Diversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT. ECE 559 Presentation Hoa Pham Dec 3, 2007

Transcription

1 Diversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT ECE 559 Presentation Hoa Pham Dec 3, 2007

2 Introduction MIMO systems provide two types of gains Diversity Gain: each path from a transmitter to a receiver experiences independent fading Spatial Multiplexing Gain: fading creates increased degrees of freedom There is a fundamental tradeoff between diversity gain and spatial multiplexing gain

3 No CSIT: Channel Model H known at receiver, not transmitter, constant within a block of l symbols Analysis focuses on high SNR regime

4 Diversity Gain Example: Uncoded Binary PSK over single antenna: In Contrast: Transmitting the same signal to a receiver with 2 antenna: At high SNR, performance gain characterized by SNR exponent of error probability d (Diversity Gain) Define as: For m transmit and n receive antennas, max diversity gain = mn

5 Multiplexing Gain Ergodic capacity of the channel we defined, with H independently and identically distributed across blocks At high SNR: Channel capacity increases with min{m, n} logsnr, instead comparision with logsnr for single antenna channels Where we thinks of a family of codes {C(SNR)}, one at each SNR level and R(SNR) is the rate of code C(SNR)

6 Optimal Tradeoff For the case Optimal tradeoff curve is a piecewise linear function connecting the points: Where:

7 Approach to Proving the Tradeoff Curve for Formulate Outage Probability d out (r) turns out to equal d*(r) Lower bound error probability with outage probability for Upper bound error probability. It turns out Therefore

8 Outage Probability Outage occurs when mutual information is below the data rate Without loss of optimality, take input to be Gaussian Optimizing over all input distributions Using a lower and upper bound, and taking SNR to inf.

9 Outage Probability After significant mathematical manipulation, we get where d out (r) can be solved explicitly, and turns out to be equal to d*(r) in optimal trade off curve

10 Lower Bound on Probability of Error Fix a codebook C of size 2 Rl and let be the input, uniformly drawn from C For a specific realization H = H, applying Fano s inequality and rearranging, we can write, for We want an expression for P e (SNR) as SNR goes to inf. Consider scenario:

11 Lower Bound on Probability of Error Therefore, we have Applying the same method as in deriving outage probability, we can write Take δ to the limit of 0, and by continuity of d out (r), we get Suggests: if channel is in outage, error very likely

12 Upper Bound on Error Probability Choose input to be random code from i.i.d Gaussian ensemble We have Use union bound to bound the second term, by considering pairwise error probability, we get Putting the two together where

13 Optimal Tradeoff Upper and lower bounds match, therefore at some rate Moreover, d out (r) = d*(r) Therefore, for multiplexing gain r, we get diversity gain d*(r) And the optimal tradeoff curve d*(r) is a piecewise linear function connecting

14 Diversity-Multiplexing Tradeoff: Partial CSIT System model: During fading block l Yl = HlSl + Wl m max( N, N ) n min( N, N ) r r t t Conditioned on an index Ι( Hl ) = i where i 1, 2,... K, the codeword S is taken from a codebook C = S (1),..., S ( M ) with rate R = r log SNR Define: Average total transmit power M 1 i P 2 i Si( k) F TM i k = 1 L as. 2 L Sl E F H PI( ) L l= 1 T H { } { } i i i i Long-term power constraint 1 1 lim = SNR i i

15 Definitions Average rate: Diversity gain: SNR d Outage Probability P Define: I L 1 as. K R lim R = Pr( I( H ) = i) R P e H ( H, π ) log det( I + HH ) N r L I( Hl ) l i L l= 1 i= 1 Pr(log det( I + HQ H ) < R ) H out, K N I ( H) I ( H) π N F( ρ, π) Pr( I( H, π) < ρ) r Using a lower and upper bound, and taking SNR to inf., we can restrict our analysis to the case Pi Q = I where tr( Q ) P t N i N i i t t

16 Main Results Single rate transmission * Lemma 1: Outage minimizing power codebook { P } K i i = 1 solves the following optimization problem: max P s.t K K [ FRP (, ) + 1 FRP (, )] P+ [ FRP (, ) FRP (, )] P SNR 0 P... < P K 1 1 i 1 i i 1 K i= 2 Then, the optimal index mapping is given by: I * ( H) * 1 if I( H, PK ) < R = * min{ i: i {1,..., K}, I( H, Pi ) R} 0 P1 <... < P K Lemma 2: Using the Wishart s distribution of eigenvalues of matrix HH H, after some mathematical manipulation we have: Fr ( log SNR, π) SNR where, π SNR D( r, p) p n n n n α A i n i 1 i= 1 + Dr (, p) inf (2i 1 + m n) α, A { α α... α, ( p α ) < r}

17 Main Results Theorem 1: The optimal D-M tradeoff of a single rate MIMO sytem with K quantization regions in the feedback link is upper bound by: d ( r) = D( r,1 + d ( r)) where d ( r) 0 r * * * out, K out, K 1 out,0 When r is sufficiently close to 0 and p 1,the minimization solution * * is achieved by choosing α = p for i = 1,..., n 1 and α = p r,then i lim Dr (, p) = NNp, then by the theorem 1, we have r 0 t r K * k r 0 dout, K r = Nt Nr i= 1 * lim r n Dr (,1) = 0, then lim r ndout, K( r) = 0, K lim ( ) ( ) It can also verify that Similarly, in the case of adaptive rate transmission, we have: d (, r r ) = D( r,1 + d (, r r ) where d (, r r ) D(,1), r r r * * * out, K min min out, K 1 min out,1 min min K * K 1 k rmin 0 dout, K r rmin = NrNt D r + NtNr k = 1 lim (, ) ( ) (,1) ( ) n

18 Conditional Achievability of the optimal D-M Tradeoff Extended Approximately Universal Condition: For any pair of codewords, let μ... μ be n smallest squared singular values of ΔX, and 1 n n j= 1 ( β ) βi j let μ = SNR, then constrain the codebook C so that (min SNR ) SNR i Consider the following index mapping 1 if I( H, PK < ( r+ ε ) log SNR, I( H) = min{ i: i {1,..., K}, I( H, Pi ) (r+ ε )logsnr SNR SNR SNR where P1 =, P2 =,..., P = K KF(( r+ ε)logsnr, P ) KF(( r+ ε)logsnr, P ) C K 1 K 1 Pi then, construct the transmit codeword as Si ( k) = X ( k), i = 1,..., K, k = 1,..., M N ε Define the ith ε-outage-free region: οi = { H: I( H) = i, I( H, P) ( r+ ε)logsnr} Show the pairwise error probability in this region decay exponentially as SNR goes to inf., then the error probability is dominated by the outage event, which characterized by the D-M tradeoff t i +. r

19 Numerical Example

20 Conclusions MIMO system provide both diversity and multiplexing gains, but there are tradeoff between them When there is no CSI at transmitter, the maximum diversity is N t N r With only a few bit feedback of quantized CSI, we can achieve much higher diversity gain, scaled exponentially with number of quantization regions. Especially, for the adaptive rate transmission, we can get non-zero diversity gain at maximum multiplexing gain with only a few bits of feedback

21 Thank You & Questions?

22 References L. Zheng and D. Tse, Diversity and Multiplexing: A Fundamental Tradeoff in Multiple Antenna Channels, IEEE Transactions on Information Theory, vol. 49, May 2003, pp Tung T. Kim and Mikael Slogkund, Diversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT