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 = 2^nextpow2(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).

Figure 10.4: Overlay of spectrum, true envelope, and cepstral envelope.

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.