Forums

Noise estimation

Started by Doug August 12, 2010
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
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

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
On Aug 12, 11:19&#2013266080;am, Tim Wescott <t...@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. &#2013266080; Assume we have > > > y(k) = x(k) + n(k) > > > and we observe the received signal y(k). &#2013266080;I would like to know if > > there is a way to determine the autocorrelation of the noise process > > n(k), that is Rnn. &#2013266080;This is not a speech problem. &#2013266080;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? &#2013266080;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). &#2013266080;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
On 08/12/2010 08:27 AM, Vladimir Vassilevsky 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.
That gets you the noise variance, but not its autocorrelation. -- 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
On Aug 12, 11:27&#2013266080;am, Vladimir Vassilevsky <nos...@nowhere.com> wrote:
> Doug wrote: > > Hello, > > > Is there a method to estimate the statistical properties of additive > > noise from a received signal observation. &#2013266080; Assume we have > > > y(k) = x(k) + n(k) > > > and we observe the received signal y(k). &#2013266080;I would like to know if > > there is a way to determine the autocorrelation of the noise process > > n(k), that is Rnn. &#2013266080;This is not a speech problem. &#2013266080;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? &#2013266080;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
Doug  <doug.barker@itt.com> wrote:

>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.
If there is pure signal of interest and no noise, the mid-symbol vs. symbol-edge statistics are related in a way defined by the signal. If there is pure noise and no signal, this relationship is not observed. So it is logical that if there is some signal and some noise, you will see something in between these two observations. (Whether it is useful for your requirement is unclear.) S.
On 08/12/2010 12:09 PM, Doug wrote:
> On Aug 12, 11:19 am, Tim Wescott<t...@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.
If you really need the noise autocorrelation or at least sigma^2 before you can estimate the channel, then I think you're screwed (perhaps its time for Vladimir's story about the girl, the boy, and the rabbi). Perhaps you should relate your long story about why you feel you need the noise statistics before you can estimate the ISI -- my understanding of channel estimation is that as long as you have a long enough sequence, and at least a partially reliable set of symbol decisions, then eventually you can estimate the ISI. -- 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
On 08/12/2010 12:13 PM, Doug wrote:
> On Aug 12, 11:27 am, Vladimir Vassilevsky<nos...@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??
If things are too messed up to get good symbol detection I don't think this will work. Just making a plain old filter around a zero in the signal spectrum and estimating energy content should, however, make for a fairly decent measure of noise. -- 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
On Aug 12, 9:58&#2013266080;am, Doug <doug.bar...@itt.com> wrote:
> Hello, > > Is there a method to estimate the statistical properties of additive > noise from a received signal observation. &#2013266080; Assume we have > > y(k) = x(k) + n(k) > > and we observe the received signal y(k). &#2013266080;I would like to know if > there is a way to determine the autocorrelation of the noise process > n(k), that is Rnn. &#2013266080;This is not a speech problem. &#2013266080;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? &#2013266080;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
If a training sequence is feasible, send a long stream of constant amplitude pulses. Then, the channel output at the sampling instant should be a constant (after a while, where "while" depends on how long you think the ISI lasts) plus noise. Thus, you can estimate the mean from the noisy sample values, and even get a histogram of sample values that will give an estimate of the noise pdf (not necessarily Gaussian). If the stream is long enough, you can also estimate the autocorrelation function of the noise. If a training sequence is not possible, search the canoe again for a paddle while holding your nose.... Hope this helps. --Dilip Sarwate