field test unit no 001 "model T"

[Qiaozhi]

Now explain how you did it

Did what?

[Davor]

How did you create and place such neat formulas on the forum? I tried something like this and it was a royal pain.

[Aziz]

LaTeX!
https://en.wikipedia.org/wiki/Formula_editor
https://en.wikipedia.org/wiki/LaTeX

------------------------
Hi Q,
you have impressed me now. Would you be so kind and make the missing proof now? (The subtraction of integrated windows: a HP-filter?)

Cheers,
Aziz

[Davor]

I'm not completely untouched by latex, but I'd never expect posting so many formulas so neatly in a forum to be anything but a royal pain.
Thanks!

[Qiaozhi]

I'm not completely untouched by latex, but I'd never expect posting so many formulas so neatly in a forum to be anything but a royal pain.
Thanks!

You're not kidding, it was a pain! It's so easy to make a typo, and very difficult to spot in the Latex code.
I've already had to go back and correct a couple of mistakes (one of the advantages of Admin privileges).

[Qiaozhi]

In case anyone is befuddled by the mathematical gobbledygook, here's a simpler explanation of the paper by John Alldred:

All metal targets have an associated decay constant. This is usually represented by the Greek letter (tau). When a PI detector switches off its transmitter, a large magnetic pulse is generated that causes eddy currents to flow in any nearby metal object. These eddy currents die away over time in an exponential manner, but the time taken for these currents to decay depends on the target's material, size, shape and orientation. In general though, small gold nuggets have a very short decay constant, whereas a horse shoe (for example) would have a very long decay constant.

John's paper explains how the voltage at the output of the sample gate / integrator is affected when different sample pulse widths are used. Two basic (approximate) equations are derived in the paper to describe the extremes of the sample pulse width settings:

where ( << ) ..... (Eq.1)

where ( >> ) ..... (Eq.2)

Equation 1 refers to the case where the main sample pulse width is much shorter than the target delay constant.
Equation 2 refers to the case where the main sample pulse width is much longer than the target delay constant.

With reference to equation 1, and assuming that (the main sample delay), (the target decay), and T (the pulse period) are all fixed; then V is mainly affected by the choice of (the sample pulse width). Remember that equation 1 is only true when is much shorter than . If the target changes to one with a very short decay time (such as a gold nugget) then may become close to or even longer than . In which case this approximation will no longer be valid.

Equation 2, on the other hand, is only valid for the condition where is much larger than . In this case, V is mainly affected by (the target decay), and the value of is irrelevant. You can see that is not even present in the equation.

However, we must also consider (in equation 1) that the use of a small value for means that the amplitude of V will also be reduced, and this may outweigh any advantage provided by removing the dependency on variations in .

[moodz]

LaTeX!
https://en.wikipedia.org/wiki/Formula_editor
https://en.wikipedia.org/wiki/LaTeX

------------------------
Hi Q,
you have impressed me now. Would you be so kind and make the missing proof now? (The subtraction of integrated windows: a HP-filter?)

Cheers,
Aziz

The proof is redundant as the the assumptions for sampling an exponential decay curve are wrong.It works in the same way a crystal set can be called a radio. BTW if you are looking for the "proof" just google relationship between time domain series and fourier transform and some random search term like "sum of integrals". The physical implementation can be seen in analysis of state variable filter using integrators or just read the Wikipedia entry for high pass filters ie .. "Discrete time realization" .... Publishing such a proof will only add more confusion to a building built on flawed foundations ( Torre pendente di Pisa ... ). Just my advice .... to find the answer you must look at your CRO.

[Aziz]

The proof is redundant as the the assumptions for sampling an exponential decay curve are wrong.It works in the same way a crystal set can be called a radio. BTW if you are looking for the "proof" just google relationship between time domain series and fourier transform and some random search term like "sum of integrals". The physical implementation can be seen in analysis of state variable filter using integrators or just read the Wikipedia entry for high pass filters ie .. "Discrete time realization" .... Publishing such a proof will only add more confusion to a building built on flawed foundations ( Torre pendente di Pisa ... ). Just my advice .... to find the answer you must look at your CRO.

Hey Moodz,

I would be satisfied with a simplified proof by example like an excel sheet. As I have already mentioned, don't look at the exponential decay curve! We have only low frequency sine wave EMI noise here (EF, 50/60 Hz means hum, ..) and the proof isn't reduntant. It is very important. Comeon, you could do it.

I'm working on my own "Torre pendente di Pisa" (the so-called Tophat coil ).
Cheers,
Aziz

[Davor]

OK, let me try finishing the proof thing once and for all. You can observe a PI receiver with late sampling as a state variable filter. it consists of a summing node and two integrators in a sequence with feedbacks and a summing node. Now, if you can't see how these two integrators are put into sequence, go back to George's neat formulas and find it yourself. In short, late sampling per se is the form of integration because it decimates high frequency content. In effect it is a Nyquist filter with cut-off frequency at 1/t (this is actually a key point here). When such signal is integrated, and fed back to the input - you have a state variable high pass filter.

A state variable filter in it's straightforward form is as follows:



And it can be put into a formula as:



(different authors use notations H(s) or G(s), where s is jω)

The second order HPF can be presented as:



As seen from the previous formula, b1 and b0 are not summed, and the factor a1 determines the Q of the filter. In case of Q being 0.71 we have a familiar Butterworth response, which is equivalent to a first order high pass filter. The Nyquist thing in this procedure is responsible for the filter Q. For the same reason a faster first sample has more of the high frequency content, and with this state variable ... thing the output is actually band pass filtered: Nyquist sets the low pass (t1) and state variable (Nyquist of t2 + integrator) sets the high pass.

Q.E.D.

P.S. whoever wishes to play with the state variable filters may do so in excel sheet that I made a slight facelift, but all credits to Mark Williamson. Link to his state variable filter page is in the sheet.

[Aziz]

Bravo Davor! Good on you.
(I personally would make my life easy by just simplifying the problem. Anyway.)

You took some steps forward but it's ok to know it now.
Thank you.
Aziz

PS: I didn't make the proof on the integrals (window integration and subtraction). But my gut did it. It's called the gut feeling. *LOL*

[Ferric Toes]

I looked at my CRO and this is what I saw with the following setup. TX pulse 60uS, delay 20uS, sample pulse 20uS, delay to 2^nd sample 100uS (Start SP1 to start SP2), and second sample 20uS. Repetition rate 5K pps.

Sine wave generator injected into RX input such that the sine wave gives pk – pk of 1V on the output of front end amplifier.. CRO connected to differential integrator output. Leaky integrator time constants 0.1uF//1M0 and input resistor 2K2. Adjusting dc offset of front end gives no change in integrator o/p with scope set to 20mV. 50Hz input give no noticeable change either. Hence 0 – 50Hz there is no pass of signal. You can of course see the ramp up during SP1 and the ramp down during SP2 but the baseline does not move. Ramps can have positive or negative values depending on the part of the cycle is being sampled. Raising the signal generator frequency one starts to see fluctuations in the baseline at about 0.5kHz. Tuning of the sig. gen. is very critical to see this but it comes and goes like a beat note on the baseline. From then on there is interference every 0.25kHz. i.e. 1.0, 1.25, 2.0, 2.5……. Severe interference occurs at multiples of the repetition frequency. This goes on right up to 200kHz although the effect seemed less at the high frequencies because you are getting integration within each sample pulse width.

This is exactly what I would expect to see from such a sampling system, whose purpose is to minimise offset, EF and powerline noise. Higher noise frequencies are best dealt with by other means. Although, by employing an adjustment of the clock frequency driving the TX and sample rate many of the problems with higher frequency CW interference can be overcome because it then becomes asynchronous. I once tried a system where the clock rate was swept over a range so that the TX/sampling pulse rate was continually changing. Results were inconclusive though.

I should point out that the detector I used for the above test has considerable filtering after the integrator i.e 1^st order LP immediately after and switched 1^st/2^nd LP later.

I’m, not quite sure what is meant by a “flawed foundation” in describing something that has worked just fine for over 40 years. Not to say that there are not other alternatives that may do things better. The reciprocating internal combustion engine could be said to be a flawed foundation for driving the wheels of a car. It is still the majority power plant.

My Magnetic Viscosity Meter uses a similar sampling and integration method but one can produce accurate graphs of the decay of ferrous and non-ferrous objects, but magnetic soil in particular.

Eric.

[Aziz]

Thanks Eric. Can we conclude, that subtracting integrated windows is a phase-sensitive high-pass filter with typical characteristics of a HP-filter up to it's cut-off frequency (-3 dB mark) and on higher frequencies, it then becomes a comb transfer function?

The phase (time) sensitivity can be eliminated, if one takes always the worst case condition.
Aziz

[Davor]

Please note that the Nyquist filter has a comb transfer function. We are talking about the same thing.

[Aziz]

BTW guys,

one can code a small C/C++ program and can do all the demodulator steps in software (simulation) and can evolve the transfer function easily. You have to make a Monte-Carlo method (varying frequency and phase of the low frequency EMI) and have to take the worst case output. Voilà, that's it.
This is what I would do, if I don't know, how to do it.
Aziz

[moodz]

yawn ...