data sequences should be generated as un- The AoDs and the AoAs are obtained from a Gaussian random derlined in Sect. IV in order to comply with (11). variable with zero mean and standard deviation 90◦ and 360◦ , respectively. The angle spread for all taps at both transmitter III.E Computational Complexity and receiver are the same and ﬁxed to 4◦ (2π/9 rad). The chan- In Tab. 1 we consider the computational complexity required nel has an exponentially decaying PDP truncated to its ﬁrst Lc for channel estimation. Speciﬁcally, complexity is measured taps and normalized to one. The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08) 0 0 10 10 LS LS rr−LS rr−LS, L =1 LMMSE c Lc = 2 rr−LS, Lc=2 −1 −1 10 10 L =4 rr−LS, Lc=4 c L = 16 rr−LS, Lc=6 c rr−LS, L =8 MSE c MSE −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Average SNR [dB] Average SNR [dB] Figure 2: MSE vs Average SNR per subcarrier, Nt = Nr = 2, Figure 3: MSE vs Average SNR per subcarrier, Nt = Nr = 4, Lc = 2, 4. Random AoDs and AoAs, 4◦ angle spreads. Lc = 1, 2, 4, 6, 8. Random AoDs and AoAs, 4◦ angle spreads. In Fig. 2 we set Nt = Nr = 2 and compare the MSE of full- [2] J. 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Fig. 3 reports the MSE performance of rr-LS for a system with Nt = Nr = 4 varying the number of channel taps. Also in o [5] H. B¨lcskei, D. Gesbert, and A. J. Paulraj, “On the Capacity of OFDM- Based Spatial Multiplexing Systems,” IEEE Trans. Commun., vol. 50, this case the behavior predicted by (17) is corroborated. MSE pp. 225-234, Oct. 2001. decreases linearly with SNR and it depends on the ratio be- [6] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. tween Lc /Nt for a ﬁxed M . Frederiksen, “A Stochastic MIMO Radio Channel Model With Exper- imental Validation”, IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. VI C ONCLUSIONS 1211-1226, Aug. 2002. [7] L. Huang, J. W. M. Bergmans, and F. M. J. Willems, “Low-Complexity In this paper least-squares channel estimation for MIMO- LMMSE-Based MIMO-OFDM Channel Estimation Via Angle-Domain OFDM systems was considered. Conventional full-complexity Processing,” IEEE Trans. Signal Proces., vol. 55, pp. 5668-5680, Dec. LS solution was recalled and a reduced-complexity per- 2007. subcarrier version reported. However, these solutions present [8] A. Dowler and A. Nix, “Performance Evaluation of Channel Estimation a high estimation MSE and then we proposed a reduced-rank Techniques in a Multiple Antenna OFDM System,” in Proc. IEEE Vehic. Tech. Conf., vol. 2, Orlando, FL, Oct. 2003, pp. 1214-1218. approach having improved estimation capabilities. Complexity [9] S. Sun, I. Wiemer, C. K. Ho, and T. T. Tjhung,“Training Sequence As- and performance were analyzed and the different techniques sisted Channel Estimation for MIMO OFDM,” in Proc. IEEE Wireless compared through both analytical evaluation of the MSE and Commun. and Networking Conf., vol. 1, New Orleans, LA, Mar. 2003, computer simulations considering a practical channel model. pp. 38-43. The proposed rr-LS solution is less complex than LMMSE but [10] V. Tarokh, H. Jafarkhani, and A. R. 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