`scipy.signal._fir_filter_design:firls`

source: /scipy/signal/_fir_filter_design.py :958

Signature

   def     firls (    numtaps    ,    bands    ,    desired    ,  * ,     weight    =  None   ,    fs    =  None     )

Summary

FIR filter design using least-squares error minimization.

Extended Summary

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).

Parameters

numtaps : int: The number of taps in the FIR filter. numtaps must be odd.
bands : array_like: 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]]).
desired : array_like: A sequence the same size as bands containing the desired gain at the start and end point of each band.
weight : array_like, optional: A relative weighting to give to each band region when solving the least squares problem. weight has to be half the size of bands.
fs : float, optional: The sampling frequency of the signal. Each frequency in bands must be between 0 and fs/2 (inclusive). Default is 2.

Returns

coeffs : ndarray: Coefficients of the optimal (in a least squares sense) FIR filter.

Notes

This implementation follows the algorithm given in ^[1]. 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 $n < n u m t a p s$ :

coe f f s (n) = coe f f s (n u m t a p s - 1 - n)

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).

Array API Standard Support

firls has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ⛔                   
PyTorch               ✅                     ⛔                   
JAX                   ⚠️ no JIT             ⛔                   
Dask                  ⚠️ computes graph     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

We want to construct a band-pass filter. Note that the behavior in the frequency ranges between our stop bands and pass bands is unspecified, and thus may overshoot depending on the parameters of our filter:

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()

✓

Aliases

scipy.signal.firls