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Chris Rehorn's paper about Sinc interpolation
balnazzar:
Hi Folks.
I'd need a paper by Chris Rehorn (HP/Agilent/Keysight), that is "Sin(x)/x Interpolation: An Important Aspect of Proper Oscilloscope Measurements".
Having been published upon EE Times, the official download page would be this one: https://www.eetimes.com/sinx-x-interpolation-an-important-aspect-of-proper-oscilloscope-measurements/
but as you may see, the downloading link doesn't work.
You can actually download it from siglent.fi, here: https://siglent.fi/data/technical-common/Sin(x)x_Agilent.pdf
But, probably due to postscript issues, the formulas and the equations are badly rendered, to the point that one misses important stuff like apices, pedices, functions' arguments, etc..
Are you aware of alternative download links for a correctly rendered version of that paper?
Thanks.
pardo-bsso:
There it is: https://web.archive.org/web/20170829021855/https://m.eet.com/media/1051226/Sin(x)x_Agilent.pdf
Same rendering issues but you can also find that content in many books.
balnazzar:
--- Quote from: pardo-bsso on October 17, 2022, 01:34:26 pm ---There it is: https://web.archive.org/web/20170829021855/https://m.eet.com/media/1051226/Sin(x)x_Agilent.pdf
Same rendering issues but you can also find that content in many books.
--- End quote ---
Ah, same rendering issues... Mh, I wrote to the author 2 days ago asking for a well-rendered version, but no reply yet.
Would you please advice about the books? I'm falling in the rabbit hole by delving into google books in search of a title or two to buy.
Thanks!
TomKatt:
Reviving an old thread - I just referenced this whitepaper in another post in the Test Gear forum. I'm nowhere near smart enough to fully understand all the math, but I am a bit surprised to see a paper published by a company of such esteem as Agilent claiming that sin(x)/x interpolation can exactly reconstruct the waveform seen at the scope input, provided no frequencies are above the Nyquist frequency. Furthermore, the paper claims there isn't much benefit to anything more the 2X that Nyquist frequency.
The paper can be found at this link: https://siglent.fi/data/technical-common/Sin%28x%29x_Agilent.pdf
It would seem that although there may be merit to this claim mathematically, in real life few signals contain no high frequency components and processors always have limitations to the extent of calculations they can perform.
Given that, I find it difficult to believe that sin(x)/x results in the exact waveform seen by the input. It may be very close, but I can't understand exact.
--- Quote from: Agilent ---An oscilloscope with more sampling rate is not always better. The signal at the input of the oscilloscope is
properly reconstructed after digitization by the sin(x)/x reconstruction filter, if the input signal does not contain
frequency content above and beyond the Nyquist frequency. It may be tempting to disable interpolation to see
the “raw samples”, but this is not necessary. The interpolated waveform is not a “guess” at what the signal was
doing between samples, it is exactly what the signal was doing between samples.
--- End quote ---
tom66:
It is an exact reconstruction, provided the Nyquist input criteria are met. When oscilloscope manufacturers design, say, a 100MHz scope, they have hopefully set the ADC filter up correctly so that in the worst-case sampling configuration (e.g. all channels on) the input bandwidth at the ADC has no frequency content above the Nyquist limit (that are beyond the quantisation noise of the ADC.) If this criteria is met, a 100MHz scope with correctly implemented sinc interpolator can be said to exactly reconstruct all signals up to 100MHz input frequency, if the ADC sampling rate is at least 200MSa/s and the ADC filter is adequately designed. That is a mathematical guarantee of a sinc interpolator. Actually, you can think of a sinc interpolator as a bit of a reverse filter operation, essentially reconstructing what the ADC analog amplifier sees before digitisation, though I couldn't hope to actually explain the maths behind that.
The benefit to sampling beyond Nyquist is that it makes the requirements for the input filter more relaxed. Of course, sinc can do nothing about ADC nonlinearity and quantisation artefacts, nor can it help with jitter of the PLL or other such noise sources, but they do tend to be quite minimal issues on a modern digital scope.
Edit: To be clear, a perfectly sampling 200MSa/s scope with 100MHz bandwidth would require a brickwall filter at 100MHz and therefore would not be something you could build, as brickwall filters do not actually exist in the real world (and even in DSP land they're a bit fictitious, you can do something with FFTs to get a similar effect but it doesn't work well.) So you would realistically need something like at least a 250MSa/s ADC and design your 100MHz bandwidth to roll off to at least -48dB at 125MHz (6dB/bit rule of thumb for quantisation SNR). That is a pretty steep filter, but with good design is achievable. However, you can get better results if you only have to get rolloff to say 250MHz. I think this is what cheaper scopes like Rigol 1104Z do, as I have seen them alias in 4 channel mode with 100MHz+ inputs, though the aliasing is still lower in amplitude it does tend to fool the sinc filter somewhat.
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