Channel capacity estimation using free probability theory

Transcription

1 Channel capacity estimation using free probability theory January 008

2 Problem at hand The capacity per receiving antenna of a channel with n m channel matrix H and signal to noise ratio ρ = 1 σ is given by C = 1 ( n log det I n + 1 ) mσ HHH = 1 n log n (1 + 1 σ λ l) (1) where λ l are the eigenvalues of 1 m HHH. We would like to estimate C. To estimate C, we will use free probability tools to estimate the eigenvalues of 1 m HHH based on some observations Ĥ i l=1

3 Observation model 1 CMA 007 The following is a much used observation model: where Ĥ i = H + σx i () The matrices are n m (n is the number of receiving antennas, m is the number of transmitting antennas) Ĥ i is the measured MIMO matrix, X i is the noise matrix with i.i.d standard complex Gaussian entries.

4 Existing ways to estimate the channel capacity Several channel capacity estimators have been used in the literature: C 1 = nl 1 ( Li=1 log det In + 1 ) mσ ĤiĤH i C = n 1 ( log det In + 1 ) Li=1 Lσ m Ĥ i Ĥ H i (3) C 3 = n 1 ( log det In + 1 σ m ( L 1 Li=1 Ĥ i )( L 1 ) Li=1 Ĥ i ) H ) Why not try to formulate an estimator based on free probability instead?

5 .6.4. Capacity True capacity C 1 C C Number of observations Comparison of the classical capacity estimators for various number of observations. σ = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3.

6 The Marchenko Pastur law The Marchenko Pastur law µ c : f µc (x) = (1 1 (x a) c )+ δ 0 (x) + + (b x) +, (4) πcx where (z) + = max(0, z), a = (1 c), b = (1 + c), and δ 0 (x) is dirac measure (point mass) at 0. free cumulants: 1, c, c, c 3,... µ c is the limit eigenvalue distribution of 1 N XXH, with X an n N with independent standard complex Gaussian entries as N, and n N c.

7 Main free probability result we will use Dene Γ n = 1 N R nr H n W n = 1 N (R n + σx n )(R n + σx n ) H, where R n and X n are independent n N random matrices, X n is complex, standard, Gaussian. Theorem If e.e.d.(γ n ) ν Γ, then e.e.d.(w n ) ν W where ν W is uniquely identied by ν W µ c = (ν Γ µ c ) δ σ ( = "the opposite of ").

8 Realization of the theorem for the problem at hand Form the compound observation matrix Ĥ 1...L = H 1...L + σ X 1...L, where L Ĥ 1...L = 1 ] [Ĥ1, Ĥ,..., ĤL, L H 1...L = 1 L [H, H,..., H], X 1...L = [X 1, X,..., X L ]. For the problem at hand, the theorem takes the form ν 1 m Ĥ 1...L Ĥ H 1...L µ n ml ( ) ν 1 m H 1...L H H µ n 1...L ml δ σ (5) Since m 1 H 1...LH H 1...L = m 1 HHH, we can now estimate the moments of m 1 HHH from the moments of the observation matrix mĥ1...lĥh L, and thereby estimate the eigenvalues, and hence the channel capacity.

9 Free probability based estimator for the moments of the channel matrix Can also be written in the following way for the rst four moments: ĥ 1 = h 1 + σ ĥ = h + σ (1 + c)h 1 + σ 4 (1 + c) ĥ 3 = h 3 + 3σ (1 + c)h + 3σ ch1 +3σ 4 ( c + 3c + 1 ) h 1 +σ 6 ( c + 3c + 1 ) ĥ 4 = h 4 + 4σ (1 + c)h 3 + 8σ ch h 1 +σ 4 (6c + 16c + 6)h +14σ 4 c(1 + c)h1 +4σ 6 (c 3 + 6c + 6c + 1)h 1 +σ 8 ( c 3 + 6c + 6c + 1 ), where ĥi are the moments of the observation matrix 1 mĥ1...lĥh 1...L, h i are the moments of 1 m HHH. (6)

10 .6.4. Capacity True capacity C f C u Number of observations Comparison of C f and C u for various number of observations. σ = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3.

11 4.5 4 Capacity True capacity, rank 3 C f, rank 3 True capacity, rank 5 C f, rank 5 True capacity, rank 6 C f, rank Number of observations C f for various number of observations. No phase o-set/phase drift. σ = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3, 5 and 6.

12 How would an algorithm for free convolution look? Denition A family of unital -subalgebras (A i ) i I is called a free family if a j A ij i 1 i, i i 3,, i n 1 i n φ(a 1 a n ) = 0. (7) φ(a 1 ) = φ(a ) = = φ(a n ) = 0 A family of random variables a i is called a free family if the algebras they generate form a free family. How do we implement this in terms of moments? From the previous result, we are basically interested in computing the moments of ab, when ab are free, and b is free Poisson.

13 Implementation of main result The following formula can be used for incremental calculation of the moments of the measure µ µ c, from the moments of the measure µ: m [coef m ](cm µ µ c ) = [coef k ](cm µ )[coef m k ](1 + cm µ µ c )k. (8) Here, k=1 M µ (z) = µ 1 z + µ z +..., where µ i are the moments of µ. coef k means the coecient of z k in the polynomial. The power series coecient can be computed through k-fold discrete (classical) convolution. (8) is proved by rst proving that cm µ µ c = (cm µ) Zeta.

14 Observation model CMA 007 A more general observation model is: Ĥ i = D r i HD t i + σx i, (9) where D r i and D t i are n n and m m diagonal matrices which represent phase o-sets and phase drifts (impairments due to the antennas and not the channel) at the receiver and transmitter given respectively by D r i = diag[e jφi 1,..., e jφi n], and D t i = diag[e jθi 1,..., e jθt m] where the phases φ i j and θ i j are random. We assume all phases independent and uniformly distributed.

15 Problem when extending to phase o-set and phase drift In the compund observation matrix we now put H 1...L = 1 L [ D r i HD t i, D r i HD t i,..., D r i HD t i ], The moments of m 1 H 1...LH H 1...L are now in general dierent from the moments of m 1 HHH! In other words stacking the observations and using the free convolution framework does not give us what we want A way to resolve this: Don't stack the observations at all. Perform convolution through exact formulas for the mixed moments of matrices and Gaussian matrices of lower order. Unbiased capacity estimator.

16 Unbiased estimator for the moments of the channel matrix Let ĥi be the rst moments of the sample covariance matrix 1 mĥiĥh i. An unbiased estimator for the rst moments h i of 1 m HHH is given by ĥ 1 = h 1 + σ ĥ = h + σ (1 + c)h 1 + σ 4 (1 + c) ĥ 3 = h 3 + 3σ (1 + c)h + 3σ ch1 +3σ 4 ( c + 3c m 1 ) h1 +σ 6 ( c + 3c m 1 ) ĥ 4 = h 4 + 4σ (1 + c)h 3 + 8σ ch h 1 +σ 4 (6c + 16c m )h +14σ 4 c(1 + c)h 1 +4σ 6 (c 3 + 6c + 6c (c+1) m )h 1, ( ) +σ 8 c 3 + 6c + 6c (c+1) m (10)

17 Exact formulas for expecations of mixed moments of Gaussian and deterministic matrices We have that E [tr n (W n )] = m 1 + σ E [ ( )] tr n W n = m + σ (1 + c)m 1 + σ 4 (1 + c) E [ ( )] tr n W 3 n = m 3 + 3σ (1 + c)m + 3σ cm1 +3σ 4 ( c + 3c N 1 ) m1 +σ 6 ( c + 3c N 1 ) E [ ( )] tr n W 4 n = m 4 + 4σ (1 + c)m 3 + 8σ cm m 1 +σ 4 (6c + 16c N 16 )m +14σ 4 c(1 + c)m1 +4σ 6 ( (c 3 + 6c + 6c (c+1) N ) )m 1 +σ 8 c 3 + 6c + 6c (c+1) N, where m j = tr n ( ( 1N R n R H n ) j ). (11)

18 Derivation of the limiting distribution for 1 N XXH When x is standard complex Gaussian, we have that E ( x p) = p!. A more general statement concerns a random matrix N 1 XXH, where X is an n N random matrix with independent standard complex Gaussian entries. It is known [HT] that (( ) 1 p ) τ n N XXH = 1 N p N n k(ˆπ) n l(ˆπ), π S p where ˆπ is a permutation in S p constructed in a certain way from π, and k(ˆπ), l(ˆπ) are functions taking values in {0, 1,,...}.

19 One can show that this equals τ n (( 1 N XXH ) p ) = c l(ˆπ) 1 + ˆπ NC p k a k N k. The convergence is "almost sure".

20 .6.4. Capacity True capacity C 3 C f C u Number of observations Comparison of capacity estimators which worked when no phase o-set/drift was present, for increasing number of observations. With phase drift and phase o-set. σ = 0.1, n = 10 receive antennas, m = 10 transmit antennas. The rank of H was 3.

21 4 3.5 True capacity C u 3 Capacity σ C u for L = 1 observation, n = 10 receive antennas, m = 10 transmit antennas, with varying values of σ. With phase drift and phase o-set. The rank of H was 3.

22 8 7 True capacity C u 6 5 Capacity σ C u for L = 1 observation, n = 4 receive antennas, m = 4 transmit antennas, with varying values of σ. With phase drift and phase o-set. The rank of H was 3.

23 8 7 True capacity C u 6 Capacity σ C u for L = 10 observations, n = 4 receive antennas, m = 4 transmit antennas, with varying values of σ. With phase drift and phase o-set. The rank of H was 3.

24 10 9 True capacity C u 8 7 Capacity σ C u for L = 50 observations, n = 4 receive antennas, m = 4 transmit antennas, with varying values of σ. With phase drift and phase o-set. The rank of H was 4.

25 10 9 True capacity C u 8 7 Capacity σ C u for L = 1600 observations, n = 4 receive antennas, m = 4 transmit antennas, with varying values of σ. With phase drift and phase o-set. The rank of H was 4.

26 References CMA 007 [HT]: "Random Matrices and K-theory for Exact C -algebras". U. Haagerup and S. Thorbjørnsen. citeseer.ist.psu.edu/11410.html This talk is available at oyvindry/talks.shtml. My publications are listed at oyvindry/publications.shtml