Introduction to Random Signals and Processes2.12002/06/062002/06/17MichaelHaagmjhaag@rice.eduMichaelHaagmjhaag@rice.edudeterministicrandomrandom processrandom signalrandom signalsstochasticThe module will introduce the concepts of a random signal and a random process.
Before now, you have probably dealt strictly with the theory
behind signals and systems, as well as look at some the basic
characteristics of signals and systems. In doing so you have developed an
important foundation; however, most electrical engineers do not
get to work in this type of fantasy world. In many cases the
signals of interest are very complex due to the randomness of
the world around them, which leaves them noisy and often
corrupted. This often causes the information contained in the
signal to be hidden and distorted. For this reason, it is
important to understand these random signals and how to recover
the necessary information.
Signals: Deterministic vs. Stochastic
For this study of signals and systems, we will divide signals
into two groups: those that have a fixed behavior and those
that change randomly. As most of you have probably already
dealt with the first type, we will focus on introducing you to
random signals. Also, note that we will be dealing strictly
with discrete-time signals since they are the signals we deal
with in DSP and most real-world computations, but these same
ideas apply to continuous-time signals.
Deterministic Signals
Most introductions to signals and systems deal strictly with
deterministic signals. Each value of these
signals are fixed and can be determined by a mathematical
expression, rule, or table. Because of this, future values
of any deterministic signal can be calculated from past
values. For this reason, these signals are relatively easy
to analyze as they do not change, and we can make accurate
assumptions about their past and future behavior.
Stochastic Signals
Unlike deterministic signals, stochastic
signals, or random signals, are not so
nice. Random signals cannot be characterized by a simple,
well-defined mathematical equation and their future values
cannot be predicted. Rather, we must use probability and
statistics to analyze their behavior. Also, because of
their randomness, average
values from a collection of signals are usually
studied rather than analyzing one individual signal.
Random Process
As mentioned above, in order to study random signals, we want
to look at a collection of these signals rather than just one
instance of that signal. This collection of signals is called
a random process.
random process
A family or ensemble of signals that correspond to every
possible outcome of a certain signal measurement. Each
signal in this collection is referred to as a
realization or sample function
of the process.
As an example of a random process, let us look at the
Random Sinusoidal Process below. We use
fnAωnφ to represent the sinusoid with a given
amplitude and phase. Note that the phase and amplitude
of each sinusoid is based on a random number, thus
making this a random process.
A random process is usually denoted by
Xt or
Xn, with
xt or
xn used to represent an individual signal or waveform
from this process.
In many notes and books, you might see the following notation
and terms used to describe different types of random
processes. For a discrete random process,
sometimes just called a random sequence,
t represents time that has a
finite number of values. If t
can take on any value of time, we have a continuous
random process. Often times discrete and continuous
refer to the amplitude of the process, and process or sequence
refer to the nature of the time variable. For this study, we
often just use random process to refer to a
general collection of discrete-time signals, as seen above in
.