Quick simulation: a review of importance sampling techniques in communications systems

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.

Stratified Sampling Monte Carlo Algorithm for Efficient BER Estimation in Long-Haul Optical Transmission Systems

“…Hence, most samples in stratum 2 will not be propagated in the true system. Such a gain in efficiency is similar to the one achieved by the improved IS method proposed in [13] and known as IS with excision [8].…”

“…The idea is based on running an MC estimation after observing an approximate, but usually faster to simulate, description of the system under test, and then to use such information to drive the estimation on the true system. It was shown that our proposed implementation of the method is a new form of adaptive IS, which is similar in spirit to the "excision IS" method [8]. The algorithm showed a significant computational gain compared to standard MC at practical values of to .…”

“…We will show that our practical implementation of the SS idea is a novel form of adaptive importance sampling (IS) [8].…”

Statistical Simulator for Block Coded Channels with Long Residual Interference

“…This conversion is illustrated for one coordinate. The parity patterns present are converted to PAM2 (here, '0 → +1) and mapped to the corresponding channel coefficients as given in (2). The probability counts are extracted from the field associated with each type and normalized by p −1 = 2 k , corresponding to the the total number of possible bit patterns associated with one codeword.…”

“…Sample-size reduction methods, such as importance sampling, exist, but long channel memory reduces their effectiveness of due to the dimensionality effect [2]. For channels with long memory, analytically computing the probability distributions associated with received voltages is common practice [3]- [5] but all such simulators assume the transmitted bits to be independent.…”