Spectral Envelope by the Cepstral Windowing Method

We now compute the log-magnitude spectrum, perform the inverse FFT to obtain the real cepstrum, lowpass-window the cepstrum, and perform the FFT to obtain the smoothed log-magnitude spectrum:

Nframe = pwr2(fs/25); % frame size = 40 ms
w = hamming(Nframe)';
winspeech = w .* speech(1:Nframe);
Nfft = 4*Nframe; % factor of 4 zero-padding
sspec = fft(winspeech,Nfft);
dbsspecfull = 20*log(abs(sspec));
rcep = ifft(dbsspecfull);  % real cepstrum
rcep = real(rcep); % eliminate round-off noise in imag part
period = round(fs/f0) % 41
nspec = Nfft/2+1;
aliasing = norm(rcep(nspec-10:nspec+10))/norm(rcep) % 0.02
nw = 2*period-4; % almost 1 period left and right
if floor(nw/2) == nw/2, nw=nw-1; end; % make it odd
w = boxcar(nw)'; % rectangular window
wzp = [w(((nw+1)/2):nw),zeros(1,Nfft-nw), ...
       w(1:(nw-1)/2)];  % zero-phase version
wrcep = wzp .* rcep;  % window the cepstrum ("lifter")
rcepenv = fft(wrcep); % spectral envelope
rcepenvp = real(rcepenv(1:nspec)); % should be real
rcepenvp = rcepenvp - mean(rcepenvp); % normalize to zero mean

Figure 9.3 shows the real cepstrum of the synthetic ``ah'' vowel (top) and the same cepstrum truncated to just under a period in length. In theory, this leaves only formant envelope information in the cepstrum. Figure 9.4 shows an overlay of the spectrum, true envelope, and cepstral envelope.

Instead of simply truncating the cepstrum (a rectangular windowing operation), we can window it more gracefully. Figure 9.5 shows the result of using a Hann window of the same length. The spectral envelope is smoother as a result.

---

Previous: Signal Synthesis
Next: Spectral Envelope by Linear Prediction Method

Order a Hardcopy of Spectral Audio Signal Processing

---

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.