Statistics of linearly filtered signal

Hi all,
I would like to know "the" right method to analytically calculate the 
Probability Density Function (PDF) of the output of a FIR filter whose 
input PDF is not gaussian (the indicator function for instance)...
I learnt that when i was a student (long time ago) and i can't remember 
the way to make such calculation... :(
Can anyone help ?
Thanx in advance.
A.Herbert

Arthur Herbert wrote:

> Hi all,
> I would like to know "the" right method to analytically calculate the 
> Probability Density Function (PDF) of the output of a FIR filter whose 
> input PDF is not gaussian (the indicator function for instance)...
> I learnt that when i was a student (long time ago) and i can't remember 
> the way to make such calculation... :(
> Can anyone help ?
> Thanx in advance.
> A.Herbert

I don't think there is a short answer - since the output of the filter 
is just the weighted output of all the samples, if you have white noise 
you can just use basic statistical theory to figure it out.

If the PDF of the input is well-enough behaved, and if you have enough 
samples to significantly span it's autocorrelation function then you can 
take refuge in the central limit theorem and claim that the output is 
Gaussian with mean and variance determined by the filter properties and 
the mean and variance of the input signal.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Arthur Herbert wrote:
> Hi all,
> I would like to know "the" right method to analytically calculate the 
> Probability Density Function (PDF) of the output of a FIR filter whose 
> input PDF is not gaussian (the indicator function for instance)...

You need two facts:

1. The pdf f of the sum of two random variables X_1 and X_2 (with pdf 
f_1 and  f_2) is the convolution:

f = f_1 * f_2.


2. If X has pdf f_X, then the random variable Y = a X (a =/= 0) has pdf

f_Y(y) = 1/|a| f_X(y/a)

Therefore if the input is i.i.d. with pdf f, then the FIR output has pdf

f_FIR(x) = \convolution_{k=0}^{N-1} 1/|a_k| f(x/a_k)

Alternatively, you can calculate the inverse Fourier transform of the 
N-th power of the characteristic function of the pdf f.

Regards,
Andor

Arthur Herbert <herbert.arthur@caramail.com> writes:

> Hi all,
> I would like to know "the" right method to analytically calculate the
> Probability Density Function (PDF) of the output of a FIR filter whose
> input PDF is not gaussian (the indicator function for instance)...
> I learnt that when i was a student (long time ago) and i can't
> remember the way to make such calculation... :(
> Can anyone help ?
> Thanx in advance.
> A.Herbert

If the input samples are independent, then the output pdf is the convolution
of the input pdf's. The coefficients scale the input pdf's. 
-- 
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface, 
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

"Andor Bariska" <an2or@nospam.net> wrote in message
news:4111e4cd$1@pfaff2.ethz.ch...
>
> 1. The pdf f of the sum of two random variables X_1 and X_2 (with pdf
> f_1 and  f_2) is the convolution:
>
> f = f_1 * f_2.

As Randy Yates noted in his response, this convolution formula
applies when the random variables are independent.  More
generally, the pdf of the sum is given by the integral

integral from -infinity to +infinity f(x, z-x) dx

where f(x, y) is the joint pdf of the two variables.  This reduces
to

integral from -infinity to +infinity f_1(x).f_2(z-x) dx = f_1 * f_2

when the random variables are independent because then the
joint pdf factors into the product of the pdfs of X_1 and X_2.