Does anybody have an easy way to convert spectral data (in IPS) to an overall value (also in IPS)? The data were imported into an Excel spreadsheet (unfortunately, not with me right now). I also imported the timewave values (in G's) and was able to Root-Mean-Square those values to get an RMS but it was in G's. Willing to try anything, I RMS'ed the spectral data and came up with a value but it was lower than the peak value so I know it was incorrect. I'm guessing that the overall value of the spectrum should be the "area under the curve" but when I tried this, I came up with a very high number compared to the peak values. So, am I confused or did I maybe type an incorrect formula into Excel?
(The actual data is in my office and I can't access it until Monday.)

Thanks,
Steve

If the spectral values were displayed in rms velocity (for example mm/sec rms), then the rms overall is the square root of the sum of the squares of the spectral values.

Simple scale factors apply if the spectral peaks are not rms or you want the overall in something other than rms.

Note the resulting overall is a bandlimited overall... of course does not reflect any content above Fmax.

You will also need a scale factor related to the window function used in addition to squart root of sum of squares of the spectral values.

Good point. I forgot about that.

Steve,
You also need to know how the analyzer came up with the overall value. For example, CSI has two options, Digital vs analog. Analog takes the full fmax possible of the analyzer & senor and calculates it that way. Digital uses the spectrum Fmax to calculate the Overall value.

Bently Nevada systems also have similar differences in that Direct values can be different from spectral content.

Here is an example calc I had done.
The E-monitor program displayed 0.116181 ips pk/0 as the spectrum (bandlimited) overall.

I exported the spectrum to excel. With that data I calculted the spectrum/bandlimited overall by using square root of sum of squares (SRSS) times a window factor sqrt(1/1.5) for Hanning window.

In this case since the spectral data was presented as pk/0 and the overall was calculated as pk/0, there is no other rms-> peak type conversion (like sqrt2) required.

It recreates the same result down to the 6th decimal place (6 decimal places is all that E-monitor displays).

I can't remember where I got the window factor sqrt(1/1.5) for Hanning window.

This message has been edited. Last edited by: electricpete,

digital_overall3.xls (110 Kb, 35 downloads)

I think that the overall correction factor sqrt(2/3)=sqrt(1/1.5) is necessary as a result of an earlier correction factor that was made to make the spectral peaks the correct height, considering leakage effects of the associated window.

I think it might be indirectly related to the 1.5 bin peak width shown on slide 2 of the attachment, but I'm not sure.

If anyone has a good resource or derivation for that window correction factor I'd be interested to hear it. (I think I knew it at one time, but I sure can't figure it out now.)

Hanning.ppt (344 Kb, 28 downloads)

I could be wrong, but correction factor =1.5 for Hanning window has to do rather with actual spectral resolution as it becomes worse due to the leakage.

Hanning window, as many other windows, has amplitude uncertainty (1.4 Db) for every peak, so the digital overall computed as SRRS will also be affected.

David

OK, I think I have got it figured out (sort of). See this link:
http://www.sovitech.co.kr/technical/manual/Theory/Complete%20theory.pdf

Go to page 27/370 (marked as page 17).

There are two types of correction factors, "amplitude correction factors" and "energy correction factors".

For the spectrum, we want the individual amplitudes to be correct and we use the amplitude correction factor.

For the overall, we combine the amplitudes using a SRSS algorithm and we want the totaled "energy" of that combination to be correct, and we need the energy correction factor.

To convert from the spectrum to the overall, you take out (divide by) the amplitude correction factor that Emonitor has applied to the spectrum, and put in (multiply by) the energy correction factor.

energy correction factor / amplitude correction factor = 2.0 / 1.63 = 0.815
This is very close to the mysterious sqrt(1/1.5)=0.816

I am pretty sure that sqrt(1/1.5) is the exact value for use with Emonitor spectral data processed with Hanning window (since it produces the correct value to 6 decimal places). I think the 2.0 and/or 1.63 are approximate values.

David - you are absolutely correct that the 1.5 is directly associated with spectral leakage and how it alters the effective binwidth (as stated in the column heading). I can't give the full derivation, but I am pretty sure that it also leads to the correction factor we are seeking because
1 - it gives the right answer to 6 decimal places when used with data exported from Emonitor AND
2 - it is not just a coincidence that we arrive at similar results using the 1/sqrt(noise band width) as when using energy correction factor / amplitude correction factor... the same relationship holds for the other windows also. For example:

For Hanning, amplitude correction factor = 2, energy correciton factor = 1.63 and noise band width = 1.5 (bins)
1.63 / 2.0 = 0.815 ~ 1/sqrt(1.5)

For Hamming, amplitude correction factor = 1.85, energy correciton factor = 1.59 and noise band width = 1.36(bins)
1.59 / 1.85 = 0.859 ~ 1/sqrt(1.36)

For Blackman, amplitude correction factor = 2.8, energy correciton factor = 1.97 and noise band width = 2 (bins)
1.97 / 2.8 = 0.704 ~ 1/sqrt(2)

(the energy and amplitude correction factors came from page 27 of 370, the noise band width came from page 22 of 370)

This message has been edited. Last edited by: electricpete,

As I recall from the fog of distant memories. One applies a window by multiplying the time domain signal by a window function. This is equivalent to convolution in the frequency domain with the FT of the window function.

The transform of the rectangular window (no window, but limited non-zero terms) is a Dirichlet function. Hann to make this have better properties (lower side lobes compared to main lobe in the frequency domain) superimposed (added) two shifted versions one frequency higher and one lower as main lobe + 1/2 plus shift + 1/2 minus shift (with scaling there could be a factor of 2).

Using another property of FT’s frequency shifting equates to multiplying by complex exponentials in the time domain; one can easily determine the inverse transform this to get Hann’s window function.

If you have a frequency component and multiply by the Hann window the convolution in the frequency domain results in two copies at 1/2 amplitude at the adjacent calculated frequencies. In terms of rms, this is sqrt(1^2 +0.5^2 +0.5^2)= sqrt(1.5).

Now that that is clear as mud, one can work at filling in the specifics if one likes.

Thanks Bill. I wouldn't have thought of that, but it makes pretty good sense. . To say it in my own words rolling in the concept of amplitude correction factor and energy correction factor:

The window is 0.5*(1+cos(x))
The fourier transform of that window has a peak at f=0 magnitude 0.5 and two sidebands at f=+/-x magnitude 0.25 and an rms of sqrt(1.5)/2 as you computed.

If we input a pure sinusoid of magnitude 1 at frequency f0, we get a sinusoid of magnitude 0.5 with two sidebands magnitude 0.25

Amplitude correction factor (ACF):
To get the correct peak value, we would need to multiply the FFT by 2 to get the peak to 1 which is the magnitude of the peak that we input. Therefore ACF = 2.

Energy correction factor (ECF):
To get the correct rms value, we would need to multiply the FFT by 2/sqrt(3/2).
Then we have a center peak at 1/sqrt(3/2) with sidebands at 0.5/sqrt(3/2). RMS is sqrt(2/3 + 1/6 + 1/6) = 1 which is the same as the sinusoid that we input. Therefore ECF = 2/sqrt(3/2).

We want to get from the spectrum (FFT * ACF) to the overall (based on FFT * ECF).

The factor is ECF / ACF ={ 2/sqrt(3/2) } / (2) = 1 / sqrt(3/2)

Here is a formula from Technical Associates:

DO - Digital Overall
Nbw - noise bandwidth factor(=1.5 for Hanning)
n - number if lines
An - amplitude in line n

DO = sqrt [(A1 ^2 + A2 ^2 + ... +An ^2) / Nbw ]


I do not know how they specifically arrived to a correction factor equal to sqrt(1/Nbw), but noise due to the leakeage into the adjacent 3-4 bins (lines) could definetely be calculated for the worse case scenario.

David

That confirms the relationship identified above in my post 13 January 2008 09:54 PM. I don't know why this factor identified as "noise bandwidth" has the relationship to ACF and ECF that it does.

Take your rectangular window in the frequency domain, Dirichlet function and add to it +/- 1 frequency shifted copies to help the side lobes.

In the time domain this takes the original rectangular window (times 1/2 - depending upon initial scaling)and adds an 0.25 (or 0.5 depending upon scaling) e^(it) + 0.25 e^(-it) ==> the additions of the shifted copies in the time domain.

Add the complex exponentials and you get .5 (1+ cos(t)) as per ElPete.

quote:

Amplitude correction factor (ACF):
To get the correct peak value, we would need to multiply the FFT by 2 to get the peak to 1 which is the magnitude of the peak that we input. Therefore ACF = 2.

I think this depends upon how one views the DFT. For a frequency peak f, there is another peak corresponding to -f for a real valued transform. Together, they take care of the 2 factor.

It looks like this thread is designed to drive beginners to start using the new forum section - maybe others, too

quote:

It looks like this thread is designed to drive beginners to start using the new forum section - maybe others, too

Click"New Beginners Site"

Wow! A big thanks to all who posted. It took me a while to read all the information but I was able to get the correct answers. The SQRT(1/1.5) factor made all the difference.

Steve