No abstract

Importance sampling (IS) is a simulation technique which aims to reduce the variance (or other cost function) of a given simulation estimator. In communication systems, this usually, but not always, means attempting to reduce the variance of the bit error rate (BER) estimator. By reducing the variance, IS estimators can achieve a given precision from shorter simulation runs; hence the term "quick simulation." The idea behind IS is that certain values of the input random variables in a simulation have more impact on the parameter being estimated than others. If these "important" values are emphasized by sampling more frequently, then the estimator variance can be reduced. Hence, the basic methodology in IS is to choose a distribution which encourages the important values. This use of a "biased" distribution will, of course, result in a biased estimator if applied directly in the simulation. However, there is a simple procedure whereby the simulation outputs are weighted to correct for the use of the biased distribution, and this ensures that the new IS estimator is unbiased. Hence, the "art" of designing quick simulations via IS is entirely dependent on the choice of biased distribution. Over the last 50 years, IS techniques have flourished, but it is only in the last decade that coherent design methods have emerged. The outcome of these developments is that at the expense of increasing technical content, modern techniques can offer substantial runtime saving for a very broad range of problems. In this paper, we present a comprehensive history and survey of IS methods. In addition, we offer a guide to the strengths and weaknesses of the techniques, and hence indicate which techniques are suitable for various types of communications systems. We stress that simple approaches can still yield useful savings, and so the simulation practitioner as well as the technical researcher should consider IS as a possible simulation tool.

Let the column vectors of X: M_N, M

This paper presents an exact derivation of the statistical distribution of the signal to interference-plus-noise ratio (SENR) for optimum linear combining in wireless systems with multiple cochannel interferes, Rayleigh fading, and additive white Gaussian noise. The distribution of the SINR is shown to be remarkably simple and leads to bounds on the bit error rate and outage probabilities which are tighter, simpler, and more robust than any previous results. The simplicity of the SINR distribution permits extremely fast computation of outage probabilities for any number of interference channels and diversity levels. Hence for wireless systems it enables performance studies to be performed over a much wider range of conditions, such as shadow fading, specific channel allocation methods, etc. Previously such studies were extremely limited due to the intensive computational requirements of simulating these systems.

The paper derives bounds on the distribution of the quadratic forms Z = y H (X X H ) −1 y and W = y H (σ 2 I + X X H ) −1 y, where the elements of the M × 1 vector y and the M × N matrix X are independent identically distributed (i.i.d.) complex zero mean Normal variables, is some N × N diagonal matrix with positive diagonal elements, I is the identity, σ 2 is a constant and H denotes the Hermitian transpose. The bounds are convenient for numerical work and appear to be tight for small values of M. This work has applications in digital mobile radio for a specific channel where M antennas are used to receive a signal with N interferers. Some of these applications in radio communication systems are discussed.