Convergence and learning curve of 2nd order RLS predictor
** lab16_3.m * mw * 05/20/2007
Contents
Input dialog
prompt = {'Order of predictor p'};
dlg_title = 'lab16_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
end
[x,fs,bits]=wavread('speech'); % speech signal
%[x,fs,bits]=wavread('handel'); % audio signal
Lx = length(x); % number of samples
sound(x,fs,bits);
Linear prediction (long term)
Rxx = xcorr(x,'unbiased'); % estimate mean autocorrelation sequence r = Rxx(Lx:Lx+Np+1); % relevant coefficients for Levinson-Durbin alg. Rxx[0],Rxx[1],... [a,E,k] = levinson(r,Np); % Levinson-Durbin alg. (Signal Processing Toolbox) [e1 g] = latcfilt(k,x); % residuum e[n] (prediction error) (Signal Processing Toolbox) clear g; % delete backward lattice filter output signal Re1e1 = xcorr(e1,'unbiased'); % estimate autocorrelation sequence of residuum fprintf('lab16_3 : Reflection coefficients\n') for m=1:Np fprintf([' k',num2str(m),' = %g \n'],k(m)) end fprintf('Audio signal Rxx[0] = %g \n',Rxx(Lx)) fprintf('Residual signal Ree[0] = %g \n',Re1e1(Lx)) fprintf('Prediction error E = %g \n',E) fprintf('Prediction gain Gp = %g dB\n\n',10*log10(Rxx(Lx)/Re1e1(Lx))) pause(1+Lx/fs) sound(e1,fs,bits); % residuum
lab16_3 : Reflection coefficients k1 = -0.982861 k2 = 0.47298 k3 = 0.16021 Audio signal Rxx[0] = 0.031681 Residual signal Ree[0] = 0.000816771 Prediction error E = 0.000814322 Prediction gain Gp = 15.887 dB
Simulation of adaptiv system - RLS algorithm
p = Np; % prediction order (FIR filter order +1) rls = struct('alpha',1/512,'beta',1/128,'b',zeros(p,1),'e',0,'x',zeros(p-1,1),... 'R_1',eye(p),'Mode','e*x'); % noise = 10^(-20/10)*randn(1,Lx); % AWGN [y,e2,rls,SAVEb] = rls_algorithm([0 x(1:Lx-1)'],x,rls); pause(1+Lx/fs) sound(e2,fs,bits); % residuum
Graphics
figure('Name',['lab16_3 : pth order linear prediction -> p = ',... num2str(Np),' , fs = ',num2str(fs)],'NumberTitle','off'); L2 = 400; t1 = (0:Lx-1)/fs; t2 = 1e3*(0:L2)/fs; subplot(3,2,1), plot(t1,x(1:Lx)),grid xlabel('t in s \rightarrow'), ylabel('x(t) \rightarrow') title('Audio signal (original)') subplot(3,2,2), plot(t2,Rxx(Lx:Lx+L2)/Rxx(Lx)),grid xlabel('\tau in ms \rightarrow'), ylabel('R_{xx}(\tau) / R_{xx}(0) \rightarrow') title('ACF') subplot(3,2,3), plot(t1,e1(1:Lx)),grid xlabel('t in s \rightarrow'), ylabel('e_{LTP}(t) \rightarrow') title('Prediction error - long term prediction') subplot(3,2,4), plot(t2,Re1e1(Lx:Lx+L2)/Rxx(Lx),0,Re1e1(Lx)/Rxx(Lx),'o'),grid xlabel('\tau in ms \rightarrow'), ylabel('R_{ee,LTP}(\tau) / R_{xx}(0) \rightarrow') title('ACF with long term prediction)') subplot(3,2,5), plot(t1,e2(1:Lx)),grid xlabel('t in s \rightarrow'), ylabel('e_{RLS}(t) \rightarrow') title('Prediction error with RLS algorithm') Re2e2 = xcorr(e2,'unbiased'); % estimate mean autocorrelation sequence subplot(3,2,6), plot(t2,Re2e2(Lx:Lx+L2)/Rxx(Lx),0,Re2e2(Lx)/Rxx(Lx),'o'),grid xlabel('\tau in ms \rightarrow'), ylabel('R_{ee,RLS}(\tau) / R_{xx}(0) \rightarrow') title('ACF with RLS algorithm') figure('Name',['lab16_3 : pth order linear prediction -> p = ',... num2str(Np),' , fs = ',num2str(fs)],'NumberTitle','off'); subplot(2,1,1), plot(t1,SAVEb(1,1:Lx)),grid xlabel('t in s \rightarrow'), ylabel('b_0(t) \rightarrow') title('FIR filter coefficient') subplot(2,1,2), plot(t1,SAVEb(2,1:Lx)),grid xlabel('t in s \rightarrow'), ylabel('b_1(t) \rightarrow')
