OVERVIEW

So far we have studied Fourier analysis as a way to observe an event and analyze its constituent frequencies. Now we consider the processing of a signal: integrating, differentiating, smoothing, and filtering input signals to produce a desired outcome. Much of the processing in a digital signal processing refers to implementing solutions of differential equations, whether in a temperature control system or in speech analysis or vibration studies. The cause-and-effect relationship lends itself to the formulation of differential equations. We encounter them when we design analog electrical circuits and mechanical systems, and since digital signal processing has its roots in analog signal processing, it is imperative to understand the methodology of deriving the algorithms to solve differential equations.

In this chapter, we will devote our attention to the convolution process, as the method of solving a differential equation, while the other technique of Laplace Transforms will be deferred until the next chapter. If our goal is to design a simple integrator or differentiator, we need only derive a difference equation to be implemented as an iterative algorithm in a digital computer, but implementing a digital or analog filter requires deriving a closed form expression called a transfer function. A transfer function defines the output of a system as a function of frequency, indicating what frequencies will be suppressed while others are available without degradation. We begin our study with a refresher and establish the necessary mathematical foundation.