TABLE OF CONTENTS In Chapter 2 we stated that the impulse response is the most important feature of a linear system, because it characterises it completely. Later, in Chapter 4, we saw how differential equations and difference equations could be used to describe or derive the impulse and step responses of linear systems. However, as yet we have not shown how the impulse response may be used to replicate or predict the response of such systems to any input signal. This is where convolution enters our discussions. It is impossible to overstate the importance of this operation, because so many DSP algorithms exploit convolution in one form or another. It is so central to the entire arena of digital signal processing that it is worth stating in a formal manner as follows:
If the impulse response h( t) of a linear system is known, then it may be convolved with any input signal x( t) to produce the system s response, denoted as y( t).
Now convolution algorithms are applied for many purposes; possibly the most common is for filtering. As we saw in the last chapter, filters are linear systems, and digital equivalents of analog designs are implemented as convolution operations. Perhaps more subtly, when we use an electronic analog filter with an electrical signal, the precise effect the circuit has on the incoming signal may be both understood and modelled using the convolution integral. More generally, there is a huge range of...
