scipy.signal.remez

scipy.signal.remez(numtaps, bands, desired, weight=None, Hz=None, type='bandpass', maxiter=25, grid_density=16, fs=None)

Calculate the minimax optimal filter using the Remez exchange algorithm.

Calculate the filter-coefficients for the finite impulse response (FIR) filter whose transfer function minimizes the maximum error between the desired gain and the realized gain in the specified frequency bands using the Remez exchange algorithm.

Parameters:

numtapsint

The desired number of taps in the filter. The number of taps is the number of terms in the filter, or the filter order plus one.

bandsarray_like

A monotonic sequence containing the band edges. All elements must be non-negative and less than half the sampling frequency as given by fs.

desiredarray_like

A sequence half the size of bands containing the desired gain in each of the specified bands.

weightarray_like, optional

A relative weighting to give to each band region. The length of weight has to be half the length of bands.

Hzscalar, optional, deprecated

The sampling frequency in Hz. Default is 1.

Deprecated since version 1.0.0: remez keyword argument Hz is deprecated in favour of fs and will be removed in SciPy 1.12.0.

type{‘bandpass’, ‘differentiator’, ‘hilbert’}, optional

The type of filter:

• ‘bandpass’ : flat response in bands. This is the default.

• ‘differentiator’ : frequency proportional response in bands.

• ‘hilbert’filter with odd symmetry, that is, type III

  (for even order) or type IV (for odd order) linear phase filters.

maxiterint, optional

Maximum number of iterations of the algorithm. Default is 25.

grid_densityint, optional

Grid density. The dense grid used in remez is of size (numtaps + 1) * grid_density. Default is 16.

fsfloat, optional

The sampling frequency of the signal. Default is 1.

Returns:

outndarray

A rank-1 array containing the coefficients of the optimal (in a minimax sense) filter.

See also

firls

firwin

firwin2

minimum_phase

References

[1]

J. H. McClellan and T. W. Parks, “A unified approach to the design of optimum FIR linear phase digital filters”, IEEE Trans. Circuit Theory, vol. CT-20, pp. 697-701, 1973.

[2]

J. H. McClellan, T. W. Parks and L. R. Rabiner, “A Computer Program for Designing Optimum FIR Linear Phase Digital Filters”, IEEE Trans. Audio Electroacoust., vol. AU-21, pp. 506-525, 1973.

Examples

In these examples, remez is used to design low-pass, high-pass, band-pass and band-stop filters. The parameters that define each filter are the filter order, the band boundaries, the transition widths of the boundaries, the desired gains in each band, and the sampling frequency.

We’ll use a sample frequency of 22050 Hz in all the examples. In each example, the desired gain in each band is either 0 (for a stop band) or 1 (for a pass band).

freqz is used to compute the frequency response of each filter, and the utility function plot_response defined below is used to plot the response.

>>> import numpy as np
>>> from scipy import signal
>>> import matplotlib.pyplot as plt

>>> fs = 22050   # Sample rate, Hz

>>> def plot_response(w, h, title):
...     "Utility function to plot response functions"
...     fig = plt.figure()
...     ax = fig.add_subplot(111)
...     ax.plot(w, 20*np.log10(np.abs(h)))
...     ax.set_ylim(-40, 5)
...     ax.grid(True)
...     ax.set_xlabel('Frequency (Hz)')
...     ax.set_ylabel('Gain (dB)')
...     ax.set_title(title)

The first example is a low-pass filter, with cutoff frequency 8 kHz. The filter length is 325, and the transition width from pass to stop is 100 Hz.

>>> cutoff = 8000.0    # Desired cutoff frequency, Hz
>>> trans_width = 100  # Width of transition from pass to stop, Hz
>>> numtaps = 325      # Size of the FIR filter.
>>> taps = signal.remez(numtaps, [0, cutoff, cutoff + trans_width, 0.5*fs],
...                     [1, 0], fs=fs)
>>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
>>> plot_response(w, h, "Low-pass Filter")
>>> plt.show()

This example shows a high-pass filter:

>>> cutoff = 2000.0    # Desired cutoff frequency, Hz
>>> trans_width = 250  # Width of transition from pass to stop, Hz
>>> numtaps = 125      # Size of the FIR filter.
>>> taps = signal.remez(numtaps, [0, cutoff - trans_width, cutoff, 0.5*fs],
...                     [0, 1], fs=fs)
>>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
>>> plot_response(w, h, "High-pass Filter")
>>> plt.show()

This example shows a band-pass filter with a pass-band from 2 kHz to 5 kHz. The transition width is 260 Hz and the length of the filter is 63, which is smaller than in the other examples:

>>> band = [2000, 5000]  # Desired pass band, Hz
>>> trans_width = 260    # Width of transition from pass to stop, Hz
>>> numtaps = 63         # Size of the FIR filter.
>>> edges = [0, band[0] - trans_width, band[0], band[1],
...          band[1] + trans_width, 0.5*fs]
>>> taps = signal.remez(numtaps, edges, [0, 1, 0], fs=fs)
>>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
>>> plot_response(w, h, "Band-pass Filter")
>>> plt.show()

The low order leads to higher ripple and less steep transitions.

The next example shows a band-stop filter.

>>> band = [6000, 8000]  # Desired stop band, Hz
>>> trans_width = 200    # Width of transition from pass to stop, Hz
>>> numtaps = 175        # Size of the FIR filter.
>>> edges = [0, band[0] - trans_width, band[0], band[1],
...          band[1] + trans_width, 0.5*fs]
>>> taps = signal.remez(numtaps, edges, [1, 0, 1], fs=fs)
>>> w, h = signal.freqz(taps, [1], worN=2000, fs=fs)
>>> plot_response(w, h, "Band-stop Filter")
>>> plt.show()