From: Doug on
Hello,

Is there a method to estimate the statistical properties of additive
noise from a received signal observation. Assume we have

y(k) = x(k) + n(k)

and we observe the received signal y(k). I would like to know if
there is a way to determine the autocorrelation of the noise process
n(k), that is Rnn. This is not a speech problem. x(k) is a digital
PAM signal that is corrupted by ISI, it can be assumed
cyclostationary.

To simplify the problem, we can assume the noise is white Gaussian
noise, can we determine the noise variance, sigma^2, and hence Rnn for
this simplified case? Ideally I don't want to assume anything about
the noise, but if assuming white gaussian noise make things tractable,
I take that solution over nothing

Thanks in advance
-Doug
From: Tim Wescott on
On 08/12/2010 07:58 AM, Doug wrote:
> Hello,
>
> Is there a method to estimate the statistical properties of additive
> noise from a received signal observation. Assume we have
>
> y(k) = x(k) + n(k)
>
> and we observe the received signal y(k). I would like to know if
> there is a way to determine the autocorrelation of the noise process
> n(k), that is Rnn. This is not a speech problem. x(k) is a digital
> PAM signal that is corrupted by ISI, it can be assumed
> cyclostationary.
>
> To simplify the problem, we can assume the noise is white Gaussian
> noise, can we determine the noise variance, sigma^2, and hence Rnn for
> this simplified case? Ideally I don't want to assume anything about
> the noise, but if assuming white gaussian noise make things tractable,
> I take that solution over nothing
>
> Thanks in advance
> -Doug

Is the ISI known?

In theory you could demodulate the digital message, remodulate it back
to its ideal original, then corrupt it with the ISI to get an estimate
of x(k). Then you subtract that out of y(k), and play with n(k) to your
heart's content.

I imagine that this would only work in practice when n(k) is fairly bad,
because otherwise it'll be drowned out by the inaccuracies of the
process I mention above.

I've never tried this, so I couldn't comment on how well this would work
in practice -- other than saying that if you're designing a system from
scratch, it may be cheaper to arrange for quiet periods dedicated to
measuring the noise that it would be to do all the engineering work to
accurately measure the noise in the presence of the signal.

--

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

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html
From: Vladimir Vassilevsky on


Doug wrote:

> Hello,
>
> Is there a method to estimate the statistical properties of additive
> noise from a received signal observation. Assume we have
>
> y(k) = x(k) + n(k)
>
> and we observe the received signal y(k). I would like to know if
> there is a way to determine the autocorrelation of the noise process
> n(k), that is Rnn. This is not a speech problem. x(k) is a digital
> PAM signal that is corrupted by ISI, it can be assumed
> cyclostationary.
>
> To simplify the problem, we can assume the noise is white Gaussian
> noise, can we determine the noise variance, sigma^2, and hence Rnn for
> this simplified case? Ideally I don't want to assume anything about
> the noise, but if assuming white gaussian noise make things tractable,
> I take that solution over nothing

Synchronize to the baud rate and compare the statistics at the end and
at the 1/2 of the symbol interval.

VLV


From: Doug on
On Aug 12, 11:19 am, Tim Wescott <t...(a)seemywebsite.com> wrote:
> On 08/12/2010 07:58 AM, Doug wrote:
>
>
>
>
>
> > Hello,
>
> > Is there a method to estimate the statistical properties of additive
> > noise from a received signal observation.   Assume we have
>
> > y(k) = x(k) + n(k)
>
> > and we observe the received signal y(k).  I would like to know if
> > there is a way to determine the autocorrelation of the noise process
> > n(k), that is Rnn.  This is not a speech problem.  x(k) is a digital
> > PAM signal that is corrupted by ISI, it can be assumed
> > cyclostationary.
>
> > To simplify the problem, we can assume the noise is white Gaussian
> > noise, can we determine the noise variance, sigma^2, and hence Rnn for
> > this simplified case?  Ideally I don't want to assume anything about
> > the noise, but if assuming white gaussian noise make things tractable,
> > I take that solution over nothing
>
> > Thanks in advance
> > -Doug
>
> Is the ISI known?
>
> In theory you could demodulate the digital message, remodulate it back
> to its ideal original, then corrupt it with the ISI to get an estimate
> of x(k).  Then you subtract that out of y(k), and play with n(k) to your
> heart's content.
>
> I imagine that this would only work in practice when n(k) is fairly bad,
> because otherwise it'll be drowned out by the inaccuracies of the
> process I mention above.
>
> I've never tried this, so I couldn't comment on how well this would work
> in practice -- other than saying that if you're designing a system from
> scratch, it may be cheaper to arrange for quiet periods dedicated to
> measuring the noise that it would be to do all the engineering work to
> accurately measure the noise in the presence of the signal.
>
> --
>
> Tim Wescott
> Wescott Design Serviceshttp://www.wescottdesign.com
>
> Do you need to implement control loops in software?
> "Applied Control Theory for Embedded Systems" was written for you.
> See details athttp://www.wescottdesign.com/actfes/actfes.html- Hide quoted text -
>
> - Show quoted text -

Tim,

The ISI or equivalently, the channel impulse response is not known. I
need Rnn or at least sigma^2 in order to estimate the channel (long
story). There is no way to get reliable symbol decisions; channel
estimation must be done first.

Thanks!
-Doug
From: Doug on
On Aug 12, 11:27 am, Vladimir Vassilevsky <nos...(a)nowhere.com> wrote:
> Doug wrote:
> > Hello,
>
> > Is there a method to estimate the statistical properties of additive
> > noise from a received signal observation.   Assume we have
>
> > y(k) = x(k) + n(k)
>
> > and we observe the received signal y(k).  I would like to know if
> > there is a way to determine the autocorrelation of the noise process
> > n(k), that is Rnn.  This is not a speech problem.  x(k) is a digital
> > PAM signal that is corrupted by ISI, it can be assumed
> > cyclostationary.
>
> > To simplify the problem, we can assume the noise is white Gaussian
> > noise, can we determine the noise variance, sigma^2, and hence Rnn for
> > this simplified case?  Ideally I don't want to assume anything about
> > the noise, but if assuming white gaussian noise make things tractable,
> > I take that solution over nothing
>
> Synchronize to the baud rate and compare the statistics at the end and
> at the 1/2 of the symbol interval.
>
> VLV- Hide quoted text -
>
> - Show quoted text -

Vladimir,

It is no problem to do symbol synchronization, but I don't understand
how comparing the statistics at mid-sym and end-sym will get you
anything. Do you have more info or point me to a good reference?

Also, I just read that one way get sigma^2 is to find the minimum
eigenvalue of Ryy. This makes some sense and given the fact that x(k)
has a zero in its power spectral density I think the noise in that
region should yield a good result. Does anyone have more information
on this technique??

Thanks again
-Doug