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From: Dean on 4 Aug 2010 03:20
Hi guys,

I am studying classical signal processing techniques, such as Periodogram-based method.

I read from literature that Periodogram is asymptotically unbiased and its variance converge to square of its power spectrum density. In another word, its variance do not decrease as N->infinity.

According Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses that the variance cause erratic behaviour in using Periodogram. Therefore people went on to develop other smoothing techniques, such as Welch or Blackman Tukey methods. But I am reluctant to use because I have to lose resolution to achieve lower variance.

QUESTION:
As long as I can get a good estimation why should I care about variance? Is the variance of the frequency estimation or the power/amplitude drive people to use pwelch method in Matlab?

AN EXAMPLE:
In my study, I want to estimate two close frequency f1 and f2 with added Gaussian white noise (20 dB). For a modest N, Periodogram can resolve them with slight bias. That is about 3% deviation from true frequency.

Since Periodogram is asymptotically unbiased, I can get very good estimation by increasing N. Why should I then worry about variance?

Cheers,

Dean
From: Greg Heath on 5 Aug 2010 10:53
On Aug 4, 3:20 am, "Dean " <jiangwei0...(a)gmail.com> wrote:
> Hi guys,
>
>     I am studying classical signal processing techniques, such as Periodogram-based method.
>
> I read from literature that Periodogram is asymptotically unbiased and its variance converge to square of its power spectrum density. In another word, its variance do not decrease as N->infinity.
>
> According Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses that the variance cause erratic behaviour in using Periodogram. Therefore people went on to develop other smoothing techniques, such as Welch or Blackman Tukey methods. But I am reluctant to use because I have to lose resolution to achieve lower variance.
>
> QUESTION:
> As long as I can get a good estimation why should I care about variance? Is the variance of the frequency estimation or the power/amplitude drive people to use pwelch method in Matlab?
>
> AN EXAMPLE:
> In my study, I want to estimate two close frequency f1 and f2 with added Gaussian white noise (20 dB). For a modest N, Periodogram can resolve them with slight bias. That is about 3% deviation from true frequency.
>
> Since Periodogram is asymptotically unbiased, I can get very good estimation by increasing N. Why should I then worry about variance?
>
> Cheers,
>
> Dean

I don't know. That is why I have crossposted to sci.stat.*

Hope this helps.

Greg
From: Greg Heath on 5 Aug 2010 18:02
Newsgroups: sci.stat.math, sci.stat.edu, sci.stat.consult
Followup-To: sci.stat.math, sci.stat.edu, sci.stat.consult
From: Herman Rubin <hru...(a)skew.stat.purdue.edu>
Date: Thu, 5 Aug 2010 17:00:30 +0000 (UTC)
Local: Thurs, Aug 5 2010 1:00 pm
Subject: Re: asymptotically unbiased and variance confusion
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On 2010-08-05, Greg Heath <he...(a)alumni.brown.edu> wrote:
> On Aug 4, 3:20 am, "Dean " <jiangwei0...(a)gmail.com> wrote:
>> Hi guys,
>> I am studying classical signal processing techniques, such as
Periodogram-based method.
>> I read from literature that Periodogram is asymptotically unbiased
and its variance converge to square of its power spectrum density. In
another word, its variance do not decrease as N->infinity.
>> According Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses that the variance cause erratic behaviour in using Periodogram. Therefore people went on to develop other smoothing techniques, such as Welch or Blackman Tukey methods. But I am reluctant to use because I have to lose resolution to achieve lower variance.
>> QUESTION: >> As long as I can get a good estimation why should I
care about variance? Is the variance of the frequency estimation or
>>the power/amplitude drive people to use pwelch method in Matlab?


You do not get good estimation unless the variance, or some
other essentially equivalent measure of accuracy.


>> AN EXAMPLE:
>> In my study, I want to estimate two close frequency f1 and f2 with
added Gaussian white noise (20 dB). For a modest N, Periodogram can
resolve them with slight bias. That is about 3% deviation from true
>>frequency.


The noise in the periodogram without a VERY large sample
size may overwhelm the signal. This is a case of the
tail wagging the dog.

If you have precise frequencies, and this is all, there
are other ways to do it than using the periodogram. But
if there is more, or the frequencies are not precise, this
might not work. I suggest you consult a good mathematical
statistician in person.


>> Since Periodogram is asymptotically unbiased, I can get
>>very good estimation by increasing N. Why should I then
>>worry about variance?


See the above. The approach of unbiased estimates to the
true value is not as fast as you seem to think.


>> Cheers,
>> Dean
> I don't know. That is why I have crossposted to sci.stat.*
> Hope this helps.
> Greg


--
This address is for information only. I do not claim that these
views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...(a)stat.purdue.edu Phone: (765)494-6054 FAX:
(765)494-0558
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