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.9.1.

Figure 9.2 shows the Hamming-windowed synthesized vowel speech, and its spectrum overlaid with the true formant envelope.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.