### Spectral Envelope Examples

This section presents matlab code for computing spectral envelopes by the cepstral and linear prediction methods discussed above. The signal to be modeled is a synthetic ``ah'' vowel (as in ``father'') synthesized using three formants driven by a bandlimited impulse train [128].

#### Signal Synthesis

% Specify formant resonances for an "ah" [a] vowel: F = [700, 1220, 2600]; % Formant frequencies in Hz B = [130, 70, 160]; % Formant bandwidths in Hz fs = 8192; % Sampling rate in Hz % ("telephone quality" for speed) R = exp(-pi*B/fs); % Pole radii theta = 2*pi*F/fs; % Pole angles poles = R .* exp(j*theta); [B,A] = zp2tf(0,[poles,conj(poles)],1); f0 = 200; % Fundamental frequency in Hz w0T = 2*pi*f0/fs; nharm = floor((fs/2)/f0); % number of harmonics nsamps = fs; % make a second's worth sig = zeros(1,nsamps); n = 0:(nsamps-1); % Synthesize bandlimited impulse train: for i=1:nharm, sig = sig + cos(i*w0T*n); end; sig = sig/max(sig); soundsc(sig,fs); % Let's hear it % Now compute the speech vowel: speech = filter(1,A,sig); soundsc([sig,speech],fs); % "buzz", "ahh" % (it would sound much better with a little vibrato)

The Hamming-windowed bandlimited impulse train `sig` and its
spectrum are plotted in Fig.10.1.

Figure 10.2 shows the Hamming-windowed synthesized vowel
`speech`, and its spectrum overlaid with the true formant
envelope.

#### 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.

#### Spectral Envelope by Linear Prediction

Finally, let's do an LPC window. It had better be good because the LPC model is exact for this example.

M = 6; % Assume three formants and no noise % compute Mth-order autocorrelation function: rx = zeros(1,M+1)'; for i=1:M+1, rx(i) = rx(i) + speech(1:nsamps-i+1) ... * speech(1+i-1:nsamps)'; end % prepare the M by M Toeplitz covariance matrix: covmatrix = zeros(M,M); for i=1:M, covmatrix(i,i:M) = rx(1:M-i+1)'; covmatrix(i:M,i) = rx(1:M-i+1); end % solve "normal equations" for prediction coeffs: Acoeffs = - covmatrix \ rx(2:M+1) Alp = [1,Acoeffs']; % LP polynomial A(z) dbenvlp = 20*log10(abs(freqz(1,Alp,nspec)')); dbsspecn = dbsspec + ones(1,nspec)*(max(dbenvlp) ... - max(dbsspec)); % normalize plot(f,[max(dbsspecn,-100);dbenv;dbenvlp]); grid;

#### Linear Prediction in Matlab and Octave

In the above example, we implemented essentially the covariance method of LP directly (the autocorrelation estimate was unbiased). The code should run in either Octave or Matlab with the Signal Processing Toolbox.

The Matlab Signal Processing Toolbox has the function `lpc`
available. (LPC stands for ``Linear Predictive Coding.'')

The Octave-Forge `lpc` function (version 20071212) is a wrapper
for the `lattice` function which implements Burg's method by
default. Burg's method has the advantage of guaranteeing stability
(
is minimum phase) while yielding accuracy comparable to the
covariance method. By uncommenting lines in `lpc.m`, one can
instead use the ``geometric lattice'' or classic autocorrelation
method (called ``Yule-Walker'' in `lpc.m`). For details,
```type lpc`''.

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Additive Synthesis (Early Sinusoidal Modeling)

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Linear Prediction Spectral Envelope