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Where Does FFT Process Gain Come From?

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Re: FFT calculations
2/17/2012 10:44:25 AM
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Once, when I was using FFTs a lot, I took an advanced math course at a university (I was going back to school to complete a degree).  Transforms figured a lot in this class (LaPlace, Fourier).  When we got to Fourier Transforms (DFT) I was waiting for some insight from the professor on the FFT.  I was implementing these for some complex signal processing research projects.  Finally, I went up to the professor and asked about the FFT.  He looked at me blankly and I felt really dumb.  So, I went home and looked at my books of the FFT (I had several, and fortunately the company was paying for them).  So, next class I explained what I was interested in.  His response was that he was a theoretical mathematician.  He did not care how long it took someone like me to comptue it.  Live and learn.

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Re: Caution - you are about to enter the "no gain" zone...
12/15/2011 11:11:15 AM
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Thanks for your comments, Don.  You are correct about accounting for the jitter and settling time in ADC measurements.  It's sometimes difficult to get everything in a 400-word column, so the January column discusses where the "noise" in a signal-to-noise ratio (SNR) comes from.  I included information about jitter and can cover this aspect to measurements in other column, too.  The January column lists several good references.  Again, thanks. --Jon

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Caution - you are about to enter the "no gain" zone...
12/14/2011 8:01:09 PM
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While it is true that SNR improvements can be enjoyed at the expense of reduced bandwidth, "there ain't no free lunch". By the very definition, FFT implies a sampled system. So if we are using a sampled data set as opposed to a continuous time analysis using DFT, where is the discussion on the sampler?

Life would be pretty easy if the signal world existed of singular tones but in the real world the signals we need to resolve occupy more than 1 bin of the FFT. (can I interest you in a jar of OFDM?)

While it is true that the processing gain in dB is 10*log(BW1/BW2) if BW2 is smaller than BW1, there are 2 additional parameters that are critical to achieving the ideal improvement in the aforementioned equation.

Settling time and aperture jitter cannot be ignored.

For example if it is desired to resolve a 4.096 V full scale signal to 12 bits accuracy at 10 MHz sample rate, the analog path has to settle to within 1 mV of the final value in 100 nS. Can you say "40 volts / uS slew rate"? If the amplifiers can't do that say goodbye to best case 74 dB SNR no matter what the FFT processing gain is.

If the clock is noisy, well you know the answer by now. If you want to get the SNR out of that 12 bit 10 Ms/s ADC better be sure to keep the clock jitter to 2 pS.

Regards,

Don

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Re: Mechanical engineer's question
12/7/2011 3:27:22 PM
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When you look at the FFT plot in an ADC data sheet, it's easy to mistakenly interpret the results as showing a noise floor lower than expected, and thus a expect the ADC to deliver a larger signal-to-noise ratio. The more points you take, the better the plot looks because of the way the FFT spreads energy among the various frequency bins. The plots of FFT results don't tell the whole story, so use them with care and pay attention to the numeric specs the ADC manufacturer provides. Or better yet, get some IC samples, or ask for a "loaner" ADC module, and run some tests in your own measurement environment. I also wanted to show there's a mathematical reason why the FFT plots look the way they do.--Jon

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Mechanical engineer's question
12/7/2011 3:14:56 PM
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Jon: Lyons appears to be saying that the signal-to-noise increases as the FFT processes more points. So if I'm a mechanical engineer employing an ADC in my product, what lesson I can learn from this?

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FFT calculations
12/7/2011 9:43:18 AM
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I remember how FFT were both a bane and a boon in engineering school. Fast Fourier Transforms are at the heart of communications theory, and are actually very interesting. At the same time, the initial learning curve, while not step, is not trivial either. On the implementation side, an flexibility you can gain regarding setting the parameters you want for digital signal processing apps is a significant design advantage, so this is a valuable article by Jon.

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