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knowledge of the channel covariance matrix on the performance of a linear minimum mean-square-error (MMSE) estimator for multiple-input multiple-output ...
Linear MMSE MIMO Channel Estimation with Imperfect Channel Covariance Information Antonio Assalini, Emiliano Dall’Anese, and Silvano Pupolin University of Padova - Department of Information Engineering (DEI) Via Gradenigo 6/B, 35131, Padova, Italy e–mail: {assa, edallane, pupolin}@dei.unipd.it Abstract—In this paper, we investigate the effects of imperfect knowledge of the channel covariance matrix on the performance of a linear minimum mean-square-error (MMSE) estimator for multiple-input multiple-output (MIMO) channels. The estimation mean-square-error (MSE) is analytically analyzed by providing both a very tight lower bound and an upper bound. The proposed analysis is useful for the understanding of how estimation accuracy of the channel covariance matrix impacts on system performance, depending on the average signal-to-noise ratio (SNR) and specific propagation conditions. Conclusions are fully supported by numerical results.

I. I NTRODUCTION Channel estimation is known to play a key role in wireless systems. In fact, imperfect knowledge of the channel precludes optimal coherent signal detection and, consequently, it has a detrimental effect on the overall system capacity [1]. In MIMO systems, even more dramatically, channel estimation errors may constrict multiplexing gain at high SNRs [2], [3]. Moreover, when a feedback link between receiver and transmitter is available, the optimal MIMO power allocation depends on the accuracy of the channel state information [4]. Linear MMSE (LMMSE) channel estimation is often considered for performance analysis and comparison [2], [3] due to its good performance at any SNR value. A MMSE-based criterion requires second-order statistical knowledge of the radio channel to be estimated. In the literature it is frequently assumed that perfect channel covariance matrix information at the receiver (CCIR) is available for LMMSE implementation. Nonetheless, spatial correlation in MIMO systems has to be estimated too [5], [6] and then, in general, channel covariance matrix is imperfectly known at the receiver. In this paper, we study the case of imperfect CCIR (ICCIR) on system performance in order to get insights about the effect of channel covariance uncertainty on estimation accuracy. In particular, the performance of the LMMSE estimator is investigated by studying the behavior of the channel estimation MSE, and its relation with the eigenvalues of both real and estimated channel covariance matrices. The rest of this paper is organized as follows. Sect. II introduces the signal and channel models. Basic elements of MMSE channel estimation are provided in Sect. III, while in Sect. IV we report the study concerning with the effects of ICCIR on estimation MSE. Numerical results are given in Sect. V in order to validate, and graphically itemize, our remarks. Finally, Sect. VI concludes this paper.

Notation: Throughout this paper, the following notation is used frequently. The superscripts (·)T , (·)† , and (·)−1 represent transpose, transpose conjugate (Hermitian), and matrix inversion, respectively. E[·] represents expectation. The operator ⊗ denotes the Kronecker product. In is the n×n identity matrix. For a generic matrix A ∈ Cn×m , A(i,j) indicates the element of position (i, j). The operation vec(A) returns a column vector obtained by stacking the columns of A one below the other. For a square Hermitian matrix A, λi (A) represents its ith eigenvalue. In particular, we let λmin (A) and λmax (A) be the smallest and largest eigenvalue of A, respectively. II. S YSTEM M ODEL A. MIMO Signal Model We consider a narrowband point-to-point MIMO system with Nt transmit antennas and Nr receive antennas. The propagation links are flat Rayleigh fading channels that slowly vary in time, such that they can be assumed constant during the transmission of L consecutive pilot symbols from each transmit antenna. For the training phase the input-output signal model can be written as Y = HX + N ,

(1)

where Y is the Nr × L received signal matrix, X is the Nt × L complex matrix collecting the information symbols and H is the Nr × Nt channel matrix. N is the Nr × L matrix representing additive white Gaussian noise (AWGN) with independent zero-mean circularly-symmetric complex Gaussian entries with variance σn2 . In this paper we are interested in LMMSE channel estimation [7]. In [8] it is shown by information-theoretical means, that, for transmissions over spatially uncorrelated channels, LMMSE estimation performances are optimized by using training sequences X that are a multiple of a matrix with orthogonal columns, i.e., XX† = c L INt ,

(2)

being c ∈ R a power value dependent on the adopted  modulation format and transmit power constraint, tr XX† = cLNt . In a more general case, where spatial correlation is present, optimal training sequences and per-antenna transmit powers should be optimized by taking into account the spatial correlation matrices [9]. However, herein, we assume the transmitter has knowledge of neither the channel state nor the channel

correlation properties. Hence, training sequences are taken to meet (2) [5]. Moreover, we set L = Nt which is the minimum L so that the number of measurements is equal to the number of parameters to be estimated, XX† = X† X = cNt INt . Such a selection is optimal in many cases of interest as discussed in [8], although longer training sequences may provide smaller estimation MSE [5], but at the cost of a reduced spectral efficiency [8]. We define the average SNR per receive antenna as ρ  κ/σn2 , where κ = c Nt . B. Channel Model The entries of the channel matrix H are correlated Gaussian distributed with full Nt Nr × Nt Nr channel covariance matrix given by   † (3) R = E vec (H) vec (H) . Therefore, the spatially correlated MIMO channel can be represented as vec(H) = R1/2 vec(Hw ) ,

(4)

where Hw ∈ CNr ×Nt has i.i.d. complex Gaussian entries with zero-mean and variance 1/2 per dimension. More frequently a simpler model is considered posing [10] 1/2

H = R1/2 Hw Rt r

,

(5)

where Rt ∈ CNt ×Nt and Rr ∈ CNr ×Nr are, respectively, the transmit and receive spatial correlation matrices. Consequently, the channel correlation matrix results R = Rt ⊗ Rr ,

III. MMSE C HANNEL E STIMATION : P ERFECT CCIR In this section we review some basic results on LMMSE channel estimation with particular emphasis on the estimation error covariance matrix and the resulting estimation MSE. It is useful in the sequel to rewrite the input-output relation (1) as follows (7)

¯ is a Toeplitz matrix with first row and column equal where X to [x1,1 , 01:(Nr −1) , x2,1 , 01:(Nr −1) , . . . , xNt ,1 , 01:(Nr −1) ] and [x1,1 , 01:(Nr −1) , x1,2 , 01:(Nr −1) , . . . , x1,L , 01:(Nr −1) ]T , respectively, with xn,k = X(n,k) and 01:(Nr −1) a 1 × (Nr − 1) all zero row vector. We note that by (2), with L = Nt it can ¯ = κ INt Nr . ¯ †X ¯X ¯† = X be shown that X The linear MMSE estimation of the MIMO channel H is given by [7]  = RHY R−1 vec(Y) , vec(H) Y

  RHY = E vec(H) vec(Y)† ,   RY = E vec(Y) vec(Y)† .   Now, being E vec(N) vec(N)† = σn2 INt Nr the MMSE mation becomes    = RX ¯ † XR ¯ X ¯ † + σn2 INt Nr −1 vec(Y) , vec(H)

(8)

(9) (10) esti(11)

which can be rewritten by applying a well-known matrix inversion identity as1    = R−1 σ 2 + X ¯ −1 X ¯ † vec(Y) ¯ †X vec(H) n (12) −1 †  −1 2 ¯ vec(Y) . X = R σn + κ INt Nr  are The entries of the estimation error matrix EH  H − H known to be zero mean complex Gaussian random variables. With perfect CCIR, it is straightforward to show the error covariance matrix of EH reads   = E vec(EH ) vec(EH )† RCCIR E (13) −1  . = R−1 + ρ INt Nr Since R is Hermitian positive definite for assumption, then the inverse R−1 exists and it is still Hermitian, with eigenvalues given by 1/λi (R), i = 1, . . . , Nt Nr [14]. Consequently, the estimation MSE for a fixed R is equal to     2 MSECCIR (R) = E vec (EH ) = tr RCCIR E −1 N t Nr

1 +ρ = (14) λi (R)

(6)

with tr (R) = Nt Nr . The ability of the so called “Kronecker product model” to reflect the real channel behavior for any transmission scenario is still subject of discussion [11], [12]. In this paper we first consider the general model given by (3) and (4) with the assumption that R has full-rank (positive definite Hermitian matrix). The Kronecker model is used in Sect. V for numerical validation of the proposed analysis.

¯ vec(H) + vec(N) , vec(Y) = X

where

=

i=1 N t Nr i=1

λi (R) . 1 + ρ λi (R)

We note that the uncorrelated channel, i.e., for R = INt Nr , leads to the higher estimation MSE. IV. MMSE C HANNEL E STIMATION : I MPERFECT CCIR In practice a wireless device employing MMSE channel  of the channel covariestimation makes use of an estimate R ance matrix [5], [6]. In the following we study the effect that an imperfect CCIR has on the estimation MSE. We assume  ∈ CNt Nr ×Nt Nr is positive definite Hermitian like R is. R  in place of R the estimated Therefore, from (12) by using R channel coefficients become

−1  = R  −1 σ 2 + κ IN N ¯ † vec(Y) , X vec(H) (15) n t r which can be further detailed by using (7) as

−1    = R  −1 σ 2 + κ IN N ¯ vec (H) + vec(N) ¯† X vec(H) X n t r 1 ¯ † vec(N) , WX κ where the term defined as

−1  −1 + ρ IN N , Wρ R t r = W vec(H) +

(16) (17)

conveys the channel covariance uncertainty. 1 Woodbury identity: Let A and B be positive definite matrix then AC† (CAC† + B)−1 = (A−1 + C† B−1 C)−1 C† B−1

The next proposition reports the expression of the estimation error covariance matrix with imperfect CCIR. Proposition 1: With an imperfect CCIR (ICCIR) the estimation error of a LMMSE estimator (16) has covariance matrix given by = RICCIR E

1 2 W + (W − INt Nr )R(W − INt Nr ) , ρ

1 = R + WRW† + WW† − RW† − WR† . ρ  is positive definite Hermitian matrix and then W Now, R in (17) is also positive definite Hermitian matrix, W = W† . Hence, by these properties and after some algebra we get (18). The estimation MSE with ICCIR can be evaluated as trace of RICCIR in (18). However, such a result cannot be obtained E in closed form and then, in the next section, we propose to apply bounding techniques in order to find a lower and an upper bound on the value of the MSE. A. Bounds on the estimation MSE From Proposition 1 the estimation MSE is given by    = tr RICCIR = MSEICCIR (R, R) E

(19) 1 = tr(W2 ) + tr (W − INt Nr )R(W − INt Nr ) . ρ The following proposition shows how the use of an imperfect channel covariance matrix in LMMSE estimators impacts on estimation MSE.  be 0 < Proposition 2: Let the eigenvalues of R and R λmin (R) = λ1 (R) ≤ λ2 (R) ≤ . . . ≤ λmax (R) = λNt Nr (R)  = λ1 (R)  ≤ λ2 (R)  ≤ . . . ≤ λmax (R)  = and 0 < λmin (R)  respectively, then the estimation MSE with ICCIR λNt Nr (R), is lower and upper bounded as N t Nr i=1 pper  MSEU ICCIR (R, R) =

N t Nr i=1

 + λi (R) ρ λ2i (R) ,  2 (1 + ρ λi (R))

 MSELower ICCIR (R, R) = MSECCIR (R) , while the upper bound (21) becomes

(18)

with W defined in (17). Proof: From the definition of the estimation error covariance matrix, by (16) and being channel realization, transmitted data and noise independent of each other, it follows that 



†  ICCIR   RE = E vec(H) − vec(H) vec(H) − vec(H)

 MSELower ICCIR (R, R) =

 = R: When the chan1) Perfect covariance knowledge R nel covariance matrix is perfectly known at the receiver,  = λi (R) the lower bound (20) collapses to the real λi (R) MSE (14)

(20)

 + λN N −i+1 (R) ρ λ2i (R) t r .(21)  2 (1 + ρ λi (R))

Proof: See Appendix. B. Remarks From Proposition 2 we can make the following remarks.

pper  MSEU ICCIR (R, R) = MSECCIR (R)+

+

N t Nr i=1

λNt Nr −i+1 (R) − λi (R) , (1 + ρ λi (R))2

which gets close to MSECCIR (R) over channels having a covariance matrix with small condition number χ2 (R) = λmax (R)/λmin (R) ≥ 1. In particular it is correct over i.i.d. channels, R = INt Nr , χ2 (INt Nr ) = 1. 2) Condition number of R: We can deduce that upper and lower bounds in Proposition 2 are expected to get close to each other for channels having small condition number χ2 (R). 3) SNR: We first note that in both (20) and (21) the SNR ρ multiplies the eigenvalues of the estimated covariance matrix  but not the ones of the real R. R • For sufficiently small SNRs both MSELower and ICCIR pper MSEU ICCIR are approximately equal to tr(R) = Nt Nr and then  nor the real the MSE is affected by neither the estimated R one R. However, in general the SNR has to be very low (lower than values of practical interest for coherent communications) in order to follow such a trend. More precisely it should be  ρ  λmin (R)/λ2max (R). • For high SNRs the lower and upper bounds get closer to each other and they both decay to zero as Nt Nr /ρ = tr(Nt Nr )/ρ, i.e., linearly in a logarithmic scale with the SNR ρ measured in dB. Therefore, for asymptotically high SNRs  the MSE behavior is independent of both R and R.  4) Setting R = INt Nr : In the absence of any knowledge  = IN N and then about the channel state we might set R t r preliminary assume that the channel taps are i.i.d. . We first note that if R = INt Nr , then λi (R) = 1 and (14) becomes MSECCIR (INt Nr ) =

N t Nr i=1

Nt Nr λi (R) = . 1 + ρ λi (R) 1+ρ

 = IN N , Now, it is straightforward to verify that if we set R t r i.e., we suppose the channel has i.i.d. components, then for (20) and (21) we obtain the same result as above MSELower ICCIR (R, INt Nr ) = pper = MSEU ICCIR (R, INt Nr ) = MSECCIR (INt Nr ) ,

and, consequently MSEICCIR (R, INt Nr ) = MSECCIR (INt Nr ) , which means that despite of the actual structure of the channel covariance matrix R, if we suppose it to be as for i.i.d. chan = IN N , then the MSE of the LMMSE nels, i.e., we set R t r channel estimator is indeed the same as for transmissions over i.i.d. channels, i.e., as for R = INt Nr .

15

10

MSE [dB]

5) On the achievability of the lower bound: We note that it  share the same eigenvectors. follows from (17) that W and R Hence, the lower bound corresponds to the real MSE if the  and R share the same eigenvectors too (see in the matrices R Appendix the case of equality in Proposition 3). Equivalently, two matrices have the same eigenvectors if and only if they  R = R R.  In particular, for a channel with commute, i.e., R  r and then,  =R t ⊗R a Kronecker product structure we get R by the properties of the Kronecker product, we conclude that  r Rr = Rr R r the lower bound is achieved if and only if R   and Rt Rt = Rt Rt .

5

0 MSE

CCIR

MSEICCIR MSELower

−5

C. Least-Squares (LS) Estimation

ICCIR Upper

MSEICCIR

For comparison purpose, we recall that LS channel estimation does not require second order statistical knowledge about the channel and SNR. In particular, from (1) LS estimation reˆ LS = YX−1 , and the correspondent estimation MSE sults: H is given by Nt Nr ; (22) MSELS = ρ

MSELS −10

0

5

10

15

20

15

20

15

20

SNR [dB]

(a) ρˆt = ρˆr = 0 15

the same as for LMMSE estimators with high SNR. 10

In this section we consider a channel described by the Kronecker model recalled in Sect. II-B. Transmit and receive correlation matrices are characterized through a single correlation coefficient ρt and ρr [13], respectively, as follows |i−j|

Rt (i,j) = ρt

MSE [dB]

V. N UMERICAL R ESULTS

5

0 MSE

and Rr (i,j) = ρ|i−j| . r

CCIR

MSEICCIR

Similarly, we assume that the supposed channel covariance  has the same structure but possibly different corrematrix R lation coefficients

ICCIR Upper

MSEICCIR MSELS −10

 r(i,j) = ρˆ|i−j| .  t(i,j) = ρˆ|i−j| and R R t r

0

5

10

SNR [dB]

(b) ρˆt = ρˆr = 0.5 15

10

MSE [dB]

Hence, we consider a simplified model where the receiver has to estimate two correlation coefficients that fully define the channel correlation matrix. In general all the elements of the supposed covariance matrix have to be estimated [5]. In all the plots in Fig. 1 we fix Nt = Nr = 4 and ρt = ρr = 0.5 for R. We report MSE behavior for different values of ρˆt and ρˆr in order to validate remarks given in Sect. IV-B.  = I16 = R: The channel is assumed with • Fig. 1(a), R i.i.d. components. Hence, as expected from Sect. IV-B4 upper and lower bounds clash and then they provide the exact MSE behavior, which is exactly the same as if the channel would indeed be i.i.d. . Nevertheless, with perfect CCIR there is a gain for low SNRs over the ICCIR case. On the other hand, in the high SNR regime the performance follows the LS solution in all the cases, Sect. IV-B3  = R: The covariance matrix is perfectly • Fig. 1(b), R known at the receiver, Sect. IV-B1. The lower bound provides the correct estimation MSE while the upper bound overestimates the error.  = R: The receiver overestimates spatial cor• Fig. 1(c), R relation setting ρˆt = ρˆr = 0.8 > ρt = ρr = 0.5. In this case we see how an ICCIR may lead to a performance degradation

MSELower

−5

5

0 MSECCIR MSE

ICCIR Lower

MSEICCIR

−5

MSEUpper ICCIR

MSE

LS

−10

0

5

10

SNR [dB]

(c) ρˆt = ρˆr = 0.8 Figure 1. MSE performance comparison. Nt = Nr = 4, ρt = ρr = 0.5 . Different figures correspond to different supposed correlation values ρˆt = ρˆr .

that makes LS estimation even better than LMMSE solution. Moreover, we note that while the upper bound diverges from the real MSE, the lower bound is still very accurate. Therefore, with the considered channel model we can state that LMMSE estimator is more robust to an underestimation of the correlation coefficients rather than an overestimation. In general, the lower bound results very tight in all the cases. When no information is available about the channel covari = IN N in ance matrix, but the SNR ρ is known, to set R t r a LMMSE estimator should be a robust approach that can be adopted in the initialization phase of the receiver [5]. VI. C ONCLUSIONS In this paper, we proposed an analysis of the effect of imperfect channel covariance information at a receiver employing LMMSE channel estimation. The estimation MSE was investigated under a general framework which relates the eigenvalues of the real and supposed channel covariance matrices. A lower and an upper bound to the estimation MSE were derived and their tightness discussed. It was found that despite of using an imperfect channel covariance matrix, a gain of LMMSE over LS estimation is achievable in the low/medium SNR regime. However, an overestimation of the spatial correlation may lead to a severe performance degradation. In such cases, the adoption on an LS criterion is advisable. When there is no knowledge about the channel covariance matrix, a robust LMMSE-like approach relies on assuming the channel as having i.i.d. components. With that assumption, estimation performance cannot be worse than for transmissions over channels that are really i.i.d. . A PPENDIX P ROOF OF P ROPOSITION 2 We first recall the following result. Proposition 3 (von Neumann trace inequality [14]): Let A ∈ Cn×n and B ∈ Cn×n be positive semidefinite Hermitian matrices with eigenvalues (real and non and negative) 0 < λ1 (A) ≤ λ2 (A) ≤ . . . ≤ λn (A) 0 < λ1 (B) ≤ λ2 (B) ≤ . . . ≤ λn (B), respectively, then n

λi (A)λn−i+1 (B) ≤ tr(AB) ≤

i=1

n

λi (A)λi (B) .

i=1

(A.1) n Equality holds on the right when B = i=1 λi (B)ui u†i and n equality holds on the left when B = i=1 λn−i+1 (B)ui u†i , where ui is a right eigenvector of A for the eigenvalue λi (A), i = 1, 2, . . . , n. Therefore, we start by evaluating the first term on the right hand side of (19): tr(W2 ). From (17) by exploiting some well-known properties of Hermitian matrices (see for instance [14]), the eigenvalues of W result  −1 + ρ IN N )−1 ) λi (W) = ρ λi ((R t

r

−1  = ρ(λ−1 = i (R) + ρ)

 ρ λi (R) < 1.  1 + ρ λi (R)

(A.2)

Hence, being W Hermitian then W2 is Hermitian too and  2 N N t Nr t Nr  ρ λi (R) 2 2 tr(W ) = λi (W ) = . (A.3)  1 + ρ λi (R) i=1 i=1 Now, since tr((W − INt Nr )R(W − INt Nr )) = tr((W − INt Nr )2 R), we find the eigenvalues of (W − INt Nr )2 before applying Proposition 3  2. λi ((W − INt Nr )2 ) = (λi (W) − 1)2 = 1/(1 + ρ λi (R)) (A.4)  are ordered in increasing We note that if the eigenvalues of R order, then the eigenvalues λi ((W − INt Nr )2 ) result ordered in decreasing order. Therefore we can apply Proposition 3 as follows N t Nr λi (R) ≤ ...  2 i=1 (1 + ρ λi (R)) (A.5) N t Nr λNt Nr −i+1 (R) 2 . ≤ tr((W − INt Nr ) R) ≤  2 (1 + ρ λi (R)) i=1

Finally, using (A.3) and (A.5) in (19) we obtain the lower and upper bounds in the form given in (20) and (21), respectively. R EFERENCES [1] M. Médard, “The Effect Upon Channel Capacity in Wireless Communications of Perfect and Imperfect Knowledge of the Channel,” IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 933-946, May 2000. [2] T. Yoo and A. Goldsmith, “Capacity of Fading MIMO Channels with Channel Estimation Error,” In Proc. Allerton’03, Monticello, Illinois, Oct. 2003. [3] L. Musavian, M.R. Nakhai, M. Dohler, and A.H. Agvami, “Effect of Channel Uncertainty on the Mutual Information of MIMO Fading Channels,” IEEE Trans. Veh. Tech., vol. 56, no. 5, pp. 2798–2806, Sept. 2007. [4] S. Serbetli and A. Yener “MMSE Transmitter Design for Correlated MIMO Systems with Imperfect Channel Estimates: Power Allocation Trade-offs,” IEEE Trans. Wireless Commun., vol. 5, no. 8, pp. 2295– 2304, Aug. 2006. [5] N. Czink, G. Matz, D. Seethaler, and F. Hlawatsch, “Improved MMSE Estimation of Correlated MIMO Channels Using a Structured Correlation Estimator,” In Proc. SPAWC ’05, pp. 595-599, 2005. [6] K. Werner and M. Jansson, “Estimating MIMO Channel Covariances from Training Data Under the Kronecker Model,” Elsevier, Signal Process., vol. 89, no. 1, pp. 1–13, Jan. 2009. [7] S. M. Kay, Fundamentals Of Statistical Signal Processing: Estimation Theory, Prentice Hall, NJ, 1993. [8] B. Hassibi and B. M. Hochwald, “How Much Training is Needed in Multiple-antenna Wireless Links?,” IEEE Trans. Inf. Theory, vol. 49, no. 4, pp. 951-963, Apr. 2003. [9] J. Pang, J. Li, L. Zhao, and Z. Lü, “Optimal Training Sequences for Frequency-Selective MIMO Correlated Fading Channels,” In Proc. AINA’07, Niagara Falls, Canada, May 2007. [10] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederiksen, “A Stochastic MIMO Radio Channel Model with Experimental Validation”, IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1211-1226, Aug. 2002. [11] H. Ozcelik, M. Herdin, W. Weichselberger, J. Wallace, and E. Bonek, “Deficiencies of "Kronecker" MIMO Radio Channel Model,” IEE Electron. Lett., vol. 39, no. 16, pp. 1209-1210, Aug. 7, 2003. [12] V. Raghavan, J. Kotecha, and A. Sayeed, “Does the Kronecker Model Result in Misleading Capacity Estimates?,” Submitted to IEEE Trans. Inf. Theory, arXiv:0808.0036v1, July 2008. [13] A. van Zelst and J. S. Hammerschmidt, “A Single Coefficient Spatial Correlation Model for Multiple-Input Multiple-Output (MIMO) Radio Channels,” In Proc. of 27th General Assembly of the URSI, pp. 1-4, Maastricht, the Netherlands, Aug. 2002. [14] G. A. F. Seber, A Matrix Handbook for Statisticians, John Wiley & Sons, Inc., NJ, 2008.