Spectral Envelope by Linear Prediction Method

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;

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Order a Hardcopy of Spectral Audio Signal Processing

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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.