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 the Fast Fourier Transform (FFT) to implement cyclic convolution when its length is a power of 2.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.