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# Discussion Groups | Comp.DSP | how to pass a random process through a non-linear system?

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# how to pass a random process through a non-linear system? - kiki - 2004-12-26 01:45:00

```Hi all,

I understood how to pass a stationary random process through a LTI syste...
the autocorrelation of the output has a beautiful relation with the input's
autocorrelations, and so does the power spectrum density...

But what if my stationary zero mean Gaussian random process x(t) go through
a non-linear system, for example, a signum device(output 1 for input x(t)>0
and output -1 for input x(t)<=0))

How to find the autocorrelation and the power spectrum density of the output
if I was given an input autocorrelation?

Thanks a lot!

```
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# Re: how to pass a random process through a non-linear system? - ted - 2004-12-26 02:10:00

```"kiki" <l...@yahoo.com> wrote in message
news:cqlmms\$epv\$1...@news.Stanford.EDU...
> Hi all,
>
> I understood how to pass a stationary random process through a LTI
syste...
> the autocorrelation of the output has a beautiful relation with the
input's
> autocorrelations, and so does the power spectrum density...
>
> But what if my stationary zero mean Gaussian random process x(t) go
through
> a non-linear system, for example, a signum device(output 1 for input
x(t)>0
> and output -1 for input x(t)<=0))
>
> How to find the autocorrelation and the power spectrum density of the
output
> if I was given an input autocorrelation?
>

I don't think you can.

> Thanks a lot!
>
>
>

```
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# Re: how to pass a random process through a non-linear system? - Tim Wescott - 2004-12-26 11:44:00

```kiki wrote:

> Hi all,
>
> I understood how to pass a stationary random process through a LTI syste...
> the autocorrelation of the output has a beautiful relation with the input's
> autocorrelations, and so does the power spectrum density...
>
> But what if my stationary zero mean Gaussian random process x(t) go through
> a non-linear system, for example, a signum device(output 1 for input x(t)>0
> and output -1 for input x(t)<=0))
>
> How to find the autocorrelation and the power spectrum density of the output
> if I was given an input autocorrelation?
>
> Thanks a lot!
>
>
>
If you have a stationary process with known distribution and
autocorrelation, then you know:

a: the probability distribution of the process at any point in time
(this is also the case if you have a non-stationary process with a known
time dependence).

b: the conditional probability distribution of the process at any point
in time given a sample of the process at some other point in time.

If you have the above two bits of information and you pass the process
through a memoryless nonlinearity (such as your signum function), then
you should be able to extract:

c:  The probability distribution of the process at any point in time.

d:  The conditional probability distribution of the process at any point
in time given a sample of the process at some other point in time.

If you know (c) then you can specify whether your output process is
stationary (it will be for a stationary input process -- it may not
necessarily be for a non-stationary input, depending on the nature of
the time dependence and your nonlinearity, I just don't know in general
but in specific you should be able to do the math and find out).  If you
know (d) then you can extract the autocorrelation function.  Then you

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```
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# Re: how to pass a random process through a non-linear system? - Country_Chiel - 2004-12-29 00:47:00

```"kiki" <l...@yahoo.com> wrote in message
news:cqlmms\$epv\$1...@news.Stanford.EDU...
> Hi all,
>
> I understood how to pass a stationary random process through a LTI
syste...
> the autocorrelation of the output has a beautiful relation with the
input's
> autocorrelations, and so does the power spectrum density...
>
> But what if my stationary zero mean Gaussian random process x(t) go
through
> a non-linear system, for example, a signum device(output 1 for input
x(t)>0
> and output -1 for input x(t)<=0))
>
> How to find the autocorrelation and the power spectrum density of the
output
> if I was given an input autocorrelation?
>
> Thanks a lot!
>
>
>
Yes I believe this has been done by Wiener. I think it is called the Wiener
expansion and such work has been done by others independently eg Barrett
back in the 1940s!
It will take you quite a while to track down the refs but most of it is
there.

Country Chiel

```
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# Re: how to pass a random process through a non-linear system? - Mike - 2004-12-30 00:23:00

```On Sat, 25 Dec 2004 22:45:48 -0800, kiki wrote:

> Hi all,
>
> I understood how to pass a stationary random process through a LTI syste...
> the autocorrelation of the output has a beautiful relation with the input's
> autocorrelations, and so does the power spectrum density...
>
> But what if my stationary zero mean Gaussian random process x(t) go through
> a non-linear system, for example, a signum device(output 1 for input x(t)>0
> and output -1 for input x(t)<=0))
>
> How to find the autocorrelation and the power spectrum density of the output
> if I was given an input autocorrelation?
>
> Thanks a lot!

In general this is not an easy problem. In the case you have presented,
however, there is a solution.

The solution I have comes from Lawson and Uhlenbeck, "Threshold Signals,"
MIT Radiation Laboratory Series, McGraw-Hill, 1950, page 57.

They reference an earlier Radiation Lab Report (Van Vleck, "The Spectrum of
Clipped Noise," RRL Report 51, July 21, 1943), which you can purchase from
the MIT library, last I checked. It's not cheap. Van Vleck's work was also
published in the IEEE Proceedings a few years ago (1966?), with Middleton
as a coauthor.

The final result is Ry(t) = (2/pi)asin(Rx(t)/Rx(0)), where Ry is the output
autocorrelation, and Rx is the input autocorrelation. The power spectrum of
the noise is the Fourier Transform of Ry(t), which is often easy to
calculate using numerical math packages. Note that if your input is not
band-limited, then Rx(t)/Rx(0) = 1 when t = 0, and 0 elsewhere, making the
Fourier Transform easy to calculate.

-- Mike --
```
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