#### Pictorial View of Acyclic Convolution

Figure 8.2 shows schematically the result of convolving
two zero-padded signals
and
. In this case, the signal
starts some time after
, say at
. Since
begins at
time **0**
, the output starts promptly at time
, but it takes some
time to ``ramp up'' to full amplitude. (This is the *transient
response* of the FIR filter
.) If the length of
is
, then
the transient response is finished at time
. Next, when
the input signal goes to zero at time
, the output reaches
zero
samples later (after the filter ``decay time''), or time
. Thus, the total number of nonzero output samples is
.

If we don't add enough zeros, some of our convolution terms ``wrap
around'' and add back upon others (due to modulo indexing). This can
be called *time-domain aliasing*. Zero-padding in the time
domain results in more samples (closer spacing) in the frequency
domain, *i.e.*, a higher `sampling rate' in the frequency domain. If we
have a high enough spectral sampling rate, we can avoid time aliasing.

The motivation for implementing acyclic convolution using a zero-padded cyclic convolution is that we can use a Cooley-Tukey Fast Fourier Transform (FFT) to implement cyclic convolution when its length is a power of 2.

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Acyclic Convolution in Matlab