Frequency resolution and amplitude in FFT
Frequency resolution and amplitude in FFT
(OP)
Hi,
I am looking at improving the frequency resolution in FFT through zero padding. Attached is an excel sheet showing an attempt to study the frequency resolution with padding of zeros to data. Any comments ?
Another basic (and obviously could be silly and stupid) question: What does the amplitude of FFT spectrum indicate ? I asked many people, read many books, etc. but this doubt remained. All books just escape saying it is amplitude, they won't give units. For example if I have acceleration (m/sec2) vs time (sec) data and I take FFT, the units of amplitude is (m/sec2). Right ? For example, a time domain acceleration shows maximum acceleration of the order of 50 m/s2. FFT shows amplitude of the order of 1. I am puzzled with this. What is the physical meaning of this? I tried summing up magnitude of all the frequencies which obviously goes beyond 50 m/s2. So what is relation of time domain amplitude to FFT amplitude ?
Attaching a picture of this data and its FFT in another worksheet of attached excel workbook.
Many thanks
Geoff
I am looking at improving the frequency resolution in FFT through zero padding. Attached is an excel sheet showing an attempt to study the frequency resolution with padding of zeros to data. Any comments ?
Another basic (and obviously could be silly and stupid) question: What does the amplitude of FFT spectrum indicate ? I asked many people, read many books, etc. but this doubt remained. All books just escape saying it is amplitude, they won't give units. For example if I have acceleration (m/sec2) vs time (sec) data and I take FFT, the units of amplitude is (m/sec2). Right ? For example, a time domain acceleration shows maximum acceleration of the order of 50 m/s2. FFT shows amplitude of the order of 1. I am puzzled with this. What is the physical meaning of this? I tried summing up magnitude of all the frequencies which obviously goes beyond 50 m/s2. So what is relation of time domain amplitude to FFT amplitude ?
Attaching a picture of this data and its FFT in another worksheet of attached excel workbook.
Many thanks
Geoff





RE: Frequency resolution and amplitude in FFT
I would argue that your peak frequency is converging to the best estimate of the frequency possible from the set of time data that you started with. If the purpose is to get the most accurate estimate of frequency, using a longer actual time record is preferable to zero padding as Hacksaw metnioned.
Zero padding (pre-FFT) is roughly interchangeable to the variety of methods that can be done post-FFT to estimate the peak frequency by interpolation
There are magnitude correction that can be applied to undo any magnitude changes associated with zero padding. They have to coordinate with the choice of windowing used. The correction factor would be different if you put the zero's on one end than if you split them onto each end. I don't have any formula ready but it should be shown in most DSP textbooks. I think the generally preferred location to put the zero's would be split among both ends for a windowed signal.
I view the FFT magnitude as the square root of the energy per bandwidth.
To interpret, we add up the energy over the frequency band of interest. (equivalence of energy in frequency and time domains suggested by Parseval's theorem). For example if you see several non-zero bins in a clump that you suspect are associated with a single sinusoidal peak, than the magnitude of that peak would be square root of sum of the squares of the individual FFT bin magnitudes. It transforms to a single sinusoid in the time domain with that magnitude (under the assumption that these are all associated with a single sinusoid).
The small noise floor between peaks tends to complicate this view, but less so when you remember that SRSS tends to be dominated by bins with large magnitudes and the contribution of bins with smaller magnitude is double-smaller (smaller squared).
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
Thank you for detailed reply. Very useful.
Zero padding -
Yes. You are right. Padding applied after windowing doesn't affect which way we add zeros - left, right or both sides. Also the peak frequency remains same whether we pad before window or after. However the peak amplitude changes. I am attaching a graph here which shows variation of FFT peak amplitude with padding - applied before and after windowing. Though after windowing has more reduction in amplitude it seems to be following a smooth trend (to me looks like exponential!) which can be easily corrected as you said using a correction factor.
Amplitude of FFT -
First things first - now I understood that at least how the units remain same. If my time domain signal has m/s2 units, I can confidently put same units on FFT amplitude as it is equal to square root of sum of squares.
The concept of energy per bandwidth is more appealing. I will try to validate this SRSS concept on a known sinusoidal signal and will report the result here. But I also see this concept is very difficult or impossible to validate on a real time signal which can be represented as a summation of many sines and cosines.
Many thanks
Geoff
RE: Frequency resolution and amplitude in FFT
consider the simple signal 1,1,1,1
now zero pad it to give twice the frequency resolution
1,1,1,1,0,0,0,0
If you are trying to tell me that the FFT of a DC signal is identical to that of a step function then something is very wrong. And a quick check says it ain't. Even if you use two leading zeroes and two after it still ain't.
So not very surprisingly we discover that there is no free lunch. Yes, you will increase your apparent frequency resolution but in doing so you will be altering the displayed spectrum in ways that you may not be able to predict.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Frequency resolution and amplitude in FFT
TTFN
FAQ731-376: Eng-Tips.com Forum Policies
RE: Frequency resolution and amplitude in FFT
I stand by all the statements made here:
thread384-325307: Do resonant frequencies produce harmonics
and here
thread384-208992: "Interpolating" to estimate exact frequency of peak from FFT results
(particularly as summarized 9 Feb 08 19:02)
All my links disappeared when I left comcast. Attached is the spreadsheet that I posted 27 Feb 08 22:12
If there is a point you disagree with, quote it and lets discuss it.
Again the issue is not free lunch, but how much of your pb&j sandwich do you throw in the trash.
If you simply estimate your frequency simply by choosing the bin center of the highest magnitude FFT point, you are throwing away information.
The amount of information thrown away in arriving at an estimate (for a fixed time record), can be reduced by the three techniques mentioned in the post 9 Feb 08 19:02.
Regarding the step change example, two thoughts come to mind
1 – I’ll bet if you worked out the example fully, at the frequency points present in the original FFT, there is a match (positive at zero frequency and the zero’s of the sinc function at all other frequencies of the original domain.... the additional in-between frequencies would have non-zero content). It should match the original results at the original frequencies, it just adds new frequency points that don’t match what you’d expect.
2 – If you used a window, you wouldn’t have had the sharp change in the time domain or the extreme behavior and the result would be closer to what you expect. My 2008 spreadsheet has example of zero-padding applied with a window (although I did not adjust the magnitudes, my interest was only frequency determination). It is technically possible to smack your kneecap with a hammer or insert a screwdriver into your eye-socket, but one shouldn't conclude from this that a hammer and a screwdriver are bad tools
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Frequency resolution and amplitude in FFT
You assume that there is only one contributor in each 1/T band. Fourier does not.
If one were to restrict oneself to rotating machinery then this is usually fair enough. But then you'd be better off using synchronous sampling in the first place. An example where it would be horribly misleading is where a 3rd order firing engine is coupled to a hookes joint via a 1.47 overdrive gear ratio. BTDT got the T shirt.
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Frequency resolution and amplitude in FFT
- Steve
RE: Frequency resolution and amplitude in FFT
Yes, it is an assumption that I made and stated. I agree it may or may not hold depending on your situation and like any assumption that needs to be considered. This assumption was clearly stated at the very beginning of my summary posted 9 Feb 08 19:02, which is the specific summary post that I referenced by date/time on 10 Jul 12 22:50.
My comments apply to estimating the frequency of a single peak to the highest possible accuracy from existing data, they do not apply to the unsolveable (without more data) task of separating peaks which fall within a single bin of an FFT of the available time record (frequencies separated by less than 1/T where T is sample duration), and I have highilighted that my comments about resolution do not apply to the case where the multiple frequencies in the input overlap.
For the rotating machinery I work with, my interest is not limited to synchronous frequencies. There are bearing defects, twice line frequeny vibrations of motor, whirls, and other phenomenon that make us interested in all frequencies. If I see a peak in my collected data at roughly 3 times running speed, I want to know whether it is 3.000 three times running speed or whether it is something like 3.014 times running speed such as a typical outer race frequency for an 8-ball deep groove bearing. Does that mean these tools are useless? Nope. They give us the best estimate of a given peak from the given data. They don’t do the impossible (separate the inseparable) but that does not make them useless. For starters, maybe there is just 3.014and no adjacent 3.000. Higher resolution helps see it. So, what about that case where there are 3.000 and 3.014 in the same bin or so cloe that they can’t be separated? The combined frequency when I try to label it will still differ from 3.000 (maybe 3.007...halfway between). If I label all my harmonic peaks on a log scale and see 1.000, 2.001, 3.007, 4.002, then I have a suspicion about that 3.007. The fact that I have higher resolution on the other peaks (1, 2, 4) makes it easier to see that outlier near 3. The point is: higher resolution always helps separates peaks and it’s always valuable to label your peaks with the highest resolution available for this reason. The higher, the better we can distinguish. I’d love to have 0.001hz resolution on every machine but then the data collection time would balloon. So I take what I get and make the best of it (using the valuable tools I have higlighted) and don’t throw any of it away.
I think your implying that zero padding and the other tools mentioned are useless and again I will strenuously disagree.
First I will mention that I consider all three tools (frequency interpolation, reconstruction, zero-padding) are of the same nature. There is one exactly precise estimate of the peak (subject to assumptions discussed), and all three will help us get toward that one precise number to various degrees depending on how much computational effort we want to put in.
These are well accepted tools, nothing I invented (ok, I came up with the particular quadratic interpolation myself but I’ll bet someone else has one it too... Entek/E-monitor must use some tool in the frequency interpolation category because they discard time waveform and phase and store only magnitudes and come up with the estimate from that).
To brush them off in this manner would be quite unwise imo. Artifacts are not created by zero-padding, they are created by windowing, which is present whether you zero-pad or not. Yes, you can come up with one example of pure dc where there is no windowing effect when performing FFT of the original signal and the windowing becomes important when you zero-pad, but it is not an important or representative sample. It is not important because those of us in vibration don’t generally use FFT to determine a dc component. It is not representative because any other signal will have a window effect.
If we stick with rectangular window for simplicity, we have the length of the time window will be the same regardless of whether we zero pad or not. So when we multiply our original time waveform by a broad rectangular window, we convolve the resulting frequency spectrum by the sync function (fourier transform of rectangular window). The width of the sinc function in frequency varies inversely with the width of the rectangular window in time. Since the width of the multiplying-rectangular window is the same in both cases, we convolve by the same sync function and get the same spreading. The only difference is that the zero-padded signal provides a higher resolution sampling in frequency of that resulting continuous function.
So the sync function by which the original signal gets multipled will be the same. The only difference is how finely in frequency we sample the output (the zero padded FFT provides finer frequency sampling of that result (the FFT points are frequency samples of the continuous DTFT... same DTFT either way, finder frequency samples with zero padding).
In summary, the zero pading does nothing to reduce the leakage We have broadening of the peak created by that leakage which is not solved or reduced by these techniques. (I have said as much in my previous linked thread). When combined with possible presence of interfering noise, these widening creates problems in estimating the frequency and we might describe as an uncertainty band. The techniques do nothing to reduce the width of that uncertainty band but they DO help us center the uncertainty band on the exact frequency where it should be centered.
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
"The point is: higher resolution always helps separates peaks and it’s always valuable to label your peaks with the highest resolution available for this reason."
should've been:
"The point is: higher resolution always helps identify peaks and it’s always valuable to label your peaks with the highest resolution available for this reason."
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
I agree, exploiting the shape of the shoulders of the bins is an effective way of helping to increase the apparent resolution for the frequency of a peak, but it relies on assumptions, again, not for the faint hearted in the general case.
I'm not too sure why we got side tracked into this curve fitting discussion, I thought the topic was zero padding.
Cheers
Greg Locock
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RE: Frequency resolution and amplitude in FFT
Let's start with #3 (reconstruction) because it gives a very good framework to view the previous two.
Reconstruction of the DTFT provides a mapping from the discretely-spaced FFT results to a continuous DTFT function. That mapping again can be found for example equation 6.17 and 6.18 here:
http://ens.ewi.tudelft.nl/education/courses/et2405...
It involves a sum of weighted, frequency-shifted continuous sinc functions. Each FFT point factors into one term. The complex value of the FFT is the weighting, the frequency of the FFT point figures into the frequency shift.
The peak of the continuous-frequency DTFT is the peak which is our best available estimate of the frequency of interest (assuming not two peaks in a bin in which case there is no technique that will separate them... I don't consider the inability to achieve the impossible as a disadvantage AND and as I showed above 1.000, 2.001, 3.007, 4.002... even where one of the peaks in the pattern is merged with another peak, we can usually recognize the pattern and the outliers better if we have the best available estimate of all the frequencies in our spectrum).
So the reconstruction provides this continous function and it’s peak the the most precise estimate we can possibly come up with. That is by my view the true best answer with available data, but it is computationally intensive. So we can try the other two approaches to try to get part way to the same answer. How does that work?
First start with zero padding. As we know, the FFT is simply samples of the DTFT at discrete intervals corresponding to the bin width. As we showed in previous discussion, the underlying DTFT is the same regardless of whether we zero pad or not. All we do with zero padding is get finer samples of that underlying DTFT function. With finer samples we will likely get closer to finding the true peak.
But zero padding is also a little bit compuationally intensive. And it has to be done before we do the FFT, not really suitable to be done when we’re poking around an FFT result. So that leads to the other approach: frequency interpolation (curve fitting as you called it).
In contrast to the other two techniques, I don’t know if any rigorous proof of the frequency interpolation techniques are available. I have my own intuitive fuzzy proof. It is that the sinc functions that we use to build the DTFT from are relatively smooth. Looking at zero crossings of the sinc to estimate it’s frequency content, the highest frequency posible occurs away from the center lobe where we see maximum rate of 180 degrees per bin-width. (By the way if we were looking at samples of a time function rather than samples of a frequency function, that would be the Nyquist frequency..... the idea that we can reconstruct the entire continuous DTFT from the discrete FFT points is analogous to the idea that we can reconstruct a time waveform from it’s samples if and only if the relation betwen sampling frequency and original waveform meets Nyquist limit). The fact that we build this final DTFT out of bandlimited sinusoids imposes a certain kind of smoothness on the resulting complex function and its’ real magnitude. It cannot vary too eratically between the known points, it is limited to varying at the rate of it’s components. This characteristic I believe is what helps the frequency interolation work.
The quadratic interpolation approach is to select the highest bin magnitude and one on each side. With three data points we can solve the A,B, C constants in a quadratic form: Y = A + B*f + C*f^2. Obviously we know dY/df = B + 2*C*F and so the maximum occurs where dY/df = 0 (f = -B/[2*C]). You can solve all the algebra ahead and you end up with a fairly simple result (I can provide the algebra/results if anyone wants). That solution is built into the vba of my spreadsheet along with some other stuff... you can graphically some results works on the first two tabs on left of the workbook... fill out the green input blocks with three frequencies spaced equally and three magnitudes (the center has to be the higest of the three).
Here is an empirical study of results of quadratic method (I’m not positive if it’s identical to the method I described):
http://www.ericjacobsen.org/fe.htm
I have done my own empircal study in the spreadsheet. I used a fixed sinusoid of varying frequencies from 74 to 79 sampled at interval of 0.0005sec as input to a 512-point FFT. Bin width is 1/(512*0.0005) ~ 3.9hz. The bin centers in the neighborhood are 74.2,
78.1, 82.0. Look at the first seven rows of the table in the summary tab which use nothing other than quadratic interpolation:
Tab / Freq / Estimate / Error / BinWidth / Fraction Of BinWidth
Trial74 74 74.05 0.05 3.90625 1.4%
Trial 75 75 74.83 0.17 3.90625 4.4%
Ttial 76 76 75.91 0.09 3.90625 2.4%
Trial 77 77 77.20 0.20 3.90625 5.2%
Trial 79.1 79.1 78.91 0.19 3.90625 5.0%
Trial76NW 76 75.35 0.65 3.90625 16.6%
Trial77NW 77 77.86 0.86 3.90625 21.9%
The accuracy of the first 5 trials in determining the frequency is 5% of the bin width. The accuracy of the last two is up to 22%, but we didn’t use a window (These last two don’t count in my book, we really should be using a window). It’s a pretty darned good improvement in precision (beyond just picking the highest bin) and it’s pretty cheap computationally. (If you want to go the other extreme and number crunch using the DTFT reconstruction, you can see the 77hz input sinusoid was estimated from FFT output to be 77.00001015).
The Entek E-monitor data base must use frequency interpolation (since they work from FFT magnitudes only, they cannot accomlish the other techniques). I don’t know exactly which technique they use, but I have reasons to suspect it’s pretty close to the quadratic method. For the user it is as simple as putting your cursor on the peak and pressing p.
I have been using that program to look at machinery spectra for probably an average of 10 hours per month for the last 12 years. Knowing the frequency is important to accomplish my job and I spend time studying the patterns to estimate how accurate they are (to how many decimal places are my harmonics exact multiples of the fundamental). I can say without hesitation that the Entek peak label feature is substantially better than just picking the highest bin center (exactly as we expect from my empirical study). I can also state my opinion that anyone assigned to do my job who didn’t use that tool based on vague unfounded fears or objections as expressed in this thread would be just plain misinformed. I have to say it that way to conviction that I feel about this subject. I definitely am not saying it that way to reflect on you, Greg. (I have learned a lot from you Greg and still am light-years away from knowing half the stuff you do about vibration).
How the three approaches tie together is that none of them accepts the false premise that the highest bin center is the best estimate and all of them move us in the direction of the same ideal best answer given available info.
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
you can take the inverse transform of the result and quantify the mean square error relative to the original data easliy enough
RE: Frequency resolution and amplitude in FFT
I'm not sure what you mean. Inverse transform will give back the original data. Mean squared error had better be zero. Note that I have compared the effects of these techniques to known input in my spreadsheet which perhaps accomplishes what you were referring to... I'm not sure.
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
1 - inverse FFT. 2 - divide by the window (to undo original window multiplication).
Perhaps you meant to do only step 1 and compare to original data. Then you would get difference associated with the window. It will tell you something about the window, but I'm not sure what it really tells us about the match. As far as validating the techniques I mentioned, comparing against known input as in my spreadsheet makes sense to me. I'm open to comment since perhaps I have misunderstood where you were heading with this comment.
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
I'll agree that if you presented someone a spectrum and didn't tell them that it was derived from zero padded TWF, that it would be misleading. Since this thread is about zero padding (and related tools that accomplish exactly the same thing), I don't think there is a concern that anyone reading this thread would fall into the category of not knowing the specttrum that they zero padded was zero padded. It's worth discussing potential misuses of tools but imo it's not the primary basis for judging those tools.
Here's the first one that pops up for me, the only one in site from Stanford:
https://ccrma.stanford.edu/~jos/st/Zero_Padding.ht...
Please tell me specifically where in this article is it emphasized that zero padding is not a simple process.
As far as I can tell this doesn't say anything resembling anything you have said in this entire thread.. I would argue that if you read it carefully it will show why zero padding has nothing to do with a "free lunch" and everything to do with getting the most out of the lunch that you have in front of you.
Here's the second one.
http://www.ni.com/white-paper/4880/en
Title "Zero Padding does not buy Spectral Resolution"
This article is intended to show the limitations of zero padding, that's fine. Read it and study it if you want. There's nothing to contradict anything I've said.
The basis for that title statement is apparently that zero padding does not improve the ability to separate closely-spaced peaks as discussed in the middle of the article. That is completely true. But separating closely spaced peaks is not the only thing we do with FFT's and not the only reason we need the best available accuracy on our peaks as I've explained (1.000, 2.001, 3.007, 4.002).
Maybe there is a terminology aspect to this discussion. Perhaps this author's definition of frequency resolution is ability to separate closely spaced peaks, and I would not fault him for that definition. I have that term in a different way and explained what I meant.
How much do you need to know about a machinery signal in order to decide you'd like to get the best estimate possible from your existing data? I don't study my data to decide whether to apply my peak label tool (which is equivalent to zero padding in result of estimating frequency closer to the ideal). For me the decision is a easy one (a.k.a. "no-brainer")
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
Walt
RE: Frequency resolution and amplitude in FFT
https://ccrma.stanford.edu/~jos/st/Zero_Padding_Ap...
" Sometimes people will say that zero-padding in the time domain yields higher spectral resolution in the frequency domain. However, signal processing practitioners should not say that, because ``resolution'' in signal processing refers to the ability to ``resolve'' closely spaced features in a spectrum analysis (see Book IV [70] for details). The usual way to increase spectral resolution is to take a longer DFT without zero padding--i.e., look at more data. In the field of graphics, the term resolution refers to pixel density, so the common terminology confusion is reasonable. However, remember that in signal processing, zero-padding in one domain corresponds to a higher interpolation-density in the other domain--not a higher resolution. "
Maybe not saying it is dangerous, but certainly not giving me a warm fuzzy feeling.
Cheers
Greg Locock
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RE: Frequency resolution and amplitude in FFT
He is saying that zero padding doesn’t improve the ability to separate frequencies, which
I agree as discussed above. For him, the ability to separate frequencies is the definition of frequency resolution, different than the way I used the term (also discussed above). For me, it brings nothing new that I didn’t already address above. Of course I was a little verbose I guess. I’ll attempt to atone by making this a short post and signing off now.
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
Kind regards
Geoff
RE: Frequency resolution and amplitude in FFT
Walt
RE: Frequency resolution and amplitude in FFT
TTFN
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RE: Frequency resolution and amplitude in FFT
There is one huge obstacle in this discussion: TERMINOLOGY.
Specifically, the word frequency “RESOLUTION”
Let’s go over that obstacle once and for all. People use that term differently. I used that term “resolution” wrong compared to how must in the vibration world would use it (my usage comes from the instrumetnation world.. where resolution is how finely can you read a result). In my defense, although I used the wrong term, I did explain exactly what I meant.
Many of the authors use that term to mean the ability to separate frequencies. The basis for all the statements about zero padding not improving resolution is that it does not improve the ability to separate frequencies (same for any of the techniques). I agree! (*) I have said so from the beginning.
* IRstuff makes a good point, zero padding probably does give some help for the task of closely spaced frequencies based on signal to noise considerations (excluding those spaced less than a bin width apart which can never be separated by any means). And while it’s a good point he brought up, it’s not the point I’ve been trying to make and not what I’m going to discuss here. I’m going to try to explain better what I’ve been saying all along.
The benefit I’m referring to is NOT separating closely spaced frequencies. It is getting the best possible estimate of a given frequency from the given data for all frequencies which can be improved. This excludes frequency estimates that are impossible to improve from original data (frequencies separated by less than one bin width). For the sake of argument I’ll also exclude frequencies that are close but more than a bin width because those are a grey area leading inevitably to more discussion and not necessary to make my point. I am talking about frequencies that are spaced far apart. It is STILL important to me to estimate the frequency those far-apart frequencies as best we can. Why? This was illustrated by the example 11 Jul 12 21:41 culminating in the numbers 1.000, 2.001, 3.007, 4.002, which is very typical of real-life experience that tells us it’s important to have the best available estimate available for ALL frequencies in the spectrum. To review, all numbers in this example expressed as multiples of running speed. We had running speed harmonics plus a bpfo frequency occuring at 3.014. I am assuming the 3.014 closely spaced to 3.000 cannot be reliably separated into two peaks. If we had analyzed this spectrum with original FFT bin centers (NOT using the 3 tools), then expressed all the peaks as multiple of the estimated running speed peak, we might get a series of peak something like 1.000, 2.005, 3.008, 4.010 (just an example, assumes the first peak gets estimated low so the others are high multiples) . The outlier in that pattern is not really obvious. When we apply our techniques, we improve the estimates 1, 2,4 and end up with a pattern more like 1.000, 2.001, 3.007, 4.002. Now the outlier in the pattern stands out much more prominently and we can more easily recognize the 3.007 as a deviation from the pattern which deserves further investigation. (there are likely other clues higher in the frequency spectrum to look for.). This is not a contrived example, rather this is illustrative of the technique I use almost every time I analyze a spectrum. ...I use the software peak label feature and also use the software order label feature such that each (interpolated) peak frequency is shown as a multiple of the fundamental (interpolated). I look at how close the harmonics are to exact integers and use this as a basis to identify peaks that may not fall in the pattern. And by the way even though we call it order label the same feature can be used with fundamental frequencies other than running speed.
Now that we have established that there is value to precisely identifying the frequency of peaks other than those closely spaced, let me get to my point.
There is leakage created by the windowing process. None of the three techniques I’ve mentioned does anything to reduce that (because they do not change the length of the time window hence do not change the width of the frequency sinc that gets convolved with the true spectrum). When we combine this spreading with noise that can be present, we can describe the resulting uncertainty simplistically as an uncertainty band.
Here’s what the three techniques can’t do:
They CAN’T reduce the width of that uncertainty band
Here’s what the three techniques can do:
They CAN tell us where the center of that uncertainty band should be
(And typically center of uncertainty band is also our most accurate estimate).
In the case of DTFT and zero-padding these tools can provide virtually unlimited ability to locate where the center of the uncertainty band should be.
In the case of zero padding, more zeros gets us closer to that ideal. In the case of frequency interpolation, our estimate of the peak is moved toward the same ideal point that would be predicted by the other two techniques.
This last paragraph applies where peaks are not closely spaced. Closely spaced peaks deserve separate discussion. The usefulenss is in getting best possible estimate of these peaks from available time record when more data is not available.
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(2B)+(2B)' ?
RE: Frequency resolution and amplitude in FFT
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(2B)+(2B)' ?