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# scipy.signal.cheby1

scipy.signal.cheby1(N, rp, Wn, btype='low', analog=False, output='ba', fs=None)

  Chebyshev type I digital and analog filter design.

  Design an Nth-order digital or analog Chebyshev type I filter and return the filter coefficients.

  Parameters:
    N int

      The order of the filter.

    rp float

      The maximum ripple allowed below unity gain in the passband. Specified in decibels, as a positive number.

    Wn array_like

      A scalar or length-2 sequence giving the critical frequencies. For Type I filters, this is the point in the transition band at which the gain first drops below -rp.

      For digital filters, Wn are in the same units as fs. By default, fs is 2 half-cycles/sample, so these are normalized from 0 to 1, where 1 is the Nyquist frequency. (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’ for backwards compatibility, but ‘sos’ should be used for general-purpose filtering.

    fs float, optional

      The sampling frequency of the digital system.

      New in version 1.2.0.

  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

  `cheb1ord`, `cheb1ap`

  Notes

  The Chebyshev type I filter maximizes the rate of cutoff between the frequency response’s passband and stopband, at the expense of ripple in the passband and increased ringing in the step response.

  Type I filters roll off faster than Type II (`cheby2`), but Type II filters do not have any ripple in the passband.

  The equiripple passband has N maxima or minima (for example, a 5th-order filter has 3 maxima and 2 minima). Consequently, the DC gain is unity for odd-order filters, or -rp dB for even-order filters.

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

  Examples

  Design an analog filter and plot its frequency response, showing the critical points:

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

      >>> b, a = signal.cheby1(4, 5, 100, 'low', analog=True)
      >>> w, h = signal.freqs(b, a)
      >>> plt.semilogx(w, 20 * np.log10(abs(h)))
      >>> plt.title('Chebyshev Type I frequency response (rp=5)')
      >>> 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.axhline(-5, color='green') # rp
      >>> plt.show()

  Generate a signal made up of 10 Hz and 20 Hz, sampled at 1 kHz

      >>> t = np.linspace(0, 1, 1000, False)  # 1 second
      >>> sig = np.sin(2*np.pi*10*t) + np.sin(2*np.pi*20*t)
      >>> fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)
      >>> ax1.plot(t, sig)
      >>> ax1.set_title('10 Hz and 20 Hz sinusoids')
      >>> ax1.axis([0, 1, -2, 2])

  Design a digital high-pass filter at 15 Hz to remove the 10 Hz tone, and apply it to the signal. (It’s recommended to use second-order sections format when filtering, to avoid numerical error with transfer function (`ba`) format):

      >>> sos = signal.cheby1(10, 1, 15, 'hp', fs=1000, output='sos')
      >>> filtered = signal.sosfilt(sos, sig)
      >>> ax2.plot(t, filtered)
      >>> ax2.set_title('After 15 Hz high-pass filter')
      >>> ax2.axis([0, 1, -2, 2])
      >>> ax2.set_xlabel('Time [seconds]')
      >>> plt.tight_layout()
      >>> plt.show()