A digital filter is simply a discrete-time, discrete-amplitude
convolver. An example is shown in Figure 1 for three-bit
amplitude quantization. Basic Fourier transform theory states
that the linear convolution of two sequences in the time
domain is the same as multiplication of two corresponding
spectral sequences in the frequency domain. Filtering is in
essence the multiplication of the signal spectrum by the
frequency domain impulse response of the filter. For an ideal
lowpass filter the pass band part of the signal spectrum is
multiplied by one and the stopband part of the signal by zero.
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