Complex cepstral analysis
Return the complex cepstrum of the input vector.
cceps(x)
x |
input data, specified as a real vector. |
Cepstral analysis is a nonlinear signal processing technique that is applied most commonly in speech and image processing, or as a tool to investigate periodic structures within frequency spectra, for instance resulting from echos/reflections in the signal or to the occurrence of harmonic frequencies (partials, overtones).
The cepstrum is used in many variants. Most important are the power cepstrum,
the complex cepstrum, and real cepstrum. The function cceps implements
the complex cepstrum by computing the inverse of the log-transformed FFT,
i.e.,
cceps(x) <- ifft(log(fft(x)))
However, because taking the logarithm of a complex number can lead to
unexpected results, the phase of fft(x) needs to be unwrapped before
taking the log.
Complex cepstrum, returned as a vector.
This function returns slightly different results in comparison with the 'Matlab' and 'Octave' equivalents. The 'Octave' version does not apply phase unwrapping, but has an optional correction procedure in case of zero phase at π radians. The present implementation does apply phase unwrapping so that the correction procedure is unnecessary. The 'Matlab' implementation also applies phase unwrapping, and a circular shift if necessary to avoid zero phase at π radians. The circular shift is not done here. In addition, the 'Octave' version shifts the zero frequency to the center of the series, which neither the 'Matlab' nor the present implementation do.
Geert van Boxtel, G.J.M.vanBoxtel@gmail.com.
## Generate a sine of frequency 45 Hz, sampled at 100 Hz. fs <- 100 t <- seq(0, 1.27, 1/fs) s1 <- sin(2 * pi * 45 * t) ## Add an echo with half the amplitude and 0.2 s later. s2 <- s1 + 0.5 * c(rep(0L, 20), s1[1:108]) ## Compute the complex cepstrum of the signal. Notice the echo at 0.2 s. cep <- cceps(s2) plot(t, cep, type="l")
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