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Spectral Envelope by the Cepstral Windowing Method

We now compute the log-magnitude spectrum, perform an 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 10.3 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 10.4 shows an overlay of the spectrum, true envelope, and cepstral envelope.

Figure 10.3: Real cepstrum (top) and windowed cepstrum (bottom).
\includegraphics[width=\textwidth ]{eps/CepstrumBoxcar}

Figure 10.4: Overlay of spectrum, true envelope, and cepstral envelope.
\includegraphics[width=\textwidth ]{eps/CepstrumEnvBoxcarC}

Instead of simply truncating the cepstrum (a rectangular windowing operation), we can window it more gracefully. Figure 10.5 shows the result of using a Hann window of the same length. The spectral envelope is smoother as a result.

Figure 10.5: Overlay of spectrum, true envelope, and cepstral envelope.
\includegraphics[width=\textwidth ]{eps/CepstrumEnvHanningC}


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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