Design Nyquist Filters
- Updated2025-10-10
- 2 minute(s) read
Nyquist filters, also called Mth band filters, are a special type of multirate Finite Impulse Response (FIR) filter. Nyquist filter coefficients have periodic zero values every Mth sample, except for the middle coefficient.
Nyquist filters have the following magnitude response specifications:
| Filter Specification | Value Range |
|---|---|
| Passband edge frequency | [0, fs/2M−ε] |
| Stopband edge frequency | [fs/2M+ε, fs/2] |
In this table, M denotes the sampling frequency conversion factor and fs denotes the sampling frequency of a Nyquist filter. For an interpolation Nyquist filter, fs equals L times the sampling frequency of the input signal, where L denotes the interpolation factor. For a decimation Nyquist filter, fs equals the sampling frequency of the input signal. You can specify ε indirectly by using the roll off α, which is defined as the following equation:
The following figure illustrates the magnitude response of a Nyquist filter:
The following figure shows the coefficients of a Nyquist filter with a sampling frequency conversion factor of 4. The 26th coefficient is the middle coefficient and does not have a zero value.
The impulse response of a Nyquist filter h(n) satisfies the following equation:
where c and k are constants.
The z-transform of a Nyquist filter H(z) satisfies the following equation:
where W = e−j2π/M and c = 1/M. The frequency response of H(zWk) is the shifted version of the frequency response of H(z), so the frequency responses of all M uniformly shifted versions of H(z) add up to a constant.
Use the DFD Nyquist Design VI to design Nyquist filters. You can use Nyquist filters to remove images in interpolation. Nyquist filters modify the interpolated zeroes, but they do not change the original samples.