4.4 Convolution Theorem

As mentioned in Section 4.1 the Fourier transform of the convolution operation remarkably turns out to be a single complex multiplication of the respective Fourier transforms in the frequency domain.  The quantities multiplied in the frequency domain are the complex spectra of the convolved signals.  This simple relationship provides wonderful insight into many, many filtering operations and mathematical expressions as you will see.  Explicitly we have

(4.4.1)

 

Figure 4.7 graphically illustrates this Fourier transform pair using two rectangle functions each with amplitude A.

Figure 4.7. Graphical example of the convolution theorem.

 

There is a reciprocity relation between convolution in the time domain and its counterpart in the frequency domain.  That is, convolution in the frequency domain becomes a multiplication in the time (or space) domain.  This is sometimes called the "frequency domain convolution theorem."  The Fourier transform pair representing this result is

(4.4.2)

Figure 4.8 illustrates this result using cosine and rectangle functions in the time domain.

 

Figure 4.8. Graphical example of the frequency convolution theorem.