#### 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.

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.

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Spectral Envelope by Linear Prediction

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Signal Synthesis