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]

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.

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.

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.

[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.

[TomKatt]

So - out of curiosity - what benefit do you get choosing to view in dot mode (or for that matter, any other interpolation view)?  Verifying that aliasing isn't an issue?

[tom66]

So - out of curiosity - what benefit do you get choosing to view in dot mode (or for that matter, any other interpolation view)?  Verifying that aliasing isn't an issue?

Usually dot mode has a higher waveform update rate as the scope doesn't have to plot vectors which for a pixel spacing of more than 1 between samples is no longer a trivial operation.  For my little scope project I used dot mode only, I flirted with vector plotting but it was very hard to get good performance.  I found that my 1074Z did get about twice the waveform update rate in dot mode as to vector mode, which was interesting to say the least (though IIRC Rigol did spec the update rate in vector mode so they weren't being misleading.)

For continuous signals (sine, etc.) you may not see a disadvantage to this plotting mode but do benefit from more wfm/s.    Other than that, I can't think of an advantage to using dot mode.   It's certainly much less intuitive to understand a square wave or other signals using it so I stay in vector mode.

See also my edit re the ADC sampling criteria.

[iMo]

..
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.

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.

..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.

That is the key message - "if the input signal does not contain frequency content above and beyond the Nyquist frequency".

What type of a signal? A sine/cosine only? Not applicable for a "square wave", for example, imho..

[TomKatt]

That is the key message - "if the input signal does not contain frequency content above and beyond the Nyquist frequency".

What type of a signal? A sine/cosine only? Not applicable for a "square wave", for example, imho..

Perhaps the front end is limiting bandwidth to less that 1/2 Nyquist as suggested.  In that case, wouldn't it be possible for the sin(x)/x to exactly regenerate the waveform the front end is passing to the adc?  In other words, it is showing what it's actually measuring because it has filtered the input prior to measurement?

[bdunham7]

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.

Your suspicions are well-founded and despite the explanations offered here and elsewhere, I think Agilent's position is mostly marketing wankery.  Perhaps they were at a disadvantage regarding sample rates.

The statements made regarding Nyquist are true if your system completely meets the requirements stated.  The problem is that no system exactly meets those requirements.  There are always noise, linearity and quantization effects, and with a typical 8-bit oscilloscope, those are not trivial issues.  Then there is the seemingly innocuous Nyquist bandwidth requirement, but the problem is that since you are typically using an oscilloscope to look at unknown (or imperfectly known) signals and no reasonable filter can have the sharp cutoffs needed to get close to maximal Nyquist performance without some pretty extreme effects elsewhere, like group delay and step response. 

Regarding dot mode, as long as the test signal doesn't have a period at or very near a simple fraction of  the oscilloscope sampling rate, it will typically function as an ad-hoc ETS (equivalent time sampling) and may provide a cleaner display than the sinc interpolation.  The easiest way to demonstrate this is with a fast edge where the trigger point and most of the rise are within two samples (well above Nyquist, of course). The sinc interpolation will give you a variety of results and rise times depending on where the samples line up with the edge, but dot mode will give you an array of widely spaced dots that all overlay nicely.  Whether the resulting display is 'correct' or not is another matter.

[bdunham7]

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.

Really?  That seems like an extraordinarily difficult task to me and almost certainly more difficult and expensive than simply quadrupling the sample rate. 

[tom66]

Really?  That seems like an extraordinarily difficult task to me and almost certainly more difficult and expensive than simply quadrupling the sample rate.

So perhaps I had better rephrase:  It is achievable with good design, at some cost.   May not necessarily be done on a 100MHz scope as you note, a faster ADC could be cheaper especially when the tolerance of the required components is, plus any factory calibration.   It would definitely be a steep filter but there are oscilloscope architectures which include similar complexity filters, for instance Siglent 5104X samples at 2.5GSa/s in some channel modes, with a 1GHz bandwidth.  As far as I know and could test there was no obvious aliasing on that scope at the bandwidth limit so it presumptively must achieve something close to -48dB rolloff at 1.25x its input frequency.

That is the key message - "if the input signal does not contain frequency content above and beyond the Nyquist frequency".

What type of a signal? A sine/cosine only? Not applicable for a "square wave", for example, imho..

Yes and no.  A 100MHz scope is widely understood to be not very useful at looking at square waves >>20MHz frequency, because the components to construct that wave are lost to the bandwidth limit of the instrument, regardless of the ADC sampling rate.  However, if you do acknowledge your instrument has a 100MHz bandwidth limit, then if its ADC filter is correctly designed and it uses correctly implemented sinc interpolation, it does accurately and exactly show the picture of what your signal looks like, up to that bandwidth limit.   I guess the way to see this is if you had an ideal 100MHz CRO and compared it to a 100MHz DSO, a well implemented DSO (with intensity-grading and the like) would show a waveform equivalent to the CRO.

What the real world implications of this are depends on what you are doing with the scope.  Are you using it for looking at square waves?  In theory, there is no upper bandwidth limit for a square wave so... you can never see the true picture, though in practice more than 10-15x the frequency for the scope bandwidth has little additional benefit. 

I suspect the Agilent engineer is writing this from a communications theory background.

[bdunham7]

It would definitely be a steep filter but there are oscilloscope architectures which include similar complexity filters, for instance Siglent 5104X samples at 2.5GSa/s in some channel modes, with a 1GHz bandwidth.  As far as I know and could test there was no obvious aliasing on that scope at the bandwidth limit so it presumptively must achieve something close to -48dB rolloff at 1.25x its input frequency.

I have not had the opportunity to test an SDS5104X, but on all the other Siglent models I've seen they just alias away and appear to have a -6dB response at least to 2X their rated BW.  -48dB at 1.25X BW would imply a 144dB/octave rolloff with an analog design.   I'd be interested to know exactly how you tested your example.  The way I would do it would be to compare equal-amplitide 1GHz and 1.5GHz inputs. 

[gf]

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.

Really?  That seems like an extraordinarily difficult task to me and almost certainly more difficult and expensive than simply quadrupling the sample rate.

A 7th order elliptic filter is likely realizable, but I guess you won't like its impuse and step response.

Btw, at the end you can never see the original signal if is not band-limited in the first place. If you need an anti-aliasing filter in order to enable "perfect reconstruction" from the samples, then you still cannot reconstruct the original signal, but only the filtered one.

[tom66]

I have not had the opportunity to test an SDS5104X, but on all the other Siglent models I've seen they just alias away and appear to have a -6dB response at least to 2X their rated BW.  -48dB at 1.25X BW would imply a 144dB/octave rolloff with an analog design.   I'd be interested to know exactly how you tested your example.  The way I would do it would be to compare equal-amplitide 1GHz and 1.5GHz inputs.

Been a while as we had it on loan as a demo unit pre-COVID, but I ran a DDS into it and swept up to about 700MHz on a square wave (IIRC that was the limit of our generator for square)  then looked at the FFT response to see if there were any obvious aliases as I moved the frequency up and down.  None appeared, though I suppose it's possible this test didn't capture real aliasing.

We've got an SDS6000 on loan (the 2GHz, 4 channel one) coming next week which will be interesting to test.

[bdunham7]

Been a while as we had it on loan as a demo unit pre-COVID, but I ran a DDS into it and swept up to about 700MHz on a square wave (IIRC that was the limit of our generator for square)  then looked at the FFT response to see if there were any obvious aliases as I moved the frequency up and down.  None appeared, though I suppose it's possible this test didn't capture real aliasing.

I'm not sure how you expect to observe aliasing with a fast edge like that in the mix, but I assume you'd hope to spot the 3rd harmonic reflected down at some point.  And what is the output filter BW of your DDS that can put out a 700MHz square wave? 

On your 2GHz model, perhaps try alternately feeding it 2GHz and 3GHz clean sine signals at the same amplitude.