Capacity of multivariate channels with multiplicative noise: I.Random matrix techniques and large-N expansions for full transfer matrices

Sengupta, Anirvan Mayukh, Mitra, Partha Pratim (October 2000) Capacity of multivariate channels with multiplicative noise: I.Random matrix techniques and large-N expansions for full transfer matrices. (Submitted)

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Abstract

We study memoryless, discrete time, matrix channels with additive white Gaussian noise and input power constraints of the form Y_i = \sum_j H_{ij} X_j + Z_i, where Y_i ,X_j and Z_i are complex, i=1..m, j=1..n, and H is a complex m\times n matrix with some degree of randomness in its entries. The additive Gaussian noise vector is assumed to have uncorrelated entries. Let H be a full matrix (non-sparse) with pairwise correlations between matrix entries of the form E[H_{ik} H^*_{jl}] = {1\over n}C_{ij} D_{kl} , where C,D are positive definite Hermitian matrices. Simplicities arise in the limit of large matrix sizes (the so called large-N limit) which allow us to obtain several exact expressions relating to the channel capacity. We study the probability distribution of the quantity f(H) = \log \det (1+P H^{\dagger}S H) . S is non-negative definite and hermitian, with Tr S=n. Note that the expectation E[f(H)], maximised over S, gives the capacity of the above channel with an input power constraint in the case H is known at the receiver but not at the transmitter. For arbitrary C,D exact expressions are obtained for the expectation and variance of f(H) in the large matrix size limit. For C=D=I, where I is the identity matrix, expressions are in addition obtained for the full moment generating function for arbitrary (finite) matrix size in the large signal to noise limit. Finally, we obtain the channel capacity where the channel matrix is partly known and partly unknown and of the form \alpha I+ \beta H, \alpha,\beta being known constants and entries of H i.i.d. Gaussian with variance 1/n. Channels of the form described above are of interest for wireless transmission with multiple antennae and receivers.

Item Type:	Paper
CSHL Authors:	Mitra, Partha P.
Communities:	CSHL labs > Mitra lab
SWORD Depositor:	CSHL Elements
Depositing User:	CSHL Elements
Date:	31 October 2000
Date Deposited:	13 Oct 2023 14:51
Last Modified:	13 Oct 2023 14:51
Related URLs:	Author
URI:	https://repository.cshl.edu/id/eprint/41235

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