Reply by Thomas Arildsen●October 4, 20072007-10-04

Jerry Avins skrev:

> Thomas Arildsen wrote:
>> I have an ARMA process with known parameters. Are there general
>> expressions for variance and possibly autocorrelation of the output
>> for such processes?
>> I can find the above for an order-(1,1) process in for example
>> Shanmugan & Breipohl's "Random Signals" (Wiley, 1988), but is it
>> possible for higher orders? If it helps, I would be happy to confine
>> the processess in question to order (p,p).
>
> Would http://www.polyx.com/demo_covf.htm help?
>

Thanks, I think there might be some ideas to help me solve the problem
as well. I actually already have one method, but I would like to check
it against some other known method.
Thomas Arildsen
--
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Reply by Jerry Avins●October 3, 20072007-10-03

Thomas Arildsen wrote:

> I have an ARMA process with known parameters. Are there general
> expressions for variance and possibly autocorrelation of the output for
> such processes?
> I can find the above for an order-(1,1) process in for example Shanmugan
> & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for
> higher orders? If it helps, I would be happy to confine the processess
> in question to order (p,p).

Would http://www.polyx.com/demo_covf.htm help?
Jerry
--
Engineering is the art of making what you want from things you can get.
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Reply by Thomas Arildsen●October 3, 20072007-10-03

Andor skrev:

> Thomas Arildsen wrote:
>> I have an ARMA process with known parameters. Are there general
>> expressions for variance and possibly autocorrelation of the output for
>> such processes?
>
> Yes. If x[n] is a WSS stochastic process, and h[n] the impulse
> response of your (stable) ARMA filter, and y[n] the output of the ARMA
> filter, ie.
>
> y[n] = x[n] * h[n],
>
> then the autocorrelation r_y[n] is
>
> r_y[n] = r_x[n] * r_h[n],
>
> where r_h[n] = h[n] * h[-n] is the autocorrelation of the impulse
> response. The mean m_y of the process y[n] is the mean m_x of x[n]
> multiplied by the DC response of the ARMA filter,
>
> m_y = m_x sum_k h[k].
>
> The variance is sigma_y^2 = r_y[0].
>
> Regards,
> Andor
>

I have the PSD expression for the filter; I think I will try to inverse
transform from that to the autocorrelation in stead.
Thomas Arildsen
--
All email to sender address is lost.
My real adress is at es dot aau dot dk for user tha.

Reply by Thomas Arildsen●October 3, 20072007-10-03

HardySpicer skrev:

> On Oct 3, 8:43 pm, Thomas Arildsen <comp-dsp.es-
> aau...@spamgourmet.com> wrote:

>

>> Would that be "Introduction to Stochastic Control Theory" from 1970?
>> What is meant by 'a set of tables'? Are they tables of calculated
>> variances and/or correlation values for specific ARMA configurations or
>> expressions for calculating these? I am looking for the latter.
>>
>> Thomas Arildsen
>
> Yes! It's formula based on the ARMA coefficients for variance orders
> and yes it's variances.
>
> Hardy
>

Great, thank's. It seems to have been reprinted on Dover in 2006 and is
available pretty cheap on Amazon. I'll give it a try.
Thomas Arildsen
--
All email to sender address is lost.
My real adress is at es dot aau dot dk for user tha.

Reply by Andor●October 3, 20072007-10-03

Thomas Arildsen wrote:

> I have an ARMA process with known parameters. Are there general
> expressions for variance and possibly autocorrelation of the output for
> such processes?

Yes. If x[n] is a WSS stochastic process, and h[n] the impulse
response of your (stable) ARMA filter, and y[n] the output of the ARMA
filter, ie.
y[n] = x[n] * h[n],
then the autocorrelation r_y[n] is
r_y[n] = r_x[n] * r_h[n],
where r_h[n] = h[n] * h[-n] is the autocorrelation of the impulse
response. The mean m_y of the process y[n] is the mean m_x of x[n]
multiplied by the DC response of the ARMA filter,
m_y = m_x sum_k h[k].
The variance is sigma_y^2 = r_y[0].
Regards,
Andor

Reply by HardySpicer●October 3, 20072007-10-03

On Oct 3, 8:43 pm, Thomas Arildsen <comp-dsp.es-
aau...@spamgourmet.com> wrote:

> HardySpicer skrev:
>
>
>
> > On Oct 3, 1:02 am, Thomas Arildsen <comp-dsp.es-
> > aau...@spamgourmet.com> wrote:
> >> I have an ARMA process with known parameters. Are there general
> >> expressions for variance and possibly autocorrelation of the output for
> >> such processes?
> >> I can find the above for an order-(1,1) process in for example Shanmugan
> >> & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for
> >> higher orders? If it helps, I would be happy to confine the processess
> >> in question to order (p,p).
>
> >> Thomas Arildsen
>
> > There is a set of tables in a book by Karl Astrom - the old 1970s
> > book.
>
> > Hardy
>
> Would that be "Introduction to Stochastic Control Theory" from 1970?
> What is meant by 'a set of tables'? Are they tables of calculated
> variances and/or correlation values for specific ARMA configurations or
> expressions for calculating these? I am looking for the latter.
>
> Thomas Arildsen
>
> --
> All email to sender address is lost.
> My real adress is at es dot aau dot dk for user tha.

Yes! It's formula based on the ARMA coefficients for variance orders
and yes it's variances.
Hardy

Reply by Thomas Arildsen●October 3, 20072007-10-03

HardySpicer skrev:

> On Oct 3, 1:02 am, Thomas Arildsen <comp-dsp.es-
> aau...@spamgourmet.com> wrote:
>> I have an ARMA process with known parameters. Are there general
>> expressions for variance and possibly autocorrelation of the output for
>> such processes?
>> I can find the above for an order-(1,1) process in for example Shanmugan
>> & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for
>> higher orders? If it helps, I would be happy to confine the processess
>> in question to order (p,p).
>>
>> Thomas Arildsen
>
> There is a set of tables in a book by Karl Astrom - the old 1970s
> book.
>
> Hardy
>

Would that be "Introduction to Stochastic Control Theory" from 1970?
What is meant by 'a set of tables'? Are they tables of calculated
variances and/or correlation values for specific ARMA configurations or
expressions for calculating these? I am looking for the latter.
Thomas Arildsen
--
All email to sender address is lost.
My real adress is at es dot aau dot dk for user tha.

Reply by HardySpicer●October 2, 20072007-10-02

On Oct 3, 1:02 am, Thomas Arildsen <comp-dsp.es-
aau...@spamgourmet.com> wrote:

> I have an ARMA process with known parameters. Are there general
> expressions for variance and possibly autocorrelation of the output for
> such processes?
> I can find the above for an order-(1,1) process in for example Shanmugan
> & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for
> higher orders? If it helps, I would be happy to confine the processess
> in question to order (p,p).
>
> Thomas Arildsen
> --
> All email to sender address is lost.
> My real adress is at es dot aau dot dk for user tha.

There is a set of tables in a book by Karl Astrom - the old 1970s
book.
Hardy

Reply by Thomas Arildsen●October 2, 20072007-10-02

I have an ARMA process with known parameters. Are there general
expressions for variance and possibly autocorrelation of the output for
such processes?
I can find the above for an order-(1,1) process in for example Shanmugan
& Breipohl's "Random Signals" (Wiley, 1988), but is it possible for
higher orders? If it helps, I would be happy to confine the processess
in question to order (p,p).
Thomas Arildsen
--
All email to sender address is lost.
My real adress is at es dot aau dot dk for user tha.