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 10.1: Bandlimited impulse train.
\includegraphics[width=\textwidth ]{eps/ImpulseTrain}

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

Figure 10.2: Synthetic vowel in time and frequency domains, with formant envelope overlaid.
\includegraphics[width=\textwidth ]{eps/Speech}


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Spectral Envelope by the Cepstral Windowing Method
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Summary of LP Spectral Envelopes