Function File: [y, ym] = rceps (x)

Return the cepstrum of the signal x.

If x is a matrix, return the cepstrum of each column.

If called with two output arguments, the minimum phase reconstruction of the signal x is returned in ym.

For example: 

f0 = 70; Fs = 10000;            # 100 Hz fundamental, 10kHz sampling rate
a = poly (0.985 * exp (1i * pi * [0.1, -0.1, 0.3, -0.3])); # two formants
s = 0.005 * randn (1024, 1);    # Noise excitation signal
s(1:Fs/f0:length(s)) = 1;       # Impulse glottal wave
x = filter (1, a, s);           # Speech signal
[y, ym] = rceps (x .* hanning (1024));

Reference: Programs for Digital Signal Processing, IEEE Press, John Wiley & Sons, New York, 1979.

Demonstration 1

The following code

 f0 = 70; Fs = 10000;                 # 100 Hz fundamental, 10 kHz sampling rate
 a = real (poly (0.985 * exp (1i * pi * [0.1, -0.1, 0.3, -0.3]))); # two formants
 s = 0.05 * randn (1024, 1);          # Noise excitation signal
 s(floor (1:Fs/f0:length (s))) = 1;   # Impulse glottal wave
 x = filter (1, a, s);                # Speech signal in x
 [y, xm] = rceps (x);                 # cepstrum and minimum phase x
 [hx, w] = freqz (x, 1, [], Fs);
 hxm = freqz (xm);
 figure (1);
 subplot (311);
 len = 1000 * fix (min (length (x), length (xm)) / 1000);
 plot ([0:len-1] * 1000 / Fs, x(1:len), "b;signal;", ...
       [0:len-1] * 1000 / Fs, xm(1:len), "g;reconstruction;");
 ylabel ("Amplitude");
 xlabel ("Time (ms)");
 subplot (312);
 axis ("ticy");
 plot (w, log (abs (hx)), ";magnitude;", ...
       w, log (abs (hxm)), ";reconstruction;");
 xlabel ("Frequency (Hz)");
 subplot (313);
 axis ("on");
 plot (w, unwrap (arg (hx)) / (2 * pi), ";phase;", ...
       w, unwrap (arg (hxm)) / (2 * pi), ";reconstruction;");
 xlabel ("Frequency (Hz)");
 len = 1000 * fix (length (y) / 1000);
 figure (2);
 plot ([0:len-1] * 1000 / Fs, y(1:len), ";cepstrum;");
 ylabel ("Amplitude");
 xlabel ("Quefrency (ms)");
 %-------------------------------------------------------------
 % confirm the magnitude spectrum is identical in the signal
 % and the reconstruction and that there are peaks in the
 % cepstrum at 14 ms intervals corresponding to an F0 of 70 Hz.

Produces the following figures

Figure 1 Figure 2

Package: signal