# scipy.signal.firls

scipy.signal.firls(numtaps, bands, desired, weight=None, nyq=None, fs=None)

  FIR filter design using least-squares error minimization.

  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.

    nyq float, optional, deprecated

      This is the Nyquist frequency. Each frequency in bands must be between 0 and nyq (inclusive). Default is 1.

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

    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.

  See also

  `firwin`
  `firwin2`
  `minimum_phase`
  `remez`

  Notes

  This implementation follows the algorithm given in [1]. As noted there, least squares design has multiple advantages:

  > 1. Optimal in a least-squares sense.

  > 2. Simple, non-iterative method.

  > 3. The general solution can obtained by solving a linear system of equations.

  > 4. 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 \(n < numtaps\):

  \[coeffs(n) = coeffs(numtaps - 1 - n)\]

  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).

  New in version 0.18.

  References

  [1]

  Ivan Selesnick, Linear-Phase Fir Filter Design By Least Squares. OpenStax CNX. Aug 9, 2005. http://cnx.org/contents/eb1ecb35-03a9-4610-ba87-41cd771c95f2@7

  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()