1st and 2nd order linear prediction
with prediction filter in lattice structure ** lab11_1.m * mw * 04/25/2007
Contents
Impulse response, acf and prediction
b = .242*[1 2 1]; a = [1 -.71 .25]; % numerator and denominator N = 21; % simulated lenght of impulse response h = impz(b,a,N); % impulse response Rhh = conv(h,flipud(h)); % acf % prediction coefficients and error signal power b0_1 = Rhh(N+1) / Rhh(N); % 1st order MSEopt_1 = Rhh(N) - b0_1*Rhh(N+1); D = Rhh(N)^2 - Rhh(N+1)^2; % 2nd order b0_2 = (Rhh(N)*Rhh(N+1)-Rhh(N+2)*Rhh(N+1)) / D; b1_2 = (Rhh(N)*Rhh(N+2)-Rhh(N+1)^2) / D; MSEopt_2 = Rhh(N) - b0_2*Rhh(N+1) - b1_2*Rhh(N+2); fprintf('lab11_1 : Prediction coefficients and error signal power\n') fprintf(' 1st order : b0 = %g ; MSEopt = %g \n',b0_1,MSEopt_1) fprintf(' 2nd order : b0 = %g ; b1 = %g ; MSEopt = %g \n\n',b0_2,b1_2,MSEopt_2) graph10_h(h,Rhh,'lab11_1 : Impulse response and acf') % graphics
lab11_1 : Prediction coefficients and error signal power 1st order : b0 = 0.788526 ; MSEopt = 0.378472 2nd order : b0 = 1.3168 ; b1 = -0.669952 ; MSEopt = 0.2086
Sample function
L = 10001; % number of samples w = randn(1,L); % normally distributed white noise x = filter(b,a,w); % normally distributed colored noise Rww = xcorr(w,'unbiased'); % estimate autocorrelation sequence of white noise Rxx = xcorr(x,'unbiased'); % estimate autocorrelation sequence of filtered noise fprintf('Estimated mean signal powers \n') fprintf(' Innovation : Rww[0] = %g \n',Rww(L)) fprintf(' Model process : Rxx[0] = %g \n\n',Rxx(L)) graph10_w(x,w,Rxx,Rww,L,'lab11_1 : Sample function and acf') % graphics
Estimated mean signal powers Innovation : Rww[0] = 1.00988 Model process : Rxx[0] = 1.03031
Linear prediction
e_1 = x(2:L)-b0_1*x(1:L-1); % 1st order prediction error Ree_1 = xcorr(e_1,'unbiased'); % estimate acf e_2 = x(3:L)-b0_2*x(2:L-1)-b1_2*x(1:L-2); % 2nd order prediction error Ree_2 = xcorr(e_2,'unbiased'); % estimate acf fprintf('Estimated mean prediction error powers \n') fprintf(' 1st order : Ree[0] = %g \n',Ree_1(L-1)) fprintf(' 2nd order : Ree[0] = %g \n\n',Ree_2(L-2)) % graphics graph10_e(e_1,e_2,Ree_1/Rxx(L),Ree_2/Rxx(L),L,'lab11_1 : Prediction error signal and acf')
Estimated mean prediction error powers 1st order : Ree[0] = 0.378724 2nd order : Ree[0] = 0.209594
1st order lattice filter implementation
k1 = - Rhh(N+1) / Rhh(N);
[u1,v] = latcfilt(k1,x);
Ruu_1 = xcorr(u1,'unbiased');
2nd order lattice filter implementation
k2 = -(Rhh(N)*Rhh(N+2)-Rhh(N+1)^2)/(Rhh(N)^2-Rhh(N+1)^2); [u2,v] = latcfilt([k1 k2],x); Ruu_2 = xcorr(u2,'unbiased'); fprintf('Estimated mean prediction error powers - lattice structure \n') fprintf(' 1st order : Ree[0] = %g \n',Ruu_1(L)) fprintf(' 2nd order : Ree[0] = %g \n\n',Ruu_2(L)) graph11_u(u1,u2,Ruu_1/Rxx(L),Ruu_2/Rxx(L),L,... 'lab11_1 : Prediction error signal and acf - lattice structure') % graphics
Estimated mean prediction error powers - lattice structure 1st order : Ree[0] = 0.378686 2nd order : Ree[0] = 0.209553
