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symmetrical, as would be the case with a linear phase filter, or in an optical system, the mirroring process is
superfluous.
Figure 3.6:
In the convolution of two continuous signals (the impulse response with the input), the impulse must be
time reversed or mirrored. This is necessary because the impulse will be moved from left to right, and mirroring
gives the impulse the correct time-domain response when it is moved past a fixed point. As the impulse response
slides continuously through the input waveform, the area where the two overlap determines the instantaneous
output amplitude. This is shown for five different times by the crosses on the output waveform.
The same process can be performed in the sampled, or discrete time domain as shown in
Figure 3.7
. The impulse
and the input are now a set of discrete samples which clearly must have the same sample spacing. The impulse
response only has value where impulses coincide. Elsewhere it is zero. The impulse response is therefore stepped
through the input one sample period at a time. At each step, the area is still proportional to the output, but as the
time steps are of uniform width, the area is proportional to the impulse height and so the output is obtained by
adding up the lengths of overlap. In mathematical terms, the output samples represent the convolution of the input
and the impulse response by summing the coincident cross-products.
