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## Convolution of Short Signals Figure 8.1 illustrates the conceptual operation of filtering an input signal by a filter with impulse-response to produce an output signal . By the convolution theorem for DTFTs2.3.5), (9.9)

or, (9.10)

where and are arbitrary real or complex sequences, and and are the DTFTs of and , respectively. The convolution of and is defined by (9.11)

In practice, we always use the DFT (preferably an FFT) in place of the DTFT, in which case we may write (9.12)

where now (length complex sequences). It is important to remember that the specific form of convolution implied in the DFT case is cyclic (also called circular) convolution [#!MDFT!#]: (9.13)

where means modulo .''

Another way to look at convolution is as the inner product of , and , where , i.e., (9.14)

This form describes graphical convolution in which the output sample at time is computed as an inner product of the impulse response after flipping it about time 0 and shifting time 0 to time . See [#!MDFT!#, p. 105] for an illustration of graphical convolution.

Subsections
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