uency of the signal.  Default is 1.

Returns
-------
out : ndarray
    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()

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