15-05-2014, 04:59 PM
GENERATING AND PROCESSING RANDOM SIGNALS
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INTRODUCTION
To this point we have been concerned with deterministic signals in simulations.
In all communication systems of practical interest, random effects such as channel
noise, interference, and fading, degrade the information-bearing signal as it passes
through the system from information source to the final user. Accurate simulation
of these systems at the waveform level requires that these random effects be modeled
accurately. Therefore, algorithms are required to produce these random effects. The
fundamental building block is the random number generator. While much can be
said about random number generators (several books and many research papers have
been written on the subject), the emphasis in this chapter is on the use of random
number generators in the simulation of communication systems. Thus, we restrict
our study to the essential task of generating sampled versions of random waveforms
(signals, interference, noise, etc.) for use in simulation programs. In the simulation
context, all random processes must be expressed by sequences of random variables.
Generating and testing these random sequences are the subject of this chapter.
Stationary and Ergodic Processes
When simulating a communications system, the sample functions generated to rep-
resent signals, noise, and interference will be assumed ergodic. This is required,
since we typically process time-domain samples of waveforms through the system
sequentially and, at each point in the system, there is a single waveform (sample
function).
Uniform Random Number Generators
A random variable having a uniform probability density function is easily trans-
formed to a random variable having a desired pdf other than uniform. Therefore,
the first step in the generation of a random variable having a specified pdf is to
generate a random variable that is uniformly distributed on the interval (0,1). This
is typically accomplished by first generating a sequence of numbers (integers) be-
tween 0 and M and then dividing each element of the sequence by M . The most
common technique for implementing random number generators is known as linear
congruence.
Testing Random Number Generators
The previous section provides us with the tools for generating pseudo-random num-
bers that are uniformly distributed between 0 and 1. To this point we have con-
sidered only the period of the sequence produced by an LCG. While we obviously
wish this period to be long, there are other desirable attributes to be satisfied for a
given application. At the very least, we desire the sequence to be delta correlated
(white). Other requirements may be necessitated by the application.
A number of procedures have been developed for testing the randomness of a
given sequence. Among the most popular of these are the Chi-square test, the
Kolomogorov-Smirnov test, and the spectral test. A study of these is beyond the
scope of the material presented here. The interested student is referred to the
literature [1, 2]. The spectral test appears to be the most powerful of these tests.
A brief description of the spectral test, applied to the Wichmann-Hill algorithm to
be discussed later, is given in the paper by Coates [3].