frequency estimation

frequency estimation

Post by Nith » Wed, 28 May 2003 12:29:42



I got the following eigen values of a toeplitz autocorrelation matrix
of order 5x5. The data was of length 40 samples (noise+signal).

0.7176
1.0324
1.6223
7.9404
10.4663

To estimate the number of sinusoids from these eigen values i chose
0.7176,1.0324,1.6223  as the eigen values corresponding to noise
subspace. Please let me know if this choice of mine is correct. from
this i concluded that there are two complex sinusoids in the given
data sequence of 40 samples.

Please let me know if my conclusion is correct or not.

Thanks,
Nithin

 
 
 

frequency estimation

Post by Rune Alln » Wed, 28 May 2003 21:44:21



> I got the following eigen values of a toeplitz autocorrelation matrix
> of order 5x5. The data was of length 40 samples (noise+signal).

> 0.7176
> 1.0324
> 1.6223
> 7.9404
> 10.4663

> To estimate the number of sinusoids from these eigen values i chose
> 0.7176,1.0324,1.6223  as the eigen values corresponding to noise
> subspace. Please let me know if this choice of mine is correct. from
> this i concluded that there are two complex sinusoids in the given
> data sequence of 40 samples.

> Please let me know if my conclusion is correct or not.

> Thanks,
> Nithin

Nithin,

From a first-glance, subjective *interpretation*, I agree that
it is *probable* that your conclusion is correct. As you obviously
have found out, order estimation for subspace separation has some sort
of subjectiveness to it, i.e. that it may be influenced by the person
who does the analysis. For instance, how would you judge if the second
largest eigenvalue was 4.9 and not 7.9? Some would stick to two components,
others may see only one while others again may see three or four components.

There are statistical methods that may come to your aid. Try to look
for "Akaike's Information Criterion". Any decent and recent (since ~1990
onwards) text book on statistical signal processing should have at least
a note on this and similar techniques. "Order estimation" could also be
a good key to search for.

Rune

 
 
 

frequency estimation

Post by Nith » Thu, 29 May 2003 15:02:53




> > I got the following eigen values of a toeplitz autocorrelation matrix
> > of order 5x5. The data was of length 40 samples (noise+signal).

> > 0.7176
> > 1.0324
> > 1.6223
> > 7.9404
> > 10.4663

> > To estimate the number of sinusoids from these eigen values i chose
> > 0.7176,1.0324,1.6223  as the eigen values corresponding to noise
> > subspace. Please let me know if this choice of mine is correct. from
> > this i concluded that there are two complex sinusoids in the given
> > data sequence of 40 samples.

> > Please let me know if my conclusion is correct or not.

> > Thanks,
> > Nithin

> Nithin,

> From a first-glance, subjective *interpretation*, I agree that
> it is *probable* that your conclusion is correct. As you obviously
> have found out, order estimation for subspace separation has some sort
> of subjectiveness to it, i.e. that it may be influenced by the person
> who does the analysis. For instance, how would you judge if the second
> largest eigenvalue was 4.9 and not 7.9? Some would stick to two components,
> others may see only one while others again may see three or four components.

> There are statistical methods that may come to your aid. Try to look
> for "Akaike's Information Criterion". Any decent and recent (since ~1990
> onwards) text book on statistical signal processing should have at least
> a note on this and similar techniques. "Order estimation" could also be
> a good key to search for.

> Rune

Hi
Thanks for your reply.
For a 11 x 11 toeplitz autocorrelation matrix the eigen values i got
are below.

    0.4687
    0.7427
    0.7700
    0.9202
    0.9706
    1.1414
    1.2223
    4.7889
    4.7909
   15.4860
   16.6119

Even from this, my perception is that there are just two eigen values
corresponding to the signal sub-space.

From, the FPE criterion ( calculated for autocorrelation matrices upto
order 11 x 11) i found that order =2 minimized the FPE criterion. But
i dont know if i have to check for higher orders of the autocorreltion
matrix. Won't the higher order autocorrelation matrices be in much
error? Please clarify this to me.

When i computed the eigen values of a higher order autocorrelation
matrix (23x23)i got 4 eigen values greater than 12 and the rest of the
eigen values were less than 5. This would indicate that there are 4
complex sinusoids.

Should i consider the higher order result or that of the lower order
one? Is my assumption of higher order autocorrelation matrices being
more in error right, when the data sample is of length just 40?

Thanks,
-Nithin

 
 
 

frequency estimation

Post by Rune Alln » Thu, 29 May 2003 22:28:31



> Hi
> Thanks for your reply.
> For a 11 x 11 toeplitz autocorrelation matrix the eigen values i got
> are below.

>     0.4687
>     0.7427
>     0.7700
>     0.9202
>     0.9706
>     1.1414
>     1.2223
>     4.7889
>     4.7909
>    15.4860
>    16.6119

> Even from this, my perception is that there are just two eigen values
> corresponding to the signal sub-space.

You *could* be right. On the other hand, the two ~4.8 eigenvalues
*may*
indicate that there are two more components at frequencies close to
the
two first frequencies.

Quote:> From, the FPE criterion ( calculated for autocorrelation matrices upto
> order 11 x 11) i found that order =2 minimized the FPE criterion. But
> i dont know if i have to check for higher orders of the autocorreltion
> matrix. Won't the higher order autocorrelation matrices be in much
> error? Please clarify this to me.

That's where insight, experience and voodoo comes in to play. Choosing
the "correct" order of the autocovariance/autocorrelation matrix is
all but a black art. It takes trial and error over long time to
develop
a "gut feeling" for what orders work.

There are some "rules of thumb" you may use. One strict requirement is
that the order P of your autocovariance matrix is higher than the
number D
of sinusoidals present in the data set. So you need to know a lot
about
what measurements you will make, already at the design stages of
your processing system. The other is that the order shouldn't be too
high. The rule of thumb I used was P=3/2 D, when I had played around
with my data a bit. Now, I used pre-recorded data in an off-line
processing situation, and had quite some freedom of choise. You may
not have that freedom of choise, and that would restrict your options.

Quote:> When i computed the eigen values of a higher order autocorrelation
> matrix (23x23)i got 4 eigen values greater than 12 and the rest of the
> eigen values were less than 5. This would indicate that there are 4
> complex sinusoids.

Sure it does. I would go for the AIC or some other type of order
estimator.
I don't know FPE, so I don't know how it works or its properties.

Quote:> Should i consider the higher order result or that of the lower order
> one? Is my assumption of higher order autocorrelation matrices being
> more in error right, when the data sample is of length just 40?

Ouch! This is a delicate matter. How many 40-length snapshots/frames
of
data do you have? You would want to average at least 40 frames in your
autocovariance estimate. If you have only one frame, you should be
very
careful, it may very well introduce artifacts as the "extra" number of
signals you found in the 23X23 case. I gave an outline in a post not
long
ago about efficient use of data when doung frequency estimation, check
out

http://groups.google.com/groups?hl=no&lr=&ie=UTF-8&selm=f56893ae.0303...

and make sure you also read the correction I posted a few days
afterwards.

Rune

 
 
 

frequency estimation

Post by Nith » Mon, 02 Jun 2003 13:59:18




> > Hi
> > Thanks for your reply.
> > For a 11 x 11 toeplitz autocorrelation matrix the eigen values i got
> > are below.

> >     0.4687
> >     0.7427
> >     0.7700
> >     0.9202
> >     0.9706
> >     1.1414
> >     1.2223
> >     4.7889
> >     4.7909
> >    15.4860
> >    16.6119

> > Even from this, my perception is that there are just two eigen values
> > corresponding to the signal sub-space.

> You *could* be right. On the other hand, the two ~4.8 eigenvalues
> *may*
> indicate that there are two more components at frequencies close to
> the
> two first frequencies.

> > From, the FPE criterion ( calculated for autocorrelation matrices upto
> > order 11 x 11) i found that order =2 minimized the FPE criterion. But
> > i dont know if i have to check for higher orders of the autocorreltion
> > matrix. Won't the higher order autocorrelation matrices be in much
> > error? Please clarify this to me.

> That's where insight, experience and voodoo comes in to play. Choosing
> the "correct" order of the autocovariance/autocorrelation matrix is
> all but a black art. It takes trial and error over long time to
> develop
> a "gut feeling" for what orders work.

> There are some "rules of thumb" you may use. One strict requirement is
> that the order P of your autocovariance matrix is higher than the
> number D
> of sinusoidals present in the data set. So you need to know a lot
> about
> what measurements you will make, already at the design stages of
> your processing system. The other is that the order shouldn't be too
> high. The rule of thumb I used was P=3/2 D, when I had played around
> with my data a bit. Now, I used pre-recorded data in an off-line
> processing situation, and had quite some freedom of choise. You may
> not have that freedom of choise, and that would restrict your options.

> > When i computed the eigen values of a higher order autocorrelation
> > matrix (23x23)i got 4 eigen values greater than 12 and the rest of the
> > eigen values were less than 5. This would indicate that there are 4
> > complex sinusoids.

> Sure it does. I would go for the AIC or some other type of order
> estimator.
> I don't know FPE, so I don't know how it works or its properties.

> > Should i consider the higher order result or that of the lower order
> > one? Is my assumption of higher order autocorrelation matrices being
> > more in error right, when the data sample is of length just 40?

> Ouch! This is a delicate matter. How many 40-length snapshots/frames
> of
> data do you have? You would want to average at least 40 frames in your
> autocovariance estimate. If you have only one frame, you should be
> very
> careful, it may very well introduce artifacts as the "extra" number of
> signals you found in the 23X23 case. I gave an outline in a post not
> long
> ago about efficient use of data when doung frequency estimation, check
> out

> http://groups.google.com/groups?hl=no&lr=&ie=UTF-8&selm=f56893ae.0303...

> and make sure you also read the correction I posted a few days
> afterwards.

> Rune

Hi Rune,
The power spectrum by Burg' method (with AR order 18) indicated a peak
just adjacent to the two main peaks. Can i take this as the reference
in separating the eigen values into signal and noise sub spaces and
hence estimate the number of frequencies or  should i strictly go by
the arder estimation criteria like "AIC" or "FPE"?

Thanks,
-Nithin

 
 
 

frequency estimation

Post by Rune Alln » Mon, 02 Jun 2003 21:45:40



> Hi Rune,
> The power spectrum by Burg' method (with AR order 18) indicated a peak
> just adjacent to the two main peaks. Can i take this as the reference
> in separating the eigen values into signal and noise sub spaces and
> hence estimate the number of frequencies or  should i strictly go by
> the arder estimation criteria like "AIC" or "FPE"?

> Thanks,
> -Nithin

You have 40 samples and think you have two or three spectrum lines,
right? Using high-order AR, MUSIC, ESPRIT, FBLP or what have you,
is dangerous in that these methods somehow want to use all their available
degrees of freedom to describe signal components. This is known as
"spurious roots", "spurious components" etc. In other words, these
are artefacts that are introduced by the analysis methods, but can be
interpreted by the casual/inexperienced/unaware user as actual features
present in the signal. You mentioned before that you had problems with
the 23x23 autocovariance equation, now you mention 18x18 systems.
I think some of your confucion/problems are due to spurious effects
in the data.

From my own experience, I would not go above order 8 (max 12) for
your data. Of course, I am assuming now that the "sum of sinusoids"
model is appropriate for your data, that the statistics is stationary
and that there is a reasonable signal-to-noise ratio, say, SNR > 20 dB.

I would suggest you choose one method for analysis, work through the
inner workings of this method, and apply an order estimator of your
choise to that method. Most methods have their strengths and weaknesses,
and it is far easier to discuss and evaluate results in the context
of one particular method, than by comparing methods.

Rune

 
 
 

1. Barry Quinn's / Ted Hannan's Frequency Estimation and Tracking book

It appears it's not available yet, but they state a March release date.

From the CUP web-site:

http://uk.cambridge.org/mathematics/catalogue/0521804469/default.htm

the table of contents:

Chapter Contents

Preface;
1. Introduction;
2. Statistical and probabilistic methods;
3. The estimation of a fixed frequency;
4. Techniques derived from ARMA modelling;
5. Estimation using Fourier coefficients;
6. Tracking frequency in low SNR conditions;
7. Other estimation techniques;
References;
Appendix.
Matlab programs;
Author index;
Subject index.

Ciao,

Peter K.

--
Peter J. Kootsookos          Wb: www.clubi.ie/PeterK
"Here comes the future and you can't run from it,
If you've got a blacklist I want to be on it"
- 'Waiting for the great leap forwards', Billy Bragg

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