#### 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