firls(numtaps, bands, desired, weight=None, nyq=None, fs=None)
Calculate the filter coefficients for the linear-phase finite impulse response (FIR) filter which has the best approximation to the desired frequency response described by bands and desired in the least squares sense (i.e., the integral of the weighted mean-squared error within the specified bands is minimized).
This implementation follows the algorithm given in . As noted there, least squares design has multiple advantages:
- Optimal in a least-squares sense.
- Simple, non-iterative method.
- The general solution can obtained by solving a linear system of equations.
- Allows the use of a frequency dependent weighting function.
This function constructs a Type I linear phase FIR filter, which contains an odd number of coeffs satisfying for n < numtaps:
The odd number of coefficients and filter symmetry avoid boundary conditions that could otherwise occur at the Nyquist and 0 frequencies (e.g., for Type II, III, or IV variants).
The number of taps in the FIR filter. numtaps must be odd.
A monotonic nondecreasing sequence containing the band edges in Hz. All elements must be non-negative and less than or equal to the Nyquist frequency given by nyq. The bands are specified as frequency pairs, thus, if using a 1D array, its length must be even, e.g., np.array([0, 1, 2, 3, 4, 5]). Alternatively, the bands can be specified as an nx2 sized 2D array, where n is the number of bands, e.g, np.array([[0, 1], [2, 3], [4, 5]]).
A sequence the same size as bands containing the desired gain at the start and end point of each band.
The sampling frequency of the signal. Each frequency in bands must be between 0 and fs/2 (inclusive). Default is 2.
Coefficients of the optimal (in a least squares sense) FIR filter.
FIR filter design using least-squares error minimization.
import numpy as np
from scipy import signal
import matplotlib.pyplot as plt
fig, axs = plt.subplots(2)
fs = 10.0 # Hz
desired = (0, 0, 1, 1, 0, 0)
for bi, bands in enumerate(((0, 1, 2, 3, 4, 5), (0, 1, 2, 4, 4.5, 5))):
fir_firls = signal.firls(73, bands, desired, fs=fs)
fir_remez = signal.remez(73, bands, desired[::2], fs=fs)
fir_firwin2 = signal.firwin2(73, bands, desired, fs=fs)
hs = list()
ax = axs[bi]
for fir in (fir_firls, fir_remez, fir_firwin2):
freq, response = signal.freqz(fir)
hs.append(ax.semilogy(0.5*fs*freq/np.pi, np.abs(response))[0])
for band, gains in zip(zip(bands[::2], bands[1::2]),
zip(desired[::2], desired[1::2])):
ax.semilogy(band, np.maximum(gains, 1e-7), 'k--', linewidth=2)
if bi == 0:
ax.legend(hs, ('firls', 'remez', 'firwin2'),
loc='lower center', frameon=False)
else:
ax.set_xlabel('Frequency (Hz)')
ax.grid(True)
ax.set(title='Band-pass %d-%d Hz' % bands[2:4], ylabel='Magnitude')
fig.tight_layout()
plt.show()
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.signal._fir_filter_design:remez
scipy.signal._fir_filter_design:firwin
scipy.signal._fir_filter_design:firwin2
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