Nth order linear prediction for 2nd order AR process
** lab11_3.m * mw * 04/20/2007
Contents
Input dialog, delay
prompt = {'Order of predictor p'};
dlg_title = 'lab11_3'; num_lines = 1; def = {'3'};
answer = inputdlg(prompt,dlg_title,num_lines,def);
if isempty(answer)
Np = 3; % default
else
Np = str2num(answer{1}); % prediction order input
end
M = 1; % delay
Impulse response and acf
b = .7969; a = [1 -.71 .25]; % numerator and denominator coefficients N = 41; % simulated lenght of impulse response h = impz(b,a,N); % impulse response Rhh = conv(h,flipud(h)); % acf graph10_h(h,Rhh,'lab11_3 : Impulse response and acf') % graphics
N-th order prediction coefficients and error signal power
RR = zeros(Np,Np); % time autocorrelation matrix for m=0:Np-1 RR(1+m,1+m) = Rhh(N); for n=m+1:Np-1 RR(1+m,1+n) = Rhh(N+n-m); RR(1+n,1+m) = RR(1+m,1+n); end end r = Rhh(N+M:N+M+Np-1); % time cross-correlation function bp = RR\r; % prediction coefficients rr^-1 * r MSEopt = Rhh(N) - bp'*r; % mean power of prediction error fprintf('lab11_3 : Prediction coefficients\n') for k=1:Np fprintf([' b',num2str(k),' = %g \n'],bp(k)) end fprintf('Minimum MSE for M = %g : MSEopt = %g \n\n',M,MSEopt) fprintf('Prediction gain : Gp = %g \n',Rhh(N)/MSEopt)
lab11_3 : Prediction coefficients b1 = 0.71 b2 = -0.25 b3 = -5.02227e-017 b4 = 9.17829e-018 b5 = 5.97545e-019 b6 = -1.69703e-018 b7 = 2.39018e-018 Minimum MSE for M = 1 : MSEopt = 0.63505 Prediction gain : Gp = 1.5747
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 Rxx = xcorr(x,'unbiased'); % estimate autocorrelation sequence fprintf('Estimated mean signal powers \n') fprintf(' Innovation : Rww[0] = %g \n',Rww(L)) fprintf(' Model process: Rxx[0] = %g \n',Rxx(L)) graph10_w(x,w,Rxx,Rww,L,'lab11_3 : Sample function and acf') % graphics
Estimated mean signal powers Innovation : Rww[0] = 0.987717 Model process: Rxx[0] = 0.991522
Linear prediction with lattice structure
r = Rhh(N:N+Np+1); k = lev_dur(r,Np); % reflexion coefficients % [A,E,K] = levinson(r,Np) % MATLAB Signal Processing Toolbox [e G] = latcfilt(k,x); clear G; Ree = xcorr(e,'unbiased'); % estimate autocorrelation sequence fprintf('Reflexion coefficients\n') for m=1:Np fprintf([' k',num2str(m),' = %g \n'],k(m)) end fprintf(' Model process: Ree[0] = %g \n\n',Ree(L))
Reflexion coefficients k1 = -0.568 k2 = 0.25 k3 = 8.74123e-017 k4 = -3.44782e-017 k5 = 3.831e-018 k6 = 5.20126e-019 k7 = 2.29457e-018 Model process: Ree[0] = 0.627249
Graphics - Model process and prediction error
FIG1 = figure('Name',['lab11_3 : pth order linear prediction with lattice structure -> p = ',... num2str(Np)],'NumberTitle','off'); L1 = 60; L2 = 20; subplot(2,2,1), stem(0:L1,x(1:L1+1)),grid xlabel('n \rightarrow'), ylabel('x[n] \rightarrow') title('Model process') subplot(2,2,2), stem(0:L2,Rxx(L:L+L2)/Rxx(L)),grid xlabel('l \rightarrow'), ylabel('R_{xx}[l] / R_{xx}[0] \rightarrow') title('ACF') subplot(2,2,3), stem(0:L1,e(Np+1:Np+L1+1)),grid xlabel('n \rightarrow'), ylabel('e[n] \rightarrow') title('Prediction error') subplot(2,2,4), stem(0:L2,Ree(L:L+L2)/Rxx(L)),grid xlabel('l \rightarrow'), ylabel('R_{ee}[l] / R_{xx}[0] \rightarrow') title('ACF')
