How do filter function work?

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Treant
Treant 2012-8-26
编辑： Sergio Cuadros 2020-6-18
I understand how the filter(b,a,X) works. What confuses me is the function with the initial conditions. The function filter(b,a,X,zi).
for example: y(n)=-0.5y(n-1)+x(n) x(n)=u(n) y(-1)=1
how will the filter function solve this? i tried to do it manually, but i get different output.

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Honglei Chen
Honglei Chen 2012-8-26
编辑：Honglei Chen 2012-8-26
MATLAB's filter() function an implementation of direction form II, the detailed algorithm can be found at
http://www.mathworks.com/help/techdoc/ref/filter.html#f83-1015962
For your case, you need to first figure out what the initial condition is. You can use filtic function.
http://www.mathworks.com/help/toolbox/signal/ref/filtic.html
e.g.,
b = 1;
a = [1 0.5];
y = filter(b,a,ones(10,1),filtic(b,a,1));
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Honglei Chen
Honglei Chen 2012-8-27
It basically perform a reverse filtering operation. If you want the details, you can look at filtic code.

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Jan
Jan 2012-8-27
编辑：Jan 2012-8-27
You find a Matlab implementation of the FILTER command at Answers: hard-code-filter. This should reveal the meaning of the initial and final conditions.
Splitting a signal into parts might be helpful also:
signal = rand(1,1000);
[b,a] = butter(3, 0.5); % Arbitrary filter parameters
inital_cond1 = zeros(1, length(a) - 1);
[filtered1, final_cond1] = filter(b, a, signal(1:500), initial_cond1);
[filtered2, final_cond2] = filter(b, a, signal(501:1000), final_cond1);
Joined = [filtered1, filtered2];
% Now filter in one step:
OnePiece = filter(b, a, signal, initial_cond1);
% Compare the results - should be tiny:
disp(mean(abs(Joined - OnePiece)));
This shows, that the inital conditions are the values of the internal status of the filter. When the signal is split into pieces, the final conditions of the former part are the inital conditions of the current part.

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