Garnering information in the frequency domain 1

Hello,

I'm doing vibration analysis at a synchrotron and I would like to know what if any information I can garner from amplitudes in the frequency domain.

I understand that the units for a integral fourier transform X{f(t)} are in [Amplitude*Time] I would think this implies [Amplitude/Freq] is that right?

I'm doing the analysis using a MathCad FFT. Are the FFT values in Amplitude/Hz or are they simply amplitudes. For example if I take a finite signal like a sine wave and do an FFT on it I end up with a two finite peaks in the frequency domain. I would think the amplitude of the frequency peaks depends on the data length because the Dirac delta function is convolved with the sinc function.

Long story short what if anything can I garner from peaks in the freq. domain or is it simply the area under the peaks that matters.

Bert

 

[peterschmidtpe]

This depends on what the problem is. If you are trying to eliminate vibration, these peaks indicate where your effort should be concentrated. The amplitude of the peaks (either relative or normalized) can give further information depending on how the data are reduced.

 

[080175]

hi ,

Yes according to parseval theorem the unit are either

amplitude*second or amplitude*(1/ hz)

bye

 

[electricpete]

I agree with the way you are looking at it... the peak does not tell the whole story.

Start with a pure sinusoid in the time domain.

Continuous Time Fourier transform of that is a pair of impulses (deltas).

Multiply by a rectangular window equal to length of sample. The delta gets "smeared" to a shorter and wider form.... now a narrow sync centered on the frequency of interest.

Sample this at > Nyquist frequency. We get a Discrete Time Fourier Transform "DTFT" which is identical to the CTFT (except that it is periodic in 0..2PI).

Now compute the Discrete Fourier Transform DFT (same result as FFT ). To me DFT this represents "samples" of the DTFT spectrum spaced at the frequency bin width ("bin width" = spacing of samples in frequency = Fmax / # samples).

So there are two distortions that the original spike has undergone: #1 it has been smeared by the windowing operation and #2 DFT/we have only equally-spaced (in frequency) samples of what started as a continuous DTFT spectrum.

In vibration they sometimes look at overalls magnitudes which are square-root-of-sum of squares of magnitudes of invididual frequency amplitudes. They also combine multiple frequency points in similar manner under a spectral band. In vibration terminology these amplitudes are invariably referred to in the same units as the time signal (displacement, inches per second, or g's). That approach seems reasonable since any statement of magnitude represents a SRSS-type integral/sum over some frequency band. In integrating over frequency, we retrieve the original time units.

When they refer to the magnitude of a single peak, they pick the single frequency bin with highest magnitude and call it the peak magnitude. That corresponds to integrating over frequency for one bin-width. Possibly some small errors in this approach due to factors discussed above... but note that if the adjacent bins which we neglected have smaller magnitudes, these will change the result very little when combined in a SRSS manner. In the special and infrequent case where peaks are for some reason several bin-width's wide near the top of the peak (remember a broad base doesn't really matter due to SRSS), an experienced analyst would do well to characterize this peak by the overall magnitude of a narrow spectral envelope which contains the peak... rather than just reporting the highest value of peak.

 

[electricpete]

I'm a little disappointed that there were no more comments. I'm relatively new to vib monitoring but
took some signal processing courses in college. Whenever I discuss this type stuff with the folks that have lots of vib experience but not much formal signal processing, it usually generates a lot of controversy. I'm not saying that I'm right and they're wrong... only that there is a lot of different ways to look at the meaning of the height of a peak.

Anyone else have any comments on the subject?

 

[WIM32]

Bert20,

Considering a simple sine with amplitude A and an FFT window that is n*T (with T is time of 1 sine), the following can be said:
When you see two finite peaks in the frequency domain, you have calculated the fourier-spectrum (two sided), and the peaks should have an ampitude of 0.5*A.
When calculating the amplitude spectrum, you should see 1 finite peak with amplitude A and the unit is considered the same as that of the sine (p.e. displacement, velocity, acceleration etc.). Each frequency component (except DC-component)is twice as large due to the summation of the symmetric frequencies.
best regards
Wim32

 

[GregLocock]

I would just add that the scaling of amplitude units in MathCAD's FFT routines is a little odd (at least up to v5) and that you should read the help file on that subject.

One gotcha is that if you plot the FFT vector MathcAD by default will plot the Real amplitudes of the vector and ignore the Imaginary component. I find it best to plot the magnitudes of the vectors as matter of course.

I do not understand why you are getting 2 peaks from the FFT of a sinusoid - are you applying Hanning windows?

Cheers

Greg Locock

 

[electricpete]

Greg - I think the two peaks that have been referred to represent components of the same sinusoid at positive and negative frequencies +/-k. They represent two complex-conjugate time series.... when added together we get the real (vs complex) time series. See discussion by wim32

wim32 - you make an interesting comment that the magnitude of the FFT coefficients of a sinusoid will be proportional to the magnitude of the sinusoid (factor of 1/2 as discussed above).

That brings to mind that there are at least two different ways of defining the dft (FFT) and inverse-dft functions... they differ in terms of where the 1/N shows up... but they both yield back the original time series when you apply the fft and inverse FFT. IF you select the definitions:
X[k]=sum(0..N-1,x[n]exp(-j*2Pi*k*n/N))
x[n]=(1/N)sum(0..N-1,X[k]exp(j*2Pi*k*n/N))
as given in "Discrete Time Signal Processing" 2nd ed by Oppenheim & Schaefer
THEN you would find that the magnitude of the DFT/FFT is not as directly tied to the magnitude of the time waveform sinusoid. There is a factor of 1/N which enters into the equation.... and you would conclude that wim32's statement is incorrect, a sinusoid of magnitude A time series sinusoid would not give fft coefficients of A/2, but NA/2.

In earlier publication "Signals and Systems" (1ST ED) by Oppenheim and Willsky, the factor of 1/N is moved from the 2nd equation (x[n]) to the first (X[k]). This gives the A/2 FFT magnitudes that wim32 predicted.... a more sensible and pleasing result.

It's worthwhile to repeat that either method is correct.... as long as you use the equations in pairs you can FFT and inverse-fft to retrieve the same result. It would be incorrect to mix and match the pairs of equations.

I believe that commercial vib programs (like E-monitor from Entek) use the more intuititive definition from Oppenheim and Willsky. Furthermore they combine the positive and negative coefficients into one magnitude. Also many choices for window setup are provided... commonly Hamming is used. They also permit several types of averaging to increased accuracy. Another interesting feature to me is the software's ability to predict the location of a spectral peak which lies in between FFT bin center points. For example if my FFT shows magnitude coefficients at 7185 and 7230 cpm with 7185 slightly higher than 7230. I put the cursor on 7185 and press "p" and it labels the peak frequency as 7201. In this particular case I know the actually frequency does correspond to 2*power frequency = 2*60hz=120hz=7200 cpm. If I plot the nearby points I would never be able to "connect the dots" smoothly to guess that point. I believe that maybe the program is using some phase information to assist with this calculation.

 

[steeno]

Now I have read this thread a couple of times.

I have an interesting task of measuring vibration
peaks generated by a wind mill. So for me it will
be of main interest to evaluate peaks in a freq
spectrum and say something about the height of the
peaks.

I just want to mention
Steven W Smith:
"The Scientists and Engineers guide to
Digital Signal Processing" second ed.

Visit the books webpage at

http://www.DSPguide.com

-I think the whole book is available from here.

I got my free copy through analog devices.

Specifically chapter 8: The Discrete Fourier Transform,
the last part: "Synthesis, Calculating the Inverse DFT"
is very instructive. But also "Parsevals relation"
(conclusion of chapter 10:Fourier Transform Properties)
is very good.

I would normally consider Oppenheim % Schaefer
(as mentioned and quoted by electricpete - I have
been greatly inspired by your replies to this thread )
THE book on the subject but This guy Smith takes it
all out of the books and into practical considerations.

Thanks again for an interesting thread to all.

Steeno