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Prologue

Many of the physical processes that give rise to signals can be described as deterministic. By this we mean that the value of a “signal”—measured in volts, meters, Euros, kilograms, Pascals, etc.—can be described at every point in time. There are, however, many physical processes where the value of a physical variable is not predictable. The underlying physical processes may be inherently stochastic1 (i.e. random), as in processes closely allied to quantum physics such as photon production. Alternatively, the physical processes may be so complex and involve so many variables that we are not capable of describing them in such a way that we can predict completely the time course of events. One well-known example is speech and another is the weather with physical variables being the temperature as a function of time or the wind velocity or wind direction as a function of time. Two examples are given in Chapter 5.

Nevertheless, we should like to be able to use the tools of signal processing to filter, control, or analyze such physical variables. The goal of this iBook is to extend the domain of signal processing tools to include stochastic signals, in particular, those signals that are a function of time.

Modern textbooks that develop and describe signal processing frequently present both continuous-time and discrete-time approaches to convolution, Fourier analysis, filtering, modulation, complex exponential transforms, and feedback systems. Further, continuous-time and discrete-time considerations are tied together through considerations of sampling and reconstruction which are frequently developed in the context of the Nyquist sampling theorem after Dr. Harry Nyquist (1889-1976).

In this iBook we introduce stochastic signal processing in the context of discrete-time signals and discrete-time systems, what is frequently termed digital signal processing. We do so because modern use of these techniques, with only a few exceptions, is in the discrete domain.

When appropriate, as in Chapter 7, we will develop, refer to, and/or use concepts from continuous-time signal processing but, in general, we will focus on discrete-time processing.

Finally, we will occasionally ask the question “What does it mean?”. The pioneer of this question in the 20th century was Professor Irwin Corey, the World’s Foremost Authority. After a monologue on a particular topic he would, in a plaintive voice, ask this question. It took us a long time in education to realize that he was on to something and that students (and their teachers) have to repeatedly ask this question. To get you started, consider Problem 2.1.

Problems

Problem 2.1

A well-known result in linear, time-invariant signal processing is that convolution is commutative, that is, \(h[n] \otimes x[n] = x[n] \otimes h[n]\) where “\(\otimes\)” is the convolution operation.

  1. What does this mean?
  2. What opportunities are represented through the commutative property of convolution?

  1. From the Oxford English Dictionary: Stochastic – randomly determined; that follows some random probability distribution or pattern, so that its behaviour may be analysed statistically but not predicted precisely; From the Greek: to aim at a mark, to guess. ↩