# scipy.signal.freqz_zpk

scipy.signal.freqz_zpk(z, p, k, worN=512, whole=False, fs=6.283185307179586)

  Compute the frequency response of a digital filter in ZPK form.

  Given the Zeros, Poles and Gain of a digital filter, compute its frequency response:

  \(H(z)=k \prod_i (z - Z[i]) / \prod_j (z - P[j])\)

  where \(k\) is the gain, \(Z\) are the zeros and \(P\) are the poles.

  Parameters:
    z array_like

      Zeroes of a linear filter

    p array_like

      Poles of a linear filter

    k scalar

      Gain of a linear filter

    worN{None, int, array_like}, optional

      If a single integer, then compute at that many frequencies (default is N=512).

      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 w is array_like.

    fs float, optional

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

      New in version 1.2.0.

  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

  `freqs`

    Compute the frequency response of an analog filter in TF form

  `freqs_zpk`

    Compute the frequency response of an analog filter in ZPK form

  `freqz`

    Compute the frequency response of a digital filter in TF form

  Notes

  New in version 0.19.0.

  Examples

  Design a 4th-order digital Butterworth filter with cut-off of 100 Hz in a system with sample rate of 1000 Hz, and plot the frequency response:

      >>> import numpy as np
      >>> from scipy import signal
      >>> z, p, k = signal.butter(4, 100, output='zpk', fs=1000)
      >>> w, h = signal.freqz_zpk(z, p, k, fs=1000)

      >>> import matplotlib.pyplot as plt
      >>> fig = plt.figure()
      >>> ax1 = fig.add_subplot(1, 1, 1)
      >>> ax1.set_title('Digital filter frequency response')

      >>> ax1.plot(w, 20 * np.log10(abs(h)), 'b')
      >>> ax1.set_ylabel('Amplitude [dB]', color='b')
      >>> ax1.set_xlabel('Frequency [Hz]')
      >>> ax1.grid(True)

      >>> ax2 = ax1.twinx()
      >>> angles = np.unwrap(np.angle(h))
      >>> ax2.plot(w, angles, 'g')
      >>> ax2.set_ylabel('Angle [radians]', color='g')

      >>> plt.axis('tight')
      >>> plt.show()