# Windowed FFT power and amplitude spectrum

Started by 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

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...
> 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

```