#### Audio FIR Filters

FIR filters shorter than the ear's ``integration time'' can generally be characterized by their magnitude frequency response (no perceivable ``delay effects''). The nominal ``integration time'' of the ear can be defined as the reciprocal of a critical bandwidth of hearing. Using Zwicker's definition of critical bandwidth [305], the smallest critical bandwidth of hearing is approximately 100 Hz (below 500 Hz). Thus, the nominal integration time of the ear is 10ms below 500 Hz. (Using the equivalent-rectangular-bandwidth (ERB) definition of critical bandwidth [179,269], longer values are obtained). At a 50 kHz sampling rate, this is 500 samples. Therefore, FIR filters shorter than the ear's ``integration time,''*i.e.*, perceptually ``instantaneous,'' can easily be hundreds of taps long (as discussed in the next section). FFT convolution is consequently an important implementation tool for FIR filters in digital audio applications.

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Example 1: Low-Pass Filtering by FFT Convolution

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Pictorial View of Acyclic Convolution