For example, could be a ``rectangularly windowed signal, zero-padded by
a factor of 2,'' where the signal happened to be dc (all s).
convolution, we need
which is the same as . When
, we say that is
matched filter for .7.5 In this case, is matched to look for a
``dc component,'' and also zero-padded by a factor of . The
zero-padding serves to simulate acyclic convolution using circular
convolution. Note from Eq. (7.3) that the maximum is obtained
in the convolution output at time 0. This peak (the largest
possible if all input signals are limited to in magnitude),
indicates the matched filter has ``found'' the dc signal starting at
time 0. This peak would persist even if various sinusoids at other
frequencies and/or noise were added in.