Note that the cyclic convolution operation can be expressed in terms of previously defined operators as
where and . It is instructive to interpret the last expression above graphically.
Figure 8.3 illustrates convolution of
For example, could be a “rectangularly windowed signal, zero-padded by a factor of 2,” where the signal happened to be dc (all s).
For the convolution, we need
which is the same as . When , we say that ismatched filter for .8.3 In this case, is matched to look for a “dc component,” and also zero-padded by a factor of . The zero-paddingserves to simulate acyclic convolution using circular convolution. The figure illustrates the computation of the convolution of and :
Note that a large peak is obtained in the convolution output at time . This large peak (the largest possible if all signals are limited to in magnitude), indicates the matched filter has “found” the dc signal starting at time . This peak would persist even if varioussinusoids at other frequencies and/or noise were added in.