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9.5 Example: Nyquist Filter Design

Nyquist filters are odd-length linear-phase (i.e., Type 1) FIR filters whose coefficients satisfy

stands for the integer part of x, N is the filter length, and k is the decimation/interpolation order. The property that every kth impulse response sample is zero except for the central coefficient, makes these filters particularly suitable for the implementation of interpolators and decimators. The placement of the zero samples in the filter matches locations of the nonzero samples in the upsampled signal, which allows these samples to remain unchanged as a result of the interpolation operation. The ideal Nyquist filter of order k has a frequency response with a cutoff at

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Here is the impulse response of an ideal Nyquist filter of order k=2.

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This defines the impulse response of the ideal half-band filter. First we clear any prior definitions of symbol h.

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The impulse response of an ideal half-band filter has the form of a sinc sequence.

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Here is a plot of the impulse response. Note the location of the zero-valued samples.

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Nyquist filters are also known as kth band filters. This terminology is made clear by an examination of the frequency response. The two-band filter has a passband width of

Hz, thus splitting the fundamental interval into two equal bands. It is thus also called a half-band filter. A kth band filter has a passband of width

Hz. Here is the magnitude spectrum of a half-band filter.

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Nyquist filters may be easily obtained by using the window filter design method. The design calls for a lowpass filter with cutoff frequency given by

, where k is the interpolation/decimation order (see Interpolate). Specifically, here is a Nyquist filter with f_c=0.25 (i.e., k=2) and length N=11.

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Here we load one of the example images.

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We now use filter q to decimate the example image. Here are the original and decimated images shown at the same resolution.

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