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

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 reduced 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

Random matrix ensembles: Wang-Landau algorithm for spectral densities

We propose a method based on the Wang-Landau algorithm to numerically generate the spectral densities of random matrix ensembles. The method employs Dyson's log-gas formalism for random matrix eigenvalues and also enables one to simultaneously investigate the thermodynamic properties. This approach is a powerful alternative to the conventionally used Monte Carlo simulations based on the Boltzmann sampling, and is ideally suited for investigating beta-ensembles.Comment: 8 pages, 5 figures, To appear in EP

Polynomial matrix decomposition techniques for frequency selective MIMO channels

For a narrowband, instantaneous mixing multi-input, multi-output (MIMO) communications system, the channel is represented as a scalar matrix. In this scenario, singular value decomposition (SVD) provides a number of independent spatial subchannels which can be used to enhance data rates or to increase diversity. Alternatively, a QR decomposition can be used to reduce the MIMO channel equalization problem to a set of single channel equalization problems. In the case of a frequency selective MIMO system, the multipath channel is represented as a polynomial matrix. Thus conventional matrix decomposition techniques can no longer be applied. The traditional solution to this broadband problem is to reduce it to narrowband form by using a discrete Fourier transform (DFT) to split the broadband channel into N narrow uniformly spaced frequency bands and applying scalar decomposition techniques within each band. This describes an orthogonal frequency division multiplexing (OFDM) based system. However, a novel algorithm has been developed for calculating the eigenvalue decomposition of a para-Hermitian polynomial matrix, known as the sequential best rotation (SBR2) algorithm. SBR2 and its QR based derivatives allow a true polynomial singular value and QR decomposition to be formulated. The application of these algorithms within frequency selective MIMO systems results in a fundamentally new approach to exploiting spatial diversity. Polynomial matrix decomposition and OFDM based solutions are compared for a wide variety of broadband MIMO communication systems. SVD is used to create a robust, high gain communications channel for ultra low signal-to-noise ratio (SNR) environments. Due to the frequency selective nature of the channels produced by polynomial matrix decomposition, additional processing is required at the receiver resulting in two distinct equalization techniques based around turbo and Viterbi equalization. The proposed approach is found to provide identical performance to that of an existing OFDM scheme while supporting a wider range of access schemes. This work is then extended to QR decomposition based communications systems, where the proposed polynomial approach is found to not only provide superior bit-error-rate (BER) performance but significantly reduce the complexity of transmitter design. Finally both techniques are combined to create a nulti-user MIMO system that provides superior BER performance over an OFDM based scheme. Throughout the work the robustness of the proposed scheme to channel state information (CSI) error is considered, resulting in a rigorous demonstration of the capabilities of the polynomial approach

On the sphere-decoding algorithm I. Expected complexity

The problem of finding the least-squares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worst-case complexity of the integer least-squares problem, we study its expected complexity, averaged over the noise and over the lattice. For the "sphere decoding" algorithm of Fincke and Pohst, we find a closed-form expression for the expected complexity, both for the infinite and finite lattice. It is demonstrated in the second part of this paper that, for a wide range of signal-to-noise ratios (SNRs) and numbers of antennas, the expected complexity is polynomial, in fact, often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can, in fact, be implemented in real time - a result with many practical implications