N-th order linear prediction

** lab10_3.m * mw * 04/18/2007

Contents

Input dialog

prompt = {'Order of predictor N','Delay m'};
dlg_title = 'lab10_3'; num_lines = 1; def = {'3','1'};
answer = inputdlg(prompt,dlg_title,num_lines,def);
if isempty(answer)
    Np=3; M=1;                 % default
else
    Np = str2num(answer{1});   % prediction order
    M  = str2num(answer{2});   % delay
end

Impulse response and acf

b = .24192*[1 2 1]; a = [1 -.71 .25]; % denominator and numerator coefficients
N = 41;                  % simulation lenght of impulse response
h = impz(b,a,N);         % impulse response
Rhh = conv(h,flipud(h)); % acf
% graphics
graph10_h(h,Rhh,'lab10_3 : Impulse response, acf and pds')

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 crosscorrelation function
bp = RR\r;                % prediction coefficients rr^-1 * r
MSEopt = Rhh(N) - bp'*r;  % mean power of prediction error
fprintf('lab10_3 : Prediction coefficients\n')
for k=1:Np
    fprintf([' b(',num2str(k-1),') = %g \n'],bp(k))
end
fprintf('Minimum MSE for m = %g : MSEopt = %g \n',M,MSEopt)
fprintf('Prediction gain : Gp = %g \n',Rhh(N)/MSEopt)

lab10_3 : Prediction coefficients
 b(0) = 1.66263 
 b(1) = -1.34969 
 b(2) = 0.516203 
Minimum MSE for m = 1 : MSEopt = 0.152914 
Prediction gain : Gp = 6.53952 

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))
% graphics
graph10_w(x,w,Rxx,Rww,L,'lab10_3 : Sample function and acf')

Estimated mean signal powers 
 Innovation   : Rww[0] = 1.00711 
 Model process: Rxx[0] = 0.984871 

Linear prediction error

e = x(Np+1:L);                  % prediction error
for k=1:Np
    e = e - bp(k)*x(Np+1-k:L-k);
end
Ree = xcorr(e,'unbiased');       % estimate auto-correlation sequence
fprintf(' Model process: Ree[0] = %g \n\n',Ree(L-Np))
% graphics
FIG1 = figure('Name',['lab10_3 : Nth order linear prediction -> N = ',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(1:L1+1)),grid
xlabel('n \rightarrow'), ylabel('e[n] \rightarrow')
title('prediction error')
subplot(2,2,4), stem(0:L2,Ree(L-Np:L-Np+L2)/Rxx(L)),grid
xlabel('l \rightarrow'), ylabel('R_{ee}[l] / R_{xx}[0] \rightarrow')
title('ACF')

 Model process: Ree[0] = 0.150561 

Published with MATLAB® 7.3