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Another current post ("Hanning or Flat top Window"), prompts me to ask a question that has been dogging me for awhile. Anybody that wants to help me will have to climb in my complicated head to understand me for a moment.

Pete and others, in the cited post, offered insightful examples describing the relative amplitude accuracies of the two windows. Pete's powerpoint showed, graphically, that the indicated amplitude of a single frequency peak may be under-reported if it is not located in the center of a Hanning Window bin.

Our vibration databox calculates overall Peak Velocity in a "round about" way. I believe many of the condition monitoring vendors use a similar method. It is not picked off the time domain... but instead calculated from the spectra. The amplitude of each individual bin is converted from RMS to peak by simply multiplying by 1.414. The calculated "overall" peak velocity is determined by a "square root of the sum of the squares" of the resultant amplitudes of all the bins within the spectra.

So,finally...my question. We agree that each individual frequency peak may under-reported by a Hanning Window if it is not bin-centered. Assuming, however, that the overall amplitude is calculated in the manner I described... will there be enough leakage to adjacent bins such that this calculated overall amplitude will really be pretty close to accurate?

We collect data that must comply with the ASME Boilercode. We considered this question as we asked ourselves whether we should change our digital vibe boxes from Hanning to Flattop to promote amplitude accuracy for these surveys. We are supposed to be able to collect data within +- 5%. I, conveniently, asked the people calibrating our vibe boxes to insert calibration signals to the center of our Hanning bins. I feel cheap and dirty doing this. I don't think anybody notices by deviousness.

Thoughts?

Thanks in advance...

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Short answer: the windows are taken into account when calculating the digital overall.

Previous discussion here:
http://maintenanceforums.com/e...921059853#3921059853

The spreadsheet shows example which recreates the Entek overall based on the bin info. The digital overall is not the square root of sum of squares of the bins. There is a scaling factor sqrt(2/3) for Hanning (varies by window).

The individual peaks already incorporate an amplitude correction factor which brings the magnitude ratio of a given window to 1.0 if properly centered in the bin.... slightly less if not centered, but in either case the leakage is not loss of energy ... it just spreads into adjacent bins. The energy correction factor accounts for all the energy from a given windowed peak spread into all bins.

So to display the spectra with proper magnitude (or just a little bit low) after windowing, we have to use the ACF. It's just a scaling factor that we wouldn't even need if we wrote the window function in a scaled form that gave 1.0 amplification for peaks at the center of the bin.

To compute the overall, we have to remove that ACF and apply the ECF.

So to convert exported spectral bin data into an overall you multiply by ECF/ACF of the associated window.

For Hanning it was done in that thread linked above. Also in the recent thread I provided derivation of the ACF and ECF directly from the window function. If you look at the ECF calculation you'll see it is an integration... summing up all the leakage from a single peak from it's own bin, adjacent bins, the next ones out... all the way accross the whole spectrum so none is lost.

The bottom line imo: the box considers the window before it calculates the digital overall, and it should apply the right factors to give you the right overall regardless of which window it's using.

Of course I'm sure you know any digital overall has a problem that it does not include energy above Fmax. Analogue overall is needed for that.

Sorry for so much rambling. By George, I think I'm done now.
RM
Wow...

Thanks, Pete, for your patience and careful instruction. I had been familiar with the concept of an "Amplitude Correction Factor" necessary for accurately displaying spectral peaks after windowing... but not an "Energy Correction Factor". It makes sense that its use is necessary to account for leakage to properly calculate overall amplitudes. There was a link from the previous post you cited that no longer works. I won't brag that I understand the entire discussion from that post, but I think I would have a better chance if I could see that link. Could I trouble you to try to post that link again? It was the 370 page link.

Despite your patience, I believe one last element of my question remains. I believe the discussion, to this point, has explained how the two correction factors preserve the accuracy of bin-centered signals. The Hannning window, by my reading, has an inaccuracy of about 15% when observing frequencies that lie exactly between bin-centers. When I send my vibe boxes off to the cal-guys with the NIST equipment, I specify that the Boiler Code we are supposed to comply with cites a tolerance of +- 5%. So, now... that 15% inaccuracy in the Hanning winddow doesn't seem insignificant! My intuition is that although a signal that is off bin-center may have a 15% inaccuracy in the amplitude of a particular bin peak, the resultant leakage distributed to additional bins will compensate for the 15% error once the overall amplitude is calculated. I'm not sure, however, that my intuition is truth? I tried real hard to look for this answer in the posts you cited... but am not sure it was there?

I should just quit bothering you guys and wait to get back to the plant and test the issue with our shaker table. Still curious, however, if anybody out there could corroborate what I might find with my shaker table?

Thanks in advance...
RM
quote:

My intuition is that although a signal that is off bin-center may have a 15% inaccuracy in the amplitude of a particular bin peak, the resultant leakage distributed to additional bins will compensate for the 15% error once the overall amplitude is calculated. I'm not sure, however, that my intuition is truth?

That is correct.... and the window correction factors used by your box help make it come out close to right.

When you take fourier transform of window times the rectangular function (to give finite window), you get a response curve.

The A.C.F. is what you have to multiply that response curve to get 1.0 where it crosses 0 (to give correct amplitude for a sinusoid centered in the bin... still can't get correct peak amplitude for sinusoid not centered).

The E.C.F. is what you have to multiply that response curve to get an area of 1.0 underneath the entire curve. Hmmm. The response curve is sampled at intervals 1.0 to give the leakage in adjacent bins.... I imagine that result would depend on centering within the bin also. I have to think about that.

I'll look for a suitable link.
RM
OK, I will go back to my original answer.

When evlauating FFT of pure sinusoid whose period is much less than the sample record duration, with window correction factors properly applied:
1 – there is error in spectral peak magnitude due to centering within the bin. If sinusoid is perfectly centered within the bin, then there is no error.
2 – there is NO error in computing the digital overall (of course it does not include energy above Fmax). Changing position of a given magnitude sinusoid with respect to bin does not change computed digital overall.

At the end of my last post I was wavering on the item #2, but I now believe it is true. I can't exactly prove it in words but I proved it to myself numerically.

My numerical "proof" is offered starting on page 24 attached by evaluating the overall based on adding up the energy (SRSS) from the bin which contains the peak and the 5 bins on each side of it. Note since we are using SRSS which represents (sqrt of) energy total, these 11 bins contain all the important energy for practical purposes (a bin whose magnitude is smaller by a factor of 10 in amplitude is smaller by a factor of 100 in energy). There are 12 sets of data. The first 6 sets of data are Hanning evaluated with sinusoid at bin center, then with the sinusoid 0.1 bins to the right of bin center, then 0.2, 0.3, 0.4, 0.5. The next 6 sets of data are same thing with Flattop. Each set of output data (blue) is a running total of SRSS as we add by SRSS the contibution from each bin starting from left to right. In all cases by the time we get to the right (all 11 bins added), the total is very close to 1.0.

I guess a more "straightforward" way to numerically prove the same thing would be to just use an example sinusoid, apply the window, compute FFT, compute overall including ECF, then repeat with frequency at different position in bin. That's a little more work but you're welcome to try it.

btw I am not intending to cover all the errors in the world (for example sensor error). Just talking about errors due to FFT and windowing as relates to spectral peak amplitude and digital overall for a pure sinusoidal signal.

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RM
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I've had discussions, with co-workers, of the value of people who contribute to this board. I've theorized that Electricpete and Arne Lindholm are not individual people... but are each, instead, large research organizations. Yeah... each takes up a full floor of a large office building. They have stacks and stacks of research books, this huge mainframe upstairs, rows and rows of doctorate level researchers, with interns and clerical people running around crazy, waving documents and yelling stuff at one another like "hey, have we gotten back to Machine Survey yet?... he's waiting for an answer."

Thanks very much, Pete. You answered my question... and verified by the attached math proofs. Thanks for the B&K reference. I have some reading to do now. I'm still gonna practice the notion with a shaker table and function generator when I return to work. Have a couple weeks off with the family now. I'll let you know when I do.

Thanks again...
RM
Thanks... that is certainly an entertaining compliment! But I can't let myself be put in the same zip-code as Arne. I do spend what some (my wife!) might consider a lot of time on the forum...because asking/answering helps me learn and I enjoy it. There is a lot to learn here from a lot of directions, and I certainly appreciate your suggestions in the past.

Enjoy your time off. Hope it comes with no phone calls from work. Will be interested to know how your experiment comes out.
RM
D. S.,

quote:
Our vibration databox calculates overall Peak Velocity in a "round about" way. I believe many of the condition monitoring vendors use a similar method. It is not picked off the time domain... but instead calculated from the spectra. The amplitude of each individual bin is converted from RMS to peak by simply multiplying by 1.414. The calculated "overall" peak velocity is determined by a "square root of the sum of the squares" of the resultant amplitudes of all the bins within the spectra.


I would separate instrument accuracy from measurement accuracy. There is a large measurement inaccuracy when measuring a high Crest Factor signal by the above method, since the True Peak is much larger than the "Sine Equivalent" Peak Velocity (1.414 x RMS).

Walt
RM
Ron,
From your sales experience, do you have an opinion whether most of the major vibe box vendors calculate the peak-overall from the spectra using square root sum squares w/ correction factors (as Pete explained above)... or do some pick the overall directly off the time domain with a peak detector? A friend who uses a different vibe equipment vendor than us said he is able to choose. We don't have a choice... our box just calculates from the spectra. If we want "true-peak", we must just observe it from the time domain display.
RM

I add here a simple Windowing Demo from RITEC.

In this simulator, the user can generate single or dual frequency signals, and apply several different types of windows on them to view the effects of non-bin-centered data, whereby leakage of energy to adjacent bins occurs; a result of processing finite-duration records.

There are several types of windows available in this simulator including: Hann (Hanning / Raised Cosine), Bartlett, 4-term Blackman-Harris, Flat-Top, and Hamming windows. The spectral content is amplitude corrected with the appropriate window Amplitude Correction Factor.

https://www.ritec-eg.com/Libra...se-Factor-basic.html

RM

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