How to find response of a system through convolution?
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So, i've been trying to solve a frequency domain problem in the time domain, to better understand how it works and because it offers simulation perfomance for some of my algorithms.
The algorithm i wrote gets results from even complicated systems like three or more orders, but for some reason, it's always slightly different from the correct response. To measure how accurate my algorithm is, i've been using the lsim function.
s = tf('s');
t = (0:0.01:10)';
unit_impulse = zeros(size(t));
unit_impulse(t==0) = 1;
unit_step = ones(size(t));
u = unit_step;
freqFunc = 4 / (s+1);
timeFunc = 4 * exp(-t);
% Calculates reference values
y_ref = lsim(freqFunc, u, t);
h_ref = impulse(freqFunc, t);
h = conv(unit_impulse, timeFunc);
h = h(1:1001);
y = median(diff(t)) * conv(u, h);
y = y(1:1001);
error_impulse = er(h_ref, h, t);
error_response = er(y_ref, y, t);
The error in the impulse response is zero, so i must be doing something wrong when calculating the input response. The mean error is very small, like 0.02, but still enough for it to be noticeable. I divided the y_ref by y elementwise, and noticed an interesting pattern. It goes like this: [Inf, 2, 1.5, 1.333, ... , ..., 1.005]
Now that is the (X+1) / X sequence, and normally used by filters i think.
What am i doing wrong?
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Paul
am 22 Apr. 2021
I don't think you're doing anything wrong. Rather, what you're seeing is the effect of approximating a convolution integral with a convolution sum scaled by dt, which is basically approximating the convolution integral with a rectangular rule. I don't know what the er() function is, but I suspect that the errors, however they are computed, will get smaller if you make the increment smaller in the time vector. Also, keep in mind that lsim() and impulse() are themselves approximations, though in this case very good approximations. Also, the step response can be computed using step(); no need to use lsim(). lsim(), step(), and impulse() aren't really needed at all. Closed form expressions for the step and impulse responses can be easily obtained and evaluated.
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Paul
am 22 Apr. 2021
You're very welcome.
See the Algorithm section on
doc lsim
for a brief discussion on how lsim works.
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