# Spectral envelope normalization (speech processing)

I created a synthetic speech signal x[n] of 256 samples assuming that the source signal u[n] is an impulse train (not a glottal flow signal) and also I assumed that there are no lips. The vocal tract filter h[n] is an all pole filter of order 10. So, in frequency domain we have X(w)=H(W)U(W).

I computed with the modified periodogram the PSD of X(w) and with freqz the PSD of H(w) with the following matlab code:

N = 256; % signal length

T = 90; % pitch period

Fs = 8000; % sampling frequency

u = zeros(1,N); % source signal

u([1:T:N])=1;

% computing the vocal tract filter for the vowel /a/

F = [650, 1075, 2463, 3558, 4631]; % Formant frequencies (Hz)

BW = [94, 91, 107, 198, 89]; % Formant bandwidths (Hz)

poles = exp(-pi*BW/Fs) .* exp(j*(2*pi*F/Fs)); % convert formants to poles

num = 1;

den = real(poly([poles,conj(poles)]));

% creating the speech signal x

x = filter(num,den,u);

x = x(:);

NFFT=N;

% Choose window

% win = rectwin(N);

win = hanning(N);

% win = hamming(N);

% win = blackman(N);

X = fft(win.*x,NFFT);

X = X(1:NFFT/2+1);

psdX = (1/(norm(win,2)^2)).*(abs(X).^2); % modified periodogram

psdX(2:end-1) = 2*psdX(2:end-1); % normalized periodogram

psdX = 10*log10(psdX); % in dB

f = linspace(0,Fs/2,NFFT/2+1);

psdH = freqz(num,den,f,Fs)./norm(win,2);

psdH = 20*log10(abs(psdH));

plot(psdX,'b');

hold on;

plot(psdH,'r');

The problem as you can see is that the psdH (i.e. the spectral envelope) is not exactly on the Top of psdX. The book of T.F. Quatieri (Discrete-Time Speech Signal Processing:Principles and Practice) says that the spectral envelope should be exactly on the Top of the periodogram of X(w).

I have the bad feeling that I have not normalized correctly the PSD of H(w).

I have spent more than two days on this problem and I cannot find the solution. Please let me know if I have done any mistake.

Kind regards,

Andreas

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