Optimal Transmit Strategies in MIMO Ricean Channels with MMSE Receiver

Transcription

1 Optimal Transmit Strategies in MIMO Ricean Channels with MMSE Receiver E. A. Jorswieck 1, A. Sezgin 1, H. Boche 1 and E. Costa 2 1 Fraunhofer Institute for Telecommunications, Heinrich-Hertz-Institut 2 Siemens AG, ICM N PG SP RC FR, Information & Comm. Mobile Networks R&C MIMO Ricean Channels with MMSE Receiver p. 1

2 Introduction Table of contents Scenarios: Mean feedback and LOS System model and performance metric Properties of the MSE Optimal transmit strategy and characterization of the optimal power allocation Illustration and Discussion Conclusion MIMO Ricean Channels with MMSE Receiver p. 2

3 Introduction Multiple antenna systems have been analyzed in terms of average mutual information, capacity, MSE, BER with different types of CSI at the transmitter, i.e. perfect CSI [Telatar99], covariance feedback [Goldsmith02, Jorswieck/Boche03], no CSI under spatially correlated Rayleigh fading In Ricean MIMO channels: capacity analysis by [Jayaweera/Poor02] optimal transmit strategy [Venkatesan/Simon03,Jafar02] impact of singular values of Rice component [Hoesli/Lapidoth2004] MIMO Ricean Channels with MMSE Receiver p. 3

4 Contribution We study the performance of the linear MMSE receiver: We characterize the optimal transmit strategy under the assumption that the transmitter knows the mean channel matrix and the receiver has perfect CSI. We show that it is optimal to transmit into the direction of the eigenvectors of the squared mean matrix. The optimal power allocation is characterized using the necessary and sufficient optimality conditions. We show, that for small SNR values, the optimal transmit covariance matrix has rank one. As a result, we derive the SNR range, in which only one eigenvalue is supported. MIMO Ricean Channels with MMSE Receiver p. 4

5 Scenario I / Mean feedback Scenarios Flat-fading MIMO channel with zero-mean and transmitter has imperfect CSI. Imperfect CSI due to channel estimation errors which are iid complex Gaussian distributed with variance proportional to the pilot signal length. Transmitter sees an iid complex Gaussian distributed fading matrix with mean that equals the estimated channel matrix. Scenario II / LOS component Channel model as in Scenario I plus LOS component. Receiver has perfect CSI. Feedback link to transmitter has low rate. Therefore, only a long-term statistics in terms of the channel means is fed back. Transmitter knows only the mean of the channel fading matrix. MIMO Ricean Channels with MMSE Receiver p. 5

6 System model n(k) d ( k ) x(k) Q H~CN(µ,Σ) + y(k) MMSE s(k) feedback (µ,σ) The received signal vector y in the quasi-static block flat fading MIMO channel H is given by y = Hx + n (1) Additive white Gaussian noise (AWGN) vector n CN(0,σnI) 2 MIMO Ricean Channels with MMSE Receiver p. 6

7 Channel model I The channel is modeled by a random non line-of-sight (NLOS) component W with identically independent distributed (iid) complex Gaussian entries, i.e. W CN(0, I) and by a LOS component D. The NLOS component occurs in rich multi-path fading. The LOS component depends on the geometry of the transmission scenario and can have rank equal to one up to n T [Haustein03]. The mean feedback component has in general full rank. We assume that the LOS component D has a singular value decomposition D = UΛV H with eigenvalues in Λ ordered in descending order diag (Λ) = [d 1,..., d nt ] with d 1 d 2... d nt 0. We assume that the LOS component has power equal to one, i.e. tr DD H = n T. MIMO Ricean Channels with MMSE Receiver p. 7

8 Channel model II The channel matrix H in (1) is given by H = aw + 1 ad for 0 a 1. The Ricean factor K is defined as the ratio between the LOS power and the NLOS power. It is given by K = 1 a 1. The transmit covariance matrix Q and its eigenvalue decomposition is given by Q = E [ xx H] = U Q PU H Q with eigenvalues (or powers) in P ordered in descending order diag (P) = [p 1,..., p nt ] with p 1 p 2... p nt 0. The transmit power is constrained to P = 1, i.e. tr Q P = 1. As a result, the SNR is given by ρ = P σ. n 2 MIMO Ricean Channels with MMSE Receiver p. 8

9 MMSE receiver I The linear MMSE receiver computes the data estimate from the received signal x = QH H ( I + HQH H) 1 y. (2) The covariance matrix of the estimation error ǫ is given as K ǫ = Q QH H ( HQH H + I) 1 HQ. (3) From (3), the normalised MSE follows as MSE(Q, Z) = n T tr ( HQH H [ I + HQH H] 1 ). (4) MIMO Ricean Channels with MMSE Receiver p. 9

10 MMSE receiver II The MSE is a convex function with respect to the transmit covariance matrix Q. For the average MSE as a function of the deterministic LOS component D, the Ricean number α = a and ᾱ = 1 a, and the transmit strategy Q we obtain MSE(D,a,Q) = n T n R ([ +E tr I + ρ(αw + ᾱd) Q(αW + ᾱd) H] 1). (5) MIMO Ricean Channels with MMSE Receiver p. 10

11 Properties of the average MSE The minimum MSE defined by M(D) = min tr Q P Q 0 MSE(D,a,Q) depends only on the singular values of the LOS component d 1,...,d nt, not on the eigenvectors in V. Of course, the optimal transmit covariance matrix Q does depend on the eigenvectors as well as on the singular values of D. MIMO Ricean Channels with MMSE Receiver p. 11

12 The minimum MSE is given by M(D) = min tr Q P Q 0 Sketch of proof MSE(D, a,q) = n R n T + E tr [ I + ρ(αw + ᾱd)p(αw + ᾱd) H] 1 [ = n R n T + E tr I + ρ(αw + ᾱuλv H ) P(αW + ᾱuλv H ) H] 1 = n R n T + E tr PV (αw + ᾱλ) H U H] 1 = n R n T + E tr [ I + ρu(αw + ᾱλ)v H [ I + ρ(αw + ᾱλ) V } H {{ PV } (αw + ᾱλ) H] 1 P MIMO Ricean Channels with MMSE Receiver p. 12

13 Optimal transmit strategy Theorem 1: The average normalized MSE is minimized by a transmit covariance matrix Q = U Q PU H Q with V contains the eigenvectors of DH D. The diagonal matrix P with eigenvalues p 1,...,p nt is given by the solution of the following power allocation problem [ ] 1 min nt k=1 p k P p k 0 n R n T + E tr I + ρ n T k=1 p k h k h H k (6) with h k = αd k δ k + ᾱh k and δ k is a vector with zeros but on the k-th entry which is equal to one. MIMO Ricean Channels with MMSE Receiver p. 13

14 Sketch of proof I We define a new transmit covariance matrix ˆP by an unitary matrix Γ which is a diagonal matrix with ones and some 1 entries ˆP = Γ PΓ H. Now, we can construct a matrix P which is more diagonal than the original matrix P P = 1 ( ) P + ˆP. 2 Finally, we have the following inequality by the convexity of the MSE with respect to the transmit covariance matrix MSE(Λ, α, P) 1 2 ( ) MSE(Λ, α, P) + MSE(Λ, α, ˆP). (7) MIMO Ricean Channels with MMSE Receiver p. 14

15 Sketch of proof II The MSE using transmit covariance matrix ˆP is given by MSE(Λ, α, ˆP) = n R n T + E tr = n R n T + E tr [I + ρ(αw + ᾱλ)γ PΓ H (αw + ᾱλ) H] 1 [I + ρ(αw + ᾱλ) P(αW + ᾱλ) H] 1 = M(Λ, α, P). It follows MSE(Λ, α, P) MSE(Λ, α, P). (8) Inequality (8) says, the more diagonal the transmit covariance matrix is the less is the average normalized MSE. MIMO Ricean Channels with MMSE Receiver p. 15

16 Optimal power allocation I The convex programming problem is min n R n T + E tr subject to n T k=1 [ I + ρ n T k=1 p k hk hh k ] 1 p k P and p k 0 1 k n T. (9) The Lagrangian function for the problem (9) is given by L(p, µ,λ) = E tr [ I + ρ n T k=1 p k hk hh k ] 1 + µ( P n T k=1 p k ) + n T k=1 λ k p k MIMO Ricean Channels with MMSE Receiver p. 16

17 Optimal power allocation II The necessary and sufficient optimality conditions ( Karush-Kuhn-Tucker (KKT) conditions) [Boyd03] for all 1 l n T are given by E ρ h H l [ I + ρ n T k=1 p k hk hh k ] 2 hl = µ λ k p k λ k = 0 p k 0 λ k 0 P n T k=1 µ 0 p k 0. (10) MIMO Ricean Channels with MMSE Receiver p. 17

18 Characterization of optimal power allocation From these conditions we observe the following behavior of the optimal power allocation for small and high SNR values. Theorem 2: Let the singular values of the Ricean component be ordered in descending order d 1 d 2... d nt 0. Beamforming in direction of the right eigenvectors which belongs to the largest singular value d 1 is optimal if and only if the SNR ρ is smaller than or equal to ρ in n R E [ ( ) 2 ] I + ρ P h 1 hh 1 E [ h H 2 ( I + ρ P h 1 hh 1 ) 2 h2 ]. (11) Corollary 1: If the Ricean matrix is singular and has only one singular value greater than zero, the beamforming range in (11) simplifies to E tr ( ] [ h1 hh 1 I [ I + ρp h 1 hh 1 ] 2 ) 0. (12) MIMO Ricean Channels with MMSE Receiver p. 18

19 Illustration 2 K=1, p 1 power allocation K=1, p 2 K=10, p 2 K=10, p SNR [db] In the figure, the optimal power allocation for a two times two MIMO system with Ricean factor K = 1 and K = 10 and fixed LOS component DD H = diag[1.4; 0.6] is shown. MIMO Ricean Channels with MMSE Receiver p. 19

20 Conclusion We studied a MIMO system with MMSE receiver in a Ricean fading environment. The receiver has perfect CSI and the transmitter knows only the mean, i.e. the Ricean component matrix of the channel. The properties of the average normalized MSE and its dependence on the singular values of the Ricean component were derived. We showed that the optimal transmit strategy is to transmit into direction of the eigenvectors of the squared Ricean component. The optimal power allocation using the necessary and sufficient optimality conditions was characterized. MIMO Ricean Channels with MMSE Receiver p. 20