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, asgoverned by the mathematics of convolution. This is the basis of many signal processingtechniques. For example: Digital filters are created bydesigning an appropriate impulseresponse. Enemy aircraft are detected with radar byanalyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulseresponse thatcounteracts 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, severalcommon impulse responses are discussed. Second, methods are presented for dealing withcascade and parallel combinations of linear systems. Third, the technique ofcorrelation isintroduced. Forth, a nasty problem with convolution is examined, the computation time can beunacceptably long using conventional algorithms and computers.

• Common Impulse Responses
• Mathematical Properties
• Correlation
• Speed