dorsal/arxiv
View SchemaCapacity of multivariate channels with multiplicative noise: I.Random matrix techniques and large-N expansions for full transfer matrices
| Authors | Anirvan Mayukh Sengupta, Partha Pratim Mitra |
|---|---|
| Categories | |
| ArXiv ID | physics/0010081 |
| URL | https://arxiv.org/abs/physics/0010081 |
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.
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"abstract": "We study memoryless, discrete time, matrix channels with additive white\nGaussian noise and input power constraints of the form $Y_i = \\sum_j H_{ij} X_j\n+ Z_i$, where $Y_i$ ,$X_j$ and $Z_i$ are complex, $i=1..m$, $j=1..n$, and $H$\nis a complex $m\\times n$ matrix with some degree of randomness in its entries.\nThe additive Gaussian noise vector is assumed to have uncorrelated entries. Let\n$H$ be a full matrix (non-sparse) with pairwise correlations between matrix\nentries of the form $ E[H_{ik} H^*_{jl}] = {1\\over n}C_{ij} D_{kl} $, where\n$C$,$D$ are positive definite Hermitian matrices. Simplicities arise in the\nlimit of large matrix sizes (the so called large-N limit) which allow us to\nobtain several exact expressions relating to the channel capacity. We study the\nprobability distribution of the quantity $ f(H) = \\log \\det (1+P H^{\\dagger}S\nH) $. $S$ is non-negative definite and hermitian, with $Tr S=n$. Note that the\nexpectation $E[f(H)]$, maximised over $S$, gives the capacity of the above\nchannel with an input power constraint in the case $H$ is known at the receiver\nbut not at the transmitter. For arbitrary $C$,$D$ exact expressions are\nobtained for the expectation and variance of $f(H)$ in the large matrix size\nlimit. For $C=D=I$, where $I$ is the identity matrix, expressions are in\naddition obtained for the full moment generating function for arbitrary\n(finite) matrix size in the large signal to noise limit. Finally, we obtain the\nchannel capacity where the channel matrix is partly known and partly unknown\nand of the form $\\alpha I+ \\beta H$, $\\alpha,\\beta$ being known constants and\nentries of $H$ i.i.d. Gaussian with variance $1/n$. Channels of the form\ndescribed above are of interest for wireless transmission with multiple\nantennae and receivers.",
"arxiv_id": "physics/0010081",
"authors": [
"Anirvan Mayukh Sengupta",
"Partha Pratim Mitra"
],
"categories": [
"physics.data-an",
"cond-mat"
],
"title": "Capacity of multivariate channels with multiplicative noise: I.Random matrix techniques and large-N expansions for full transfer matrices",
"url": "https://arxiv.org/abs/physics/0010081"
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