The Scientist and Engineer's Guide to
Digital Signal Processing
By Steven W. Smith, Ph.D.

Chapter 7: Properties of Convolution

A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on. This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Forth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers.

• Common Impulse Responses
• Mathematical Properties
• Correlation
• Speed