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# scipy.signal.butter

`scipy.signal.``butter`(N, Wn, btype='low', analog=False, output='ba')

  Butterworth digital and analog filter design.

  Design an Nth order digital or analog Butterworth filter and return the filter coefficients.

  Parameters:

  N : int

  > The order of the filter.

  Wn : array_like

  > A scalar or length-2 sequence giving the critical frequencies. For a Butterworth filter, this is the point at which the gain drops to 1/sqrt(2) that of the passband (the “-3 dB point”). For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s).

  btype : {‘lowpass’, ‘highpass’, ‘bandpass’, ‘bandstop’}, optional

  > The type of filter. Default is ‘lowpass’.

  analog : bool, optional

  > When True, return an analog filter, otherwise a digital filter is returned.

  output : {‘ba’, ‘zpk’, ‘sos’}, optional

  > Type of output: numerator/denominator (‘ba’), pole-zero (‘zpk’), or second-order sections (‘sos’). Default is ‘ba’.

  Returns:

  b, a : ndarray, ndarray

  > Numerator (b) and denominator (a) polynomials of the IIR filter. Only returned if `output='ba'`.

  z, p, k : ndarray, ndarray, float

  > Zeros, poles, and system gain of the IIR filter transfer function. Only returned if `output='zpk'`.

  sos : ndarray

  > Second-order sections representation of the IIR filter. Only returned if `output=='sos'`.

  See also

  `buttord`

  Notes

  The Butterworth filter has maximally flat frequency response in the passband.

  The `'sos'` output parameter was added in 0.16.0.

  Examples

  Plot the filter’s frequency response, showing the critical points:

      >>> from scipy import signal
      >>> import matplotlib.pyplot as plt

      >>> b, a = signal.butter(4, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Butterworth filter frequency response')
      >>> plt.xlabel('Frequency [radians / second]')
      >>> plt.ylabel('Amplitude [dB]')
      >>> plt.margins(0, 0.1)
      >>> plt.grid(which='both', axis='both')
      >>> plt.axvline(100, color='green') # cutoff frequency
      >>> plt.show()

  (Source code)