Contents

% impulse response and time-correlation sequence of
% 2nd order system with conjugate complex pole pair
% lab3_1.m * mw * 02/09/2007
b = [1 2 1]; a = [1 -1.4 .74]; % numerator and denominator

Impulse response

N = 40;
x = [1 zeros(1,N)];     % impulse
hm = filter(b,a,x);     % impulse response (MATLAB)
% analytical computation for conjugate complex pole pair
p = roots(a); z = p(1); r = abs(z); w = angle(z);
B = -j*(b(1)*z^2 + b(2)*z + b(3)) / (z*2*imag(z));
n = 0:N;
ha = 2*(r.^n).*(real(B)*cos(w*n) - imag(B)*sin(w*n));
ha(1) = ha(1) + b(3)/r^2;

Time-correlation sequence

Rm = conv(ha,fliplr(ha));  % numerical solution
% analytical computation for conjugate complex pole pair (right sided)
B = -j*(b(1)*z^2 + b(2)*z + b(3)) / (z*2*imag(z));
B = B*(b(1)+b(2)*z+b(3)*z^2) / (1+a(2)*z+a(3)*z^2);
Ra = 2*(r.^n).*(real(B)*cos(w*n) - imag(B)*sin(w*n));
Ra(1) = Ra(1) + b(1)*b(3)/r^2;
Ra = [Ra(N+1:-1:2) Ra]; % two sided
fprintf('lab3_1: energy of impulse response = %g\n',Ra(N+1))

lab3_1: energy of impulse response = 82.8054

Graphics - Impulse response and time acf

FIG1 = figure('Name','lab3_1 : Impulse response and time-correlation function',...
    'NumberTitle','off');
subplot(2,1,1), plot(0:N,ha,':',0:N,hm,'o'), grid
xlabel('n \rightarrow'), ylabel('h[n] \rightarrow')
title('Impulse response')
subplot(2,1,2),plot(-N:N,Rm,'o',-N:N,Ra,':'), grid
xlabel('l \rightarrow'), ylabel('R_{hh}[l] \rightarrow')
title('Time auto-correlation function')

Published with MATLAB® 7.3