Summing spectral lines to give an overall level 3

[BrianD]

I would like some advise about calculating an overall level by summing the amplitude of the individuals lines that make up a frequency spectrum. In my case I am measuring acoustic data and the frequency range I am interested in is 0 to 5kHz. When I summed the individual lines the overall level was slightly higher than I expected. The 'error' would appear to be a linked to the time windowing function that I am using (which is 'Hanning' in the majority of my tests).

Are there correction values that can be applied that take into account the effective bandwidth of the 'filters' or lines in the frequency spectrum. I would be grateful for any additional information on this subject / correction factors for the different time windowing functions (in particular Hanning).

Thanks

 

[Hatch]

The FFT window selection has a direct effect on the accuracty and resolution of a signal.

A uniform window has an Amplitude Uncertainity of 56.5%
A Hanning window has an Amplitude Uncertainity of 18.8%
A flat to window has an Amplitude Uncertainity of 1%

Therefore if you want amplitude accuracy you should use a flat top window.

Next, the choice of window has an significant effect on the signal if it has transients. A transient could be missed completely if the wrong window is chosen.

What I am trying to say is that a applying a correction factor would be complex and difficult and it would be easier to remeasure the data.

Hope that helps.

 

[electricpete]

Perhaps it is obvious/assumed, but since you said "summing", I feel compelled to point out that the basic operation for combining spectral components is the square-root of the sum of the squares. Voverall=sqrt(sum{Vi^2}) where Vi are individual spectral components.

And of course another most accurate method is the "analog overall" computed directly from time samples. I think that formula is Voverall=sqrt((1/N)*sum{vi^2}) where vi are individual time samples and N is the number of samples

 

[ivymike]

My most closely related experience is with summing vibratory responses of a system to a series of harmonic excitations, to find total response of the system vs time. If this is similar to what you're doing, then my comment would be that you should make sure that you're phasing the components properly with respect to each other. In my experience, if you forget the relative phasing of the responses and directly sum the amplitudes, the overall response you calculate is way too high.

 

[machinedoc]

thanks for the information on the uniform, hanning , and flat window.This is the first time I have seen this published . thanks again

machinedoctor

= (sample only)

 

[steeno]

Thanks for an interesting thread !

To ElectricPete (glow, glow)

Yes - the way to do it is Root Mean Square -RMS.
There was an interesting thread pointing out that
Parsevals Relation shows that RMS in time and
frequency must end up with the same result.

Its a truly wonderful feeling to see that come true.
I used Matlab to do the computations for me.

Go to

http://www.DSPguide.com

and get the book
"The Scientist and Engineer's Guide to
Signal Processing" - Particularly the final page
of chapter 10, where Parsevals Relation is
explained. The "trick" explained here is to
remember that X[0]^2 and X[N/2]^2 (first and last
stave squared in the fft of length N) have to be
divided with 2 in order to compensate for the fact
that they only contribute each one half stave width.
This trick is easier to remember once you realise
that the number of fft staves is not N/2 but N/2+1.

In floating point precision the two RMS values come
out with 3-4-5 digits alike - a truly convincing result.

Steven W. Smith who wrote the book I recommend above
ends chapter 10 saying "While Parsevals Relation is
intersting from the physics it describes (conservation
of energy) it has few practical uses in DSP. That is
a big pity, I find. Doing this simple test makes it
easy to "get on solid ground" with results from an fft.

Regards, steeno