Random variable through filter - D/A conversion

Hello,

I am trying to find how a random variable will transform once gone through
a filter. 
Specifically I am trying to find the probability density function when
Gaussian random variables are applied to a digital-to-analog conversion. 

I am considering a sample-and-hold operation, which essentially has a rect
impulse response. I want to know how the Gaussian PDF will be transformed
when sent through a filter.

Any advise or references are greatly appreciated.

Cheers.

Anybody...?
Thank you.

On Jul 16, 11:49&#4294967295;pm, "m26k9" <maduranga.liyan...@gmail.com> wrote:
> Hello,
>
> I am trying to find how a random variable will transform once gone through
> a filter.
> Specifically I am trying to find the probability density function when
> Gaussian random variables are applied to a digital-to-analog conversion.
>
> I am considering a sample-and-hold operation, which essentially has a rect
> impulse response. I want to know how the Gaussian PDF will be transformed
> when sent through a filter.
>
> Any advise or references are greatly appreciated.
>
> Cheers.

The pdf of a sum of two independent variables can be obtained by the
convolution of their respective pdf's

That doesn't answer your question I know but is the converse true?

Hardy

Thank you Hardy.

Suppose the converse is true, how could I relate it to the filter
convolution? Because I cannot find a PDF for the filter.

>> Gaussian random variables are applied to a digital-to-analog conversion.


Digital-to-analog, or analog-to-digital?  A Gaussian RV is a continuous
RV.  If analog-to-digital, the pdf should be *approximately* gaussian,
depending on the ratio of the variance to the quantization interval
(stairstep-looking).  If digital-to-analog, the pdf shouldn't be any worse
of an approximation to Gaussian than you've already made.  I am assuming an
ideal converter.


>I am trying to find how a random variable will transform once gone
through
>a filter. 
> [snip]
>I am considering a sample-and-hold operation, which essentially has a
rect
>impulse response. 

The impulse response is a largely separate problem.  You have not given
the psd or autocorrelation of your original signal.  If you know the
impulse response, though, what's the problem?

Thank you very much Michael.

This is digital-to-analog with uniform quantization I am assuming.
Basically a quantizer and a sample-and-hold filter in cascade. So the
filter will be a rect function. 

My input will be Gaussian but I am trying to find a general method to find
the output PDF of the signal after the S/H filter. 

My problem is to find the output PDF when the input PDF and the filter
characteristics are known. Is there any general method find how the input
PDF will be transformed once it goes through a filter?

Thank you very much.

"m26k9" <maduranga.liyanage@gmail.com> writes:

> From: "m26k9" <maduranga.liyanage@gmail.com>
> Subject: Re: Random variable through filter - D/A conversion
> Newsgroups: comp.dsp
> Date: Sat, 18 Jul 2009 11:49:33 -0500
>
> Thank you very much Michael.
>
> This is digital-to-analog with uniform quantization I am assuming.
> Basically a quantizer and a sample-and-hold filter in cascade. So the
> filter will be a rect function. 
>
> My input will be Gaussian but I am trying to find a general method to find
> the output PDF of the signal after the S/H filter. 
>
> My problem is to find the output PDF when the input PDF and the filter
> characteristics are known. Is there any general method find how the input
> PDF will be transformed once it goes through a filter?
>
> Thank you very much.

I believe someone has already given you the answer.

If your filter's impulse response is a unit-height rectangle with a
width that covers, say, N of your output samples, then the resulting PDF
will be the convolution of the N input PDFs, assuming the N input
samples are independent.

In the ideal S/H filter, only one sample at a time is held, so the
output distribution will be the same as the input distribution.

What *does* change, and what I haven't heard anyone bring up, is the
power spectrum of the output. If the digital samples are completely
uncorrelated, their spectrum would be white.  But the spectrum of the
S/H filter will be a sinc, so the output power spectrum will be sinc^2.

In practice, you're going to have some sort of lowpass filtering
(whether you want it or not) which will spread out your PDF and make the
output more Gaussian-like than the inputs (think Central Limit Theorem),
even if the inputs weren't Gaussian, and which will cut off the infinite
tails of the resulting power spectrum.
--
Randy Yates                      % "So now it's getting late,
Digital Signal Labs              %    and those who hesitate
mailto://yates@ieee.org          %    got no one..."
http://www.digitalsignallabs.com % 'Waterfall', *Face The Music*, ELO

>In practice, you're going to have some sort of lowpass filtering
>(whether you want it or not) which will spread out your PDF and make the
>output more Gaussian-like than the inputs (think Central Limit Theorem),
>even if the inputs weren't Gaussian, and which will cut off the infinite
>tails of the resulting power spectrum.

Thank you very much Randy.

This is what I was trying to find out. I do consider my DAC to be a
quantizer followed by a S/H filter but usually one needs a low-pass filter
after that to smooth out the sharp edges. So I am looking to find out if
there is a mathematical/analytical method to find the output PDF when the
input PDF and the filter impulse response if given.

My input PDF will be Gaussian (they are output from an IFFT block) so I
think after going through the {quantizer + S/H filter + LPF filter} the
output PDF will be Gaussian. Thank you very much for the clarification
Randy. I need to find the PDF of this output so I guess I only needs to
find the variance.

Thank you very much.

On Jul 18, 9:00&#4294967295;pm, "m26k9" <maduranga.liyan...@gmail.com> wrote:
> >In practice, you're going to have some sort of lowpass filtering
> >(whether you want it or not) which will spread out your PDF and make the
> >output more Gaussian-like than the inputs (think Central Limit Theorem),
> >even if the inputs weren't Gaussian, and which will cut off the infinite
> >tails of the resulting power spectrum.
>
> Thank you very much Randy.
>
> This is what I was trying to find out. I do consider my DAC to be a
> quantizer followed by a S/H filter but usually one needs a low-pass filter
> after that to smooth out the sharp edges. So I am looking to find out if
> there is a mathematical/analytical method to find the output PDF when the
> input PDF and the filter impulse response if given.
>
> My input PDF will be Gaussian (they are output from an IFFT block) so I
> think after going through the {quantizer + S/H filter + LPF filter} the
> output PDF will be Gaussian. Thank you very much for the clarification
> Randy. I need to find the PDF of this output so I guess I only needs to
> find the variance.
>
> Thank you very much.

Well, if I understand your question correctly, this is just a function
of two random variables: let the input be a Gaussian X, and the filter
is a uniform (for now, you may have other pdf's on your mind) Y, the
output is Z, so we have

Z(t) = X(t) * Y(t)

Or, even simpler, find the characteristic funciton of X and Y in S-
transform, you can get Z(s) = X(s) Y(s). I guess it would be easier to
avoid an integration. Except this small tricks, now gets to your
problem: what is the pdf of Z?

It is a function of two RV's X and Y. So you can use a transform
method to figure it out.

Thank you very much Verixroe.

>It is a function of two RV's X and Y. So you can use a transform
>method to figure it out.

Is Y a random variable? If Y is a RV then I can find a PDF for that and
eventually find the PDF of Z. But since Y is a filter response, which is
deterministic, how can I find a PDF for that?

Thank you very much for the input.