% Spectral factorization using Kolmogorov 1939 approach. % (code follows pp. 232-233, Signal Analysis, by A. Papoulis) % % Computes the minimum-phase impulse response which satisfies % given auto-correlation. % % Input: % r: top-half of the auto-correlation coefficients % starts from 0th element to end of the auto-corelation % should be passed in as a column vector % Output % h: impulse response that gives the desired auto-correlation function h = spectral_fact(r) % length of the impulse response sequence n = length(r); % over-sampling factor mult_factor = 100; % should have mult_factor*(n) >> n m = mult_factor*n; % computation method: % H(exp(jTw)) = alpha(w) + j*phi(w) % where alpha(w) = 1/2*ln(R(w)) and phi(w) = Hilbert_trans(alpha(w)) % compute 1/2*ln(R(w)) w = 2*pi*[0:m-1]/m; R = [ ones(m,1) 2*cos(kron(w',[1:n-1])) ]*r; alpha = 1/2*log(R); % find the Hilbert transform alphatmp = fft(alpha); alphatmp(floor(m/2)+1:m) = -alphatmp(floor(m/2)+1:m); alphatmp(1) = 0; alphatmp(floor(m/2)+1) = 0; phi = real(ifft(j*alphatmp)); % now retrieve the original sampling index = find(rem([0:m-1],mult_factor)==0); alpha1 = alpha(index); phi1 = phi(index); % compute the impulse response (inverse Fourier transform) h = real(ifft(exp(alpha1+j*phi1),n));