Cepstrum

acoustics.cepstrum.complex_cepstrum(x, n=None)

Compute the complex cepstrum of a real sequence.

x : ndarray

Real sequence to compute complex cepstrum of.

n : {None, int}, optional

Length of the Fourier transform.

ceps : ndarray

The complex cepstrum of the real data sequence x computed using the Fourier transform.

ndelay : int

The amount of samples of circular delay added to x.

The complex cepstrum is given by

\[\begin{split}c[n] = F^{-1}\\left{\\log_{10}{\\left(F{x[n]}\\right)}\\right}\end{split}\]

where \(x_[n]\) is the input signal and \(F\) and :math:`F_{-1} are respectively the forward and backward Fourier transform.

real_cepstrum: Compute the real cepstrum. inverse_complex_cepstrum: Compute the inverse complex cepstrum of a real sequence.

In the following example we use the cepstrum to determine the fundamental frequency of a set of harmonics. There is a distinct peak at the quefrency corresponding to the fundamental frequency. To be more precise, the peak corresponds to the spacing between the harmonics.

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

>>> duration = 5.0
>>> fs = 8000.0
>>> samples = int(fs*duration)
>>> t = np.arange(samples) / fs

>>> fundamental = 100.0
>>> harmonics = np.arange(1, 30) * fundamental
>>> signal = np.sin(2.0*np.pi*harmonics[:,None]*t).sum(axis=0)
>>> ceps, _ = complex_cepstrum(signal)

>>> fig = plt.figure()
>>> ax0 = fig.add_subplot(211)
>>> ax0.plot(t, signal)
>>> ax0.set_xlabel('time in seconds')
>>> ax0.set_xlim(0.0, 0.05)
>>> ax1 = fig.add_subplot(212)
>>> ax1.plot(t, ceps)
>>> ax1.set_xlabel('quefrency in seconds')
>>> ax1.set_xlim(0.005, 0.015)
>>> ax1.set_ylim(-5., +10.)

[1]	Wikipedia, “Cepstrum”. http://en.wikipedia.org/wiki/Cepstrum

[2]	M.P. Norton and D.G. Karczub, D.G., “Fundamentals of Noise and Vibration Analysis for Engineers”, 2003.

[3]	B. P. Bogert, M. J. R. Healy, and J. W. Tukey: “The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudo Autocovariance, Cross-Cepstrum and Saphe Cracking”. Proceedings of the Symposium on Time Series Analysis Chapter 15, 209-243. New York: Wiley, 1963.

acoustics.cepstrum.real_cepstrum(x, n=None)

Compute the real cepstrum of a real sequence.

x : ndarray

Real sequence to compute real cepstrum of.

n : {None, int}, optional

Length of the Fourier transform.

ceps: ndarray

The real cepstrum.

The real cepstrum is given by

\[c[n] = F^{-1}\left{\log_{10}{\left|F{x[n]}\right|}\right}\]

where \(x_[n]\) is the input signal and \(F\) and :math:`F_{-1} are respectively the forward and backward Fourier transform. Note that contrary to the complex cepstrum the magnitude is taken of the spectrum.

complex_cepstrum: Compute the complex cepstrum of a real sequence. inverse_complex_cepstrum: Compute the inverse complex cepstrum of a real sequence.

>>> from scipy.signal import real_cepstrum

[1]	Wikipedia, “Cepstrum”. http://en.wikipedia.org/wiki/Cepstrum

acoustics.cepstrum.inverse_complex_cepstrum(ceps, ndelay)

Compute the inverse complex cepstrum of a real sequence.

ceps : ndarray

Real sequence to compute inverse complex cepstrum of.

ndelay: int

The amount of samples of circular delay added to x.

x : ndarray

The inverse complex cepstrum of the real sequence ceps.

The inverse complex cepstrum is given by

\[x[n] = F^{-1}\left{\exp(F(c[n]))\right}\]

where \(c_[n]\) is the input signal and \(F\) and :math:`F_{-1} are respectively the forward and backward Fourier transform.

complex_cepstrum: Compute the complex cepstrum of a real sequence. real_cepstrum: Compute the real cepstrum of a real sequence.

Taking the complex cepstrum and then the inverse complex cepstrum results in the original sequence.

>>> import numpy as np
>>> from scipy.signal import inverse_complex_cepstrum
>>> x = np.arange(10)
>>> ceps, ndelay = complex_cepstrum(x)
>>> y = inverse_complex_cepstrum(ceps, ndelay)
>>> print(x)
>>> print(y)

[1]	Wikipedia, “Cepstrum”. http://en.wikipedia.org/wiki/Cepstrum

acoustics.cepstrum.minimum_phase(x, n=None)

Compute the minimum phase reconstruction of a real sequence.

x : ndarray

Real sequence to compute the minimum phase reconstruction of.

n : {None, int}, optional

Length of the Fourier transform.

Compute the minimum phase reconstruction of a real sequence using the real cepstrum.

m : ndarray

The minimum phase reconstruction of the real sequence x.

real_cepstrum: Compute the real cepstrum.

>>> from scipy.signal import minimum_phase

[1]	Soo-Chang Pei, Huei-Shan Lin. Minimum-Phase FIR Filter Design Using Real Cepstrum. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-II: EXPRESS BRIEFS, VOL. 53, NO. 10, OCTOBER 2006