signals and systems question

With an s-domain function for the photodiode, treating the rest of the system as linear & discrete states (i.e., keep ignoring any transmission line effects), and ignoring acquisition jitter, I'm 99.44% sure that I could come up with a closed-form solution for this.

If I couldn't, I could at least come up with a closed-form solution for an upper bound to the error, and that closed-form solution could be expanded to include acquisition jitter, injected noise, and all that lovely stuff.

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Tim Wescott 
Wescott Design Services 
http://www.wescottdesign.com

Well, maybe you could, but I can't. I'll need to present the customer with a number of cases, different waveforms and different filters, the intent being to talk him out of making us change anything. I can simulate this accurately.

El Customer prefers PowerPoint things with short sentences, things they can show to Management.

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John Larkin         Highland Technology, Inc 
picosecond timing   laser drivers and controllers 

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

Fun, Can you send in a "calibration" pulse(s) from an LED or something?

George H.

Oh, that's too bad. You can't pick your customers.

George H.

The customer has a calibration methodology that they haven't revealed to me. Probably just as well.

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John Larkin         Highland Technology, Inc 
picosecond timing   laser drivers and controllers 

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

Jan, Dont forget this is a bandlimited system. So how fast will the start ans end of rhe signal be and how many samples will be taken during the rise and fall.

Mark

A few minutes playing with an AWG and a sampling scope says it does too. Measuring the area of a BW limited pulse at the 2x the nyquist rate gives .51 times the rms error obtained at the nyquist rate. 4x sampling gives .35 times the nyquist rms error, and oversampling at about 39k nyquist gives 0.22 times the nyquist rate error. These errors based on

50 measurements each. This is consistent with your observation of a jitter in the measurement when undersampling.

It also looks like the undersampling underestimates the true area with nyquist rate sampling giving .92 times true area and 2x sampling giving .97 times true area, assuming 39k times nyquist yields the true area.

This numbers will likely change with pulse shape. This pulse was a low pass filtered square. I don't know the roll-off of the filter.

ChesterW

Yes, they do, but that does not imply that adding the samples will give you the area with no error.

Yes, and each one of those inserted values will be different from the existing samples and not *linear* like a straight line drawn through them. A linear interpolation will give you the same average or integral. Other shapes will not in general.

Saying that meeting the Shannon criteria is sufficient to preserve the area is simply hand waving. There is no connection unless you do some math to show it.

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Rick

On a sunny day (Mon, 09 Mar 2015 14:25:03 -0700) it happened John Larkin wrote in :

You are wrong.

That is is the problem of your solution.

The graph you published shows an error of 1.5 %, if that is good enough for your custard comer so be it.

Still not clear to me what you (or the custard comer) actually wants to measure.

Basically the (any) sampling system is a mixer. You get f1 + f2 and f1 - f2 added to the sample output. For a constant phase (synched sample pulse) that results in a constant DC voltage, If you start at the right point the offset will be zero.

AM modulator: 1/3 4053 CMOS switch .......... C . . audio in--||-----------------0\ . | . \-------- AM out | --------0 . R | .......... | /// | |+ | 2.5 V bias | | RF square wave ///

And it has been shown to be true once or twice in this thread already! Once the sampling criterion is satisfied, there is *no* extra information to be had. Interpolation does *not* improve the result and the phase of the samples w.r.t. the signal does

*not* matter.

Again, only *if* the Nyquist, Shannon, ... criterion is satisfied. That's never true in the mathematical sense, but we are engineers. We can get close enough for all practical purposes. ;-)

Jeroen Belleman

I saw someone show that a low pass filter will not disturb the integration, but I have not seen any proof that meeting the Nyquist?Shannon sampling criterion preserves the integral as the sum of the samples. Can you point me to that?

*Again*, stating something to be true is not the same as it being true. We don't know what the requirements are, so I don't know how you can say anything is "close enough".

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Rick

OK, I Spiced the entire signal chain

and poked in a pretty radical input pulse. 40 ns out, the difference in integrals is about 0.3%, over the four sampling phases. A softer input pulse does even better. I now need the customer to get me an actual photodiode waveform, so I can add that to the sim.

Rob did it more analytically, and got about the same answer, so we have some confidence.

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John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

jlarkin att highlandtechnology dott com 
http://www.highlandtechnology.com

I'll do some handwaving, maybe total bollocks ...

if Nyquist-Shannon is meet you should be able to get the signal in continues time with an ideal AD and an ideal reconstruction filter.

mimicicking that in in discrete time is upsampling by inserting zeros between samples and running it trough a "brickwall" filter

but since the sum of that interpolated signal is just each of the input samples times the sum of the filter impulse respense (a constant) the result is the same as if you took the sum of the original samples

-Lasse

I think it was Tim Wescott who put it somewhat this way: The original signal is reconstructed by summing sinc functions weighted and time-shifted with the samples. Since integration is a linear operation, superposition holds, and it doesn't matter if we scale and sum first and then integrate, or the reverse. If we do the latter, it's easy to see that the overall integral is simply the sum of all samples times the integral of one sinc function. There's your proof.

Jeroen Belleman

Integrals being preserved is a consequence of the signal being entirely characterised by its Nyquist rate sampled time series *IFF* it is truly band limited. There is always a trade off between time domain and frequency domain so you have to choose which one you want to have finite extent and/or functions that go to zero quickly like exp(-x^2).

In practice it is wise to oversample by at least a factor of 1.2-1.5x to avoid out of band aliasing problems with real world filters.

That is obviously false at least for a general signal. A low pass filter

*will* potentially affect the integral if it blurs a previous pulse into your integration window.

It is pretty close for an isolated pulse in a time series provided that there isn't a previous pulse with a decaying tail that blurs into the start of the next one as a result of applying the low pass filter.

Integration is itself a form of crude low pass filter so it may not be that serious. Boxcar averaging is a common quick and dirty smoothing of noisy data in realtime to allow features to be seen during acquisition.

Sinc(x) interpolation of time series is all very well from a theoretical point of view but in practice no-one ever uses it.

You have to be careful about asking the right question of the data. If you want to know the area more accurately you can sacrifice knowledge of the timing or vice versa. Often you want a decent estimate of both.

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Regards, 
Martin Brown