signals and systems question

Here's your key:

What's the ideal reconstruction? A brick wall filter. AKA sinc interpolation. Suppose you up-sample, using that interpolation. Then summate (= discrete integrate) the (now much more numerous) samples. Then you'll get the correct result.

Alternately, you can roll those together into an integral(sinc) interpolation function. Sadly, the sine integral function Si(x) = \Int_0^x(sin t dt / t) doesn't have a closed-form representation in elementary functions, so you'll probably end up with a finite polynomial approximation, optimised for a dt = Ts sample rate.

Or you can ignore the math, and do the first thing. As you're fond of saying, processing is cheap.

Tim

--
Seven Transistor Labs 
Electrical Engineering Consultation 
Website: http://seventransistorlabs.com

From what I've read, the commonest approach to doing this is to use a charge-sensitive amplifier... basically just an integrator, with a large-value "reset" resistor in parallel with the integrating capacitor. You end up with the peak height of the pulse being proportional to the total charge delivered.

Seems to be a pretty standard approach for PMT-based scintillation detectors, and I imagine the same trick would work with photodiode detectors as well.

How high a count rate are you looking at? Hundreds per second? Thousands? Tens of thousands?

We're committed to the digital acquisition. The question is whether the energy integral is accurate.

It's looking like passing a pulse waveform through an ideal lowpass filter, and then sampling above the Nyquist rate and summing, gives a perfect integral, independent of the phasing between the optical pulse and the ADC sample clock. The gotcha is that we don't have an ideal filter, so the error (integral vs sample phasing) will have to be simulated, using an approximation of the typical photodiode pulse. Essentially, the high frequency stuff that sneaks through the filter stopband violates Nyquist and jitters the integral.

The pre-filter photodiode pulse is currently about 5 ns FWHM, with a long tail out to maybe 100 ns. Rep rate is around 100 KHz.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement  

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

There should be plenty of stuff related to xray florescence

I've done some XRF with an adc and fpga as pulse processor but the sensor had a charge amplifier

-Lasse

I have a pulse from a photodiode, and I want to digitize the total energy in the pulse, which would be the integral of the photocurrent over the pulse duration.

We have a transimpedance preamp, a lowpass filter, and an ADC, and we'll sum the ADC samples to get the integrated energy value.

The lowpass filter is 60 MHz (passive, 5 pole, transitional Gaussian) and the ADC runs at 250 MHz, so we're running a bit over 2x Nyquist. Acording to Shannon, we should be able to (almost) perfectly reconstruct the bandlimited pulse from the ADC sample data. It's not an ideal LPF, but it's pretty good.

Seems to me that the signal area must be the same before and after the lowpass filter, mumble mumble.

What I don't know is whether the ADC samples add up to the correct area, and specifically whether shifting the time relationship between the pulse and the sample clock will change the sum. Intuitively, we think it won't, but my customer thinks it will.

Ignore noise and quantization here, and assume that we sum enough ADC samples to get comfortably into the pre- and post-pulse zero baseline.

Given that we don't have an ideal lowpass filter, I may just have to simulate this. But I never thought about the Shannon theorem in term of integrated area reconstruction.

--

John Larkin         Highland Technology, Inc 
picosecond timing   precision measurement 

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

Grab a numerical analysis text and look up trapezoidal and Simpson's rule  
integration.  You can get substantially better accuracy with not much more  
computational effort by stepping up to Simpson's rule. 

Regards, 
Carl Ijames carl.ijames aat deletethis verizon dott net

1) With no regard to the actual pulse shape and amplitude, one could have a pulse drive an integrator that has low positive or negative droop, and measure the final amplitude. Works pretty well if slew rate and gain bandwidth are adequate; an RC circuit could do that (or a slow scope if looking at pulse). 2) Betcha the shape is constant at all amplitudes of interest, meaning the width to (say) 5% on tail increases with increasing amplitude. That translates to a computable energy from the simple measurement of the peak amplitude with a S&H.

More than one way to get a cat out of its skin...

will work well for 2^k evenly spaced samples

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From what I see on Wikipedia, Simpson's rule is to weight three points with 1/6, 2/3, 1/6 for finding the area of the parabolic approximation to the function being integrated. I would expect this to be applied to sampled data my estimating the areas of either adjacent or overlapping regions of three samples. If estimating adjacent areas, each area will be three points summed as 1/3 f1 + 4/3 f2 + 1/3 f3. The end points f1 and f3 will be used in two adjacent areas so will appear in the total as

2/3 while the center points will appear as 4/3. That is a bit odd...

If estimating by summing overlapping areas, the sum for each segment will be the same, but now each point other than the end regions will be summed as an end point twice and a mid point once for a total of 6/3. Since the overlap is 50% a divisor of 2 will need to be used give a result the same as simply adding up all the samples. So I don't see any advantage.

Am I misunderstanding how Simpson's rule would be applied to this case?

--

Rick

I'm not sure, but for noise, the cap is the path to energy loss. (In an RC.)

George H.

Hmm. Interesting question.

Tim Williams was correct in that a theoretically correct solution "beat it with a sledgehammer" solution would be to take your train of impulses, feed them through a brick-wall filter at some high sampling rate (or better yet, in continuous time), and integrate the result.

If you proceed with this in continuous time, then the result of filtering your impulse train with a brick-wall filter is a weighted sum of sinc functions, offset in time, with the weights being your ADC readings.

The integral of that weighted sum of sinc functions, over all time, is simply the integral of a single sinc function (cuz the integral obeys superposition) times the sum of the weights (which are just your ADC readings).

If the sinc is the impulse response of a filter with a DC gain of 1, its integral over all time is 1. So the result of doing a "proper" reconstruction followed by a "proper" integration in continuous time is just the sum of the ADC readings. The only error that you're introducing is from the sampling process, and it is the aliasing of that bit of energy that passes through the filter from Nyquist on up.

So -- you're mostly right. Gaussian filters tend to have pretty broad tails, so I would check that the amount of energy that leaks through the filter from 125MHz on up is small enough not to mess you up -- you may find yourself unpleasantly surprised at the answer.

If you do find signal leakage, post again -- the actual cut-off frequency you need to worry about isn't Nyquist -- it's more like 190MHz or maybe

220MHz (it's the sampling rate minus the bandwidth of the integral over however many samples you're integrating, if I'm getting my math right). I don't want to expound on it here because (a) you may not care and it takes thought, and (b) I'm about to lay my head on the keyboard and fall asleep, so my results may not be reliable.

--

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

Iff the LP filter was perfect then I think it would be.

You might be able to test how much difference it really makes in practice by running a pair of ADCs at 250MHz but in quadrature digitising the same signal. Some differences between the ADCs are inevitable so you would also have to cross calibrate them first.

Shannon says you can reconstruct the original waveform perfectly from the optimally sampled time series *provided* that it is truly band limited in frequency space. Your signal will not be perfectly band limited but the difference may be small enough to hardly matter.

--
Regards, 
Martin Brown

Well he did say ignore noise so I suppose you are entitled to suggest this but it won't help at all and will only increase the noise floor.

It works perfectly on noise free analytic functions.

A related parabolic trick can sometimes be used to determine the exact time of a pulse from the digitised stream more precisely than one clock tick provided that the signal to noise is really good.

--
Regards, 
Martin Brown

Hi John,

Why not use an integrator instead of a TIA? That would remove the speed constraints on your ADC and still give you total energy. The signal to the ADC would be larger and you could use the oversampling ADC to improve precision.

Whatever happened on your National Ignition Source project?

ChesterW

Op I think there are two sources of error due to the imperfect lpf.

1. How much of he desired signal is lost in the f?
2. How much undesired signal is allowed to alias back in band by the im perfect lpf.

So do a spectral analysis of your worst case signal and evaluate the 2 values above.

Is any of the original signal "undesired" in the integration. Maybe the best answer is no lpf.

The aliased signal may be in the wrong place but If it contributes to the integration, so what.

Mark

The first question was whether the ability to reconstruct the (Shannon-compliant) signal implies that the summed samples is a perfect integral. Our response so far is "we think so." I was hoping that some serious math jock would say "of course, it's obvious from..."

We know that an imperfect lowpass filter can make the integral jitterey. How jittery is a task for simulation.

--

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

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

The signal chain is a preamp box feeding a filter/digitizer box. We also do waveform acquisition (globally, simultaneously across a big system) and some other waveform metrics, so we can't change the basic hardware. The question is how much the filter affects the repeatability of the energy integral. Concensus is that it does some.

--

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

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

I think this is a red herring. It doesn't matter if the filter is perfect as much as it matters that the signal you wish to integrate is already band limited to your filter bandwidth. If the filter is removing signal (other than noise you are trying to ignore) you are losing information and therefore likely losing content to the integral as well.

I'm not sure integrating a voltage signal is the same as measuring the energy of the signal, but my understanding is that aliasing does not alter the total energy content, it just makes the out of band energy show up *in band*. If the signal you wish to measure has out of band energy (above the Nyquist rate) I would think you would *want* it to alias into the pass band of the sampling and *not* be filtered. Perhaps the total bandwidth of the signal has to be within the Nyquist bandwidth but that doesn't mean below half the sample rate. These are individual pulses with no or very little low frequency content.

I think you can even do this in a spread sheet or MathCAD to determine the effects of a sub-optimal filter and the sample rate given a pulse shape and frequency content. No need to build anything to test the concept.

--

Rick

I think you are on target here. There is a need for the bandwidth of the signal to be restricted to the Nyquist bandwidth, Fs/2.

Hmmm... giving it a bit more thought, here is a thought experiment...

Signal content at Fs/2-A and and equal signal content at Fs/2+A. The content at Fs/2+A will alias to Fs/2-A. If they are in phase they add and you measure the full signal. If they are out of phase they subtract and give you zero result. :(

So it looks like the signal has to be fully within the range of 0 to Fs/2 or any one of the aliased images.

But if you filter the pulses to restrict the bandwidth, you will completely lose the content being filtered and will get a distorted result regardless of how "perfect" the filter is.

--

Rick