On Certain Large Random Hermitian Jacobi Matrices With Applications to Wireless Communications

Abstract

In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular we are interested in an uplink cellular channel which models mobile users experiencing a soft-handoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed form expression for the per-cell sum-rate of this channel in high-SNR, when an intra-cell TDMA protocol is employed.Since the matrices of interest are tridiagonal, their eigenvectors can be considered as sequences with second order linear recurrence. Therefore, the problem is r educed to the study of the exponential growth of products of two by two matrices. For the case where K users are simultaneously active in each cell, we obtain a series of lower and upper bound on the high-SNR power offset of the per-cell sum-rate, which are considerably tighter than previously known bounds.

Multi-Cell MIMO Cooperative Networks: A New Look at Interference

“…A central limit theorem is also proven in [27] along with a corresponding large deviation result, providing evidence to the fact that, given the limited randomness present in matrix H (due to the banded structure), convergence is slower than in classical random matrix theory (see, e.g., [51]). Finally, [52] characterizes the high-SNR behavior (in the sense of [53]) of (9a) as M grows large and K = 1 user and L = 1. Performance bounds are also provided for K > 1.…”

On the spectrum of large random hermitian finite-band matrices

“…Finally, α, β ∈ [0, 1] are constants. The normalized input-output mutual information of (1) conditioned on H N (also known as the Shannon transform) 1 An N × N identity matrix is denoted by I N . …”

“…the distribution of H N converges as well. This is since (3) is uniformly integrable due to the Hadamard inequality and the bounded power moment assumption, and hence the a.s. convergence implies convergence in expectation [1]. In Section II it will be realized that if the channel H N is known at the receiver and its variation over time is stationary and ergodic, then the expectation of (3) w.r.t.…”

“…Recently [1], the percell capacity high-SNR parameters for a two diagonal H N (K = 1, α = 1, and β = 0) were calculated for N → ∞ and rather general fading distributions: S∞ = 1 ; L∞ = −2 max (Eπ a log 2 |x| , Eπ b log 2 |x|) . (13) The main idea is to link the spectral properties of H N H † N with the exponential growth of the elements of its eigenvectors.…”

On information rates of the fading Wyner cellular model via the thouless formula for the strip

“…Considering non-fading channels and a "wideband" (WB) transmission scheme, where all bandwidth is available for coding (as opposed to random spreading), the throughputs obtained with optimum and linear MMSE joint processing of the received signals from all cell-sites are derived in [3]. Since it was first presented, "Wyner-like" models have provided a framework for many works analyzing various transmission schemes in both the up-link and down-link channels (see [1] [5] and references therein).…”