# Measurement Noise of PSD estimation (AR)

Started by July 13, 2009
```Hallo,

I am quite new to this topic and I hope I can get some help here.
suppose we have a system driven by white noise resulting in a Random
Process. The measurement I get is the addition of the Random process with
the measurement noise.

my question is :
Is it possible to identify the measurement noise from the recieved noisy
signal? Do anyone has a paper or a book approaching such topic?

what I am trying to do afterward is to estimate the AR parameters for the
De-Noised measured signal.

```
```On 13 Jul, 17:07, "m.alwaieh" <m.alaw...@gmail.com> wrote:
> Hallo,
>
> &#2013266080;I am quite new to this topic and I hope I can get some help here.
> suppose we have a system driven by white noise resulting in a Random
> Process. The measurement I get is the addition of the Random process with
> the measurement noise.
>
> my question is :
> Is it possible to identify the measurement noise from the recieved noisy
> signal? Do anyone has a paper or a book approaching such topic?

The best you can do is to estimate the noise power,
subject to the assumptions of the AR model. Once you
have come up with a set of parameters, you also have
an associated noise variance, which caracterizes the
noise over the data segment.

This noise will contain several components:

1) Model mis-match, in the case the data do not
comply to the AR(p) model (including additive
noise not complying to assumptions about noise
autocovariance function)
3) Instrumentation noise, like hum from power
supplies and quantiation noise

You can't separate between these components unless you

As for literature, "Random Data" by Bendat & Piersol
is a classic.

Rune
```
```m.alwaieh wrote:
> Hallo,
>
>  I am quite new to this topic and I hope I can get some help here.
> suppose we have a system driven by white noise resulting in a Random
> Process. The measurement I get is the addition of the Random process with
> the measurement noise.
>
> my question is :
> Is it possible to identify the measurement noise from the recieved noisy
> signal? Do anyone has a paper or a book approaching such topic?

Probably not, but there are more things on heaven and earth ...

> what I am trying to do afterward is to estimate the AR parameters for the
> De-Noised measured signal.

How about determining the measurement noise and subtracting its power
from your "noisy" measurement of the random process? That can be a bit
tricky, but it's certainly possible. If the measurement noise is 20 dB
below the process noise, it will have negligible effect.
log(10)/log(10.1) =.9957

Jerry
--
Engineering is the art of making what you want from things you can get.
&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
```
```On Jul 13, 8:07&#2013266080;am, "m.alwaieh" <m.alaw...@gmail.com> wrote:
> Hallo,
>
> &#2013266080;I am quite new to this topic and I hope I can get some help here.
> suppose we have a system driven by white noise resulting in a Random
> Process. The measurement I get is the addition of the Random process with
> the measurement noise.
>
> my question is :
> Is it possible to identify the measurement noise from the recieved noisy
> signal? Do anyone has a paper or a book approaching such topic?
>
> what I am trying to do afterward is to estimate the AR parameters for the
> De-Noised measured signal.

No you cannot. You may be able to find the ratio of the measurement
noise variance  to the innovations variance. Not straight forward.

Hardy
```
```*Suppose that I have the measurement noise variance, How to carry on with
power spectral density estimation without including the measurment noise?

I mean how can i obtain the transfer function (H(z)) charcterizing the
system from a noisy measurment given I know this noise?

*Do Sub-Space Estimators (like MUSIC) as they decompose the observation
into a signal subspace and a noise subspace?

>As for literature, "Random Data" by Bendat & Piersol
>is a classic.
Thanks, I will check this one.
```
```On 14 Jul, 10:10, "m.alawieh" <m.alaw...@gmail.com> wrote:
> *Suppose that I have the measurement noise variance, How to carry on with
> power spectral density estimation without including the measurment noise?
>
> I mean how can i obtain the transfer function (H(z)) charcterizing the
> system from a noisy measurment given I know this noise?

One way is to pre-whiten the noise. Another is to
subtract the noise covariance from the signal + noise
covariance. Both methods have its risks.

> *Do Sub-Space Estimators (like MUSIC) as they decompose the observation
> into a signal subspace and a noise subspace? &#2013266080;

Do what? Those kinds of things can also exploit

Rune
```
```>One way is to pre-whiten the noise. Another is to subtract the noise
covariance from the signal + noise
>covariance. Both methods have its risks.

I didn't deal with this problems since a while, I need to refresh my
information first. Thanks for the highlights though.

>Do what? Those kinds of things can also exploit
>
Sorry this wrong statement, I read that MUSIC algorithm is based on
dividing the observation space to a signal space and noise space.

The way I understand it is; this algorithm estimates the signal from the
observed noisy measurement and the noise. This is done by obtaining the
eigen-vector approximation of the auto-correlation matrix for each of the
noise and signal spaces.

Here I might be way too wrong,
1-we take the AC matrix obtained from AR estimation for example.
2-use MUSIC (or any subspace method) to get the eigen values for the corr
matrix
3- Yeooo, I have signal and noise separated spaces.

```
```On 14 Jul, 14:03, "m.alawieh" <m.alaw...@gmail.com> wrote:

> Here I might be way too wrong,
> 1-we take the AC matrix obtained from AR estimation for example.
> 2-use MUSIC (or any subspace method) to get the eigen values for the corr
> matrix
> 3- Yeooo, I have signal and noise separated spaces.

Not quite. The idea behind MUSIC is that the noise
covariance matrix is of full rank, whil ethe signal
covariance matrix is singular. Which means that the
signal+noise covariance matrix is of full rank.

What happens with MUSIC is that one isolates the
signal subspace, that also contains noise. So you
don't *remove* the noise; you *reduce* it. And
anyway, with MUSIC it's the noise-only subspace
you use in the computations.

But your general idea is correct. If you apply the
same sorts of ideas to an AR estimator, the resulting
coefficient estimate might be better.

However, there are a couple of details to watch
ut for:

1) These things require very expensive computational
routines, like eigen value decompositions or
SVDs.
2) Order estimation is a moot point, since order
estimators as developed for the Levinson recursion
do not *formally* match EVDs or SVDs.

This last point doesn't mean the order estimators
do not work (they do), but depending on the context
you need to be very cautious. This is the kind of
thing that might seed discussions if you try and
publish such a method in one of the journals.

Rune
```