We have defined the convolution of two functions for the continuous case in equation (), and have given the convolution theorem as equation (). The theorem says that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms. | 538 Chapter 13. Fourier and Spectral Applications Convolution and Deconvolution Using the FFT We have defined the convolution of two functions for the continuous case in equation and have given the convolution theorem as equation . The - -o o theorem says that the Fourier transform of the convolution of two functions is equal g. to the product of their individual Fourier transforms. Now we want to deal with the discrete case. We will mention first the context in which convolution is a useful 1 .-5 procedure and then discuss how to compute it efficiently using the FFT. The convolution of two functions r t and s t denoted r s is mathematically J 11 equal to their convolution in the opposite order s r. Nevertheless in most M S applications the two functions have quite different meanings and characters. One of the functions say s is typically a signal or data stream which goes on indefinitely in time or in whatever the appropriate independent variable may be . The other function r is a response function typically a peaked function that falls to zero in g both directions from its maximum. The effect of convolution is to smear the signal s t in time according to the recipe provided by the response function r t as shown in Figure . In particular a spike or delta-function of unit area in s which occurs at some time t0 is supposed to be smeared into the shape of the response function itself but translated from time 0 to time t0 as r t - t0 . In the discrete case the signal s t is represented by its sampled values at equal a 8 time intervals sj. The response function is also a discrete set of numbers r k with the 3 following interpretation r0 tells what multiple of the input signal in one channel one 83 p particular value of j is copied into the identical output channel same value of j . 5 r1 tells what multiple of input signal in channel j is additionally copied into output p S. I channel j 1 r-1 tells the multiple that is copied into channel j - 1
