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Windowed FFT power and amplitude spectrum

Started by Cagri July 30, 2005
Here is another fascinating question on FFT spectrum!!

I am trying to find how to incorporate the normalisation factors when
windowing is applied to the time series before FFT is applied.

When windowing (eg Hanning, flat top etc.) is applied to the time series,
the amplitude of the time series obviously changes. In this case, how do
we represent the power and amplitude spectrum in the frequency domain
after applying the FFT? 

My guess is P(i)=|x(i)|^2/N power expression will probably need to be
multiplied by a scaling factor but I am not sure how this scaling factor
will be calculated and how exactly it will be applied. This is also valid
for the amplitude expression; A(i)=|x(i)|/N

I also think that the scaling factors will depend on the type of windowing
that is used. How are those scaling factors calculated? Is there a
reference that I can have a look at to shed some light on this issue?

Thank you to all who provide feedback.

Regards,
Cagri


		
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"Cagri" <dr_cagri_tanriover@yahoo.co.uk> wrote in message
news:VI2dnR3EpomGWXbfRVn-1Q@giganews.com...
> Here is another fascinating question on FFT spectrum!! > > I am trying to find how to incorporate the normalisation factors when > windowing is applied to the time series before FFT is applied. > > When windowing (eg Hanning, flat top etc.) is applied to the time series, > the amplitude of the time series obviously changes. In this case, how do > we represent the power and amplitude spectrum in the frequency domain > after applying the FFT? > > My guess is P(i)=|x(i)|^2/N power expression will probably need to be > multiplied by a scaling factor
Not to my knowledge. You leave it the same. Shytot
Hi Cagri!

I think, what you are looking for are the power gain factors for the 
different signal windows. You find a table with the values of some important 
windows on this page:
http://www.bores.com/courses/advanced/windows/10_end.htm ( Look for the 
"Coherent Power Gain" column! )

I hope this helps.

Kind regards,
Karin

"Cagri" <dr_cagri_tanriover@yahoo.co.uk> schrieb im Newsbeitrag 
news:VI2dnR3EpomGWXbfRVn-1Q@giganews.com...
> Here is another fascinating question on FFT spectrum!! > > I am trying to find how to incorporate the normalisation factors when > windowing is applied to the time series before FFT is applied. > > When windowing (eg Hanning, flat top etc.) is applied to the time series, > the amplitude of the time series obviously changes. In this case, how do > we represent the power and amplitude spectrum in the frequency domain > after applying the FFT? > > My guess is P(i)=|x(i)|^2/N power expression will probably need to be > multiplied by a scaling factor but I am not sure how this scaling factor > will be calculated and how exactly it will be applied. This is also valid > for the amplitude expression; A(i)=|x(i)|/N > > I also think that the scaling factors will depend on the type of windowing > that is used. How are those scaling factors calculated? Is there a > reference that I can have a look at to shed some light on this issue? > > Thank you to all who provide feedback. > > Regards, > Cagri > > > > This message was sent using the Comp.DSP web interface on > www.DSPRelated.com
PS: You can find a nice explenation about coherent power gain on this page:
http://www.bores.com/courses/advanced/windows/10_cpg.htm

Karin

"Cagri" <dr_cagri_tanriover@yahoo.co.uk> schrieb im Newsbeitrag 
news:VI2dnR3EpomGWXbfRVn-1Q@giganews.com...
> Here is another fascinating question on FFT spectrum!! > > I am trying to find how to incorporate the normalisation factors when > windowing is applied to the time series before FFT is applied. > > When windowing (eg Hanning, flat top etc.) is applied to the time series, > the amplitude of the time series obviously changes. In this case, how do > we represent the power and amplitude spectrum in the frequency domain > after applying the FFT? > > My guess is P(i)=|x(i)|^2/N power expression will probably need to be > multiplied by a scaling factor but I am not sure how this scaling factor > will be calculated and how exactly it will be applied. This is also valid > for the amplitude expression; A(i)=|x(i)|/N > > I also think that the scaling factors will depend on the type of windowing > that is used. How are those scaling factors calculated? Is there a > reference that I can have a look at to shed some light on this issue? > > Thank you to all who provide feedback. > > Regards, > Cagri > > > > This message was sent using the Comp.DSP web interface on > www.DSPRelated.com
"Cagri" <dr_cagri_tanriover@yahoo.co.uk> wrote in message 
news:VI2dnR3EpomGWXbfRVn-1Q@giganews.com...
> When windowing (eg Hanning, flat top etc.) is applied to the time series, > the amplitude of the time series obviously changes. In this case, how do > we represent the power and amplitude spectrum in the frequency domain > after applying the FFT? > > My guess is P(i)=|x(i)|^2/N power expression will probably need to be > multiplied by a scaling factor but I am not sure how this scaling factor > will be calculated and how exactly it will be applied. This is also valid > for the amplitude expression; A(i)=|x(i)|/N > > I also think that the scaling factors will depend on the type of windowing > that is used. How are those scaling factors calculated? Is there a > reference that I can have a look at to shed some light on this issue?
Others have already pointed out that the scale factor does depend on which window you use. In addition, the scale factor depends on what you are trying to measure. If you're looking at a "line" spectrum and want to preserve the amplitude of the "lines" as much as possible, that gives you one scale factor. But if you're looking at noise levels and want to preserve the noise power, that gives you a different scale factor. And if you're looking at the time-domain signal, maybe you don't want to scale at all, so that the peak in the middle of the time-domain is unchanged. Often, the scale factor doesn't matter at all because what you're looking at is relative amplitudes of different parts of the spectrum. -- Eric Backus R&D Design Engineer Agilent Technologies, Inc.
Cagri wrote:
> Here is another fascinating question on FFT spectrum!! > > I am trying to find how to incorporate the normalisation factors when > windowing is applied to the time series before FFT is applied. > > When windowing (eg Hanning, flat top etc.) is applied to the time series, > the amplitude of the time series obviously changes. In this case, how do > we represent the power and amplitude spectrum in the frequency domain > after applying the FFT? > > My guess is P(i)=|x(i)|^2/N power expression will probably need to be > multiplied by a scaling factor but I am not sure how this scaling factor > will be calculated and how exactly it will be applied. This is also valid > for the amplitude expression; A(i)=|x(i)|/N > > I also think that the scaling factors will depend on the type of windowing > that is used. How are those scaling factors calculated? Is there a > reference that I can have a look at to shed some light on this issue?
I am not aware that people use such scaling factors with windows. The reason has to do with why one uses windows in the first place. The periodogram (squared magnitude of spectrogram) is the "simplest" estimator for the power spectrum. Unfortunately, the periodogram has a very large variance. The variance is on the order Var [P(f)] = P(f)^2 where P(f) is the periodogram coefficient at frequency f. So, with the periodogram one can't say very much mor than "we have measured something". Most people who do such analysis would know that beforehand... To get a slightly more useful estimate -- useful in the sense that one can tell that "in this band there is some signal present while in that band there is only noise" -- one came up with the idea of averaging nearby periodogram coefficients, which amounts to applying a time-domain weight function. The reason for using windows is that one wants to see the "structure" of the spectrum. Where is there only noise? Where is there signal and noise? What do the spectrum shapes look like? In addition, there are questions like confidence intervals and so on, so it sin't crucial to get the numbers with six significant digits. Since the numbers are of less concern and there are many other "fuzzy" aspects of nonparametric spectrum estimation, there is no need to apply these scaling factors. Rune