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scipy.signal.

# freqz

scipy.signal.freqz(b, a=1, worN=512, whole=False, plot=None, fs=6.283185307179586, include_nyquist=False)

  Compute the frequency response of a digital filter.

  Given the M-order numerator b and N-order denominator a of a digital filter, compute its frequency response:

                  jw                 -jw              -jwM
         jw    B(e  )    b[0] + b[1]e    + ... + b[M]e
      H(e  ) = ------ = -----------------------------------
                  jw                 -jw              -jwN
               A(e  )    a[0] + a[1]e    + ... + a[N]e

  Parameters:
    b array_like

      Numerator of a linear filter. If b has dimension greater than 1, it is assumed that the coefficients are stored in the first dimension, and `b.shape[1:]`, `a.shape[1:]`, and the shape of the frequencies array must be compatible for broadcasting.

    a array_like

      Denominator of a linear filter. If b has dimension greater than 1, it is assumed that the coefficients are stored in the first dimension, and `b.shape[1:]`, `a.shape[1:]`, and the shape of the frequencies array must be compatible for broadcasting.

    worN{None, int, array_like}, optional

      If a single integer, then compute at that many frequencies (default is N=512). This is a convenient alternative to:

          np.linspace(0, fs if whole else fs/2, N, endpoint=include_nyquist)

      Using a number that is fast for FFT computations can result in faster computations (see Notes).

      If an array_like, compute the response at the frequencies given. These are in the same units as fs.

    whole bool, optional

      Normally, frequencies are computed from 0 to the Nyquist frequency, fs/2 (upper-half of unit-circle). If whole is True, compute frequencies from 0 to fs. Ignored if worN is array_like.

    plot callable

      A callable that takes two arguments. If given, the return parameters w and h are passed to plot. Useful for plotting the frequency response inside `freqz`.

    fs float, optional

      The sampling frequency of the digital system. Defaults to 2*pi radians/sample (so w is from 0 to pi).

      Added in version 1.2.0.

    include_nyquist bool, optional

      If whole is False and worN is an integer, setting include_nyquist to True will include the last frequency (Nyquist frequency) and is otherwise ignored.

      Added in version 1.5.0.

  Returns:
    w ndarray

      The frequencies at which h was computed, in the same units as fs. By default, w is normalized to the range [0, pi) (radians/sample).

    h ndarray

      The frequency response, as complex numbers.

  See also

  `freqz_zpk`
  `freqz_sos`

  Notes

  Using Matplotlib’s `matplotlib.pyplot.plot` function as the callable for plot produces unexpected results, as this plots the real part of the complex transfer function, not the magnitude. Try `lambda w, h: plot(w, np.abs(h))`.

  A direct computation via (R)FFT is used to compute the frequency response when the following conditions are met:

  1. An integer value is given for worN.

  2. worN is fast to compute via FFT (i.e., `next_fast_len(worN)` equals worN).

  3. The denominator coefficients are a single value (`a.shape[0] == 1`).

  4. worN is at least as long as the numerator coefficients (`worN >= b.shape[0]`).

  5. If `b.ndim > 1`, then `b.shape[-1] == 1`.

  For long FIR filters, the FFT approach can have lower error and be much faster than the equivalent direct polynomial calculation.

  Examples

      >>> from scipy import signal
      >>> import numpy as np
      >>> taps, f_c = 80, 1.0  # number of taps and cut-off frequency
      >>> b = signal.firwin(taps, f_c, window=('kaiser', 8), fs=2*np.pi)
      >>> w, h = signal.freqz(b)

      >>> import matplotlib.pyplot as plt
      >>> fig, ax1 = plt.subplots(tight_layout=True)
      >>> ax1.set_title(f"Frequency Response of {taps} tap FIR Filter" +
      ...               f"($f_c={f_c}$ rad/sample)")
      >>> ax1.axvline(f_c, color='black', linestyle=':', linewidth=0.8)
      >>> ax1.plot(w, 20 * np.log10(abs(h)), 'C0')
      >>> ax1.set_ylabel("Amplitude in dB", color='C0')
      >>> ax1.set(xlabel="Frequency in rad/sample", xlim=(0, np.pi))

      >>> ax2 = ax1.twinx()
      >>> phase = np.unwrap(np.angle(h))
      >>> ax2.plot(w, phase, 'C1')
      >>> ax2.set_ylabel('Phase [rad]', color='C1')
      >>> ax2.grid(True)
      >>> ax2.axis('tight')
      >>> plt.show()

  Broadcasting Examples

  Suppose we have two FIR filters whose coefficients are stored in the rows of an array with shape (2, 25). For this demonstration, we’ll use random data:

      >>> rng = np.random.default_rng()
      >>> b = rng.random((2, 25))

  To compute the frequency response for these two filters with one call to `freqz`, we must pass in `b.T`, because `freqz` expects the first axis to hold the coefficients. We must then extend the shape with a trivial dimension of length 1 to allow broadcasting with the array of frequencies. That is, we pass in `b.T[..., np.newaxis]`, which has shape (25, 2, 1):

      >>> w, h = signal.freqz(b.T[..., np.newaxis], worN=1024)
      >>> w.shape
      (1024,)
      >>> h.shape
      (2, 1024)

  Now, suppose we have two transfer functions, with the same numerator coefficients `b = [0.5, 0.5]`. The coefficients for the two denominators are stored in the first dimension of the 2-D array a:

      a = [   1      1  ]
          [ -0.25, -0.5 ]

      >>> b = np.array([0.5, 0.5])
      >>> a = np.array([[1, 1], [-0.25, -0.5]])

  Only a is more than 1-D. To make it compatible for broadcasting with the frequencies, we extend it with a trivial dimension in the call to `freqz`:

      >>> w, h = signal.freqz(b, a[..., np.newaxis], worN=1024)
      >>> w.shape
      (1024,)
      >>> h.shape
      (2, 1024)