Quotient Sync Filter Better Than Lock In

Hey Bret,

are you talking from a theoretical perspective, or actually results?

Isn't it the case that even the timing signal has transition noise between on and off. In which perhaps double the processing power applied to considering the union of split timing signal (by 2)?

Multipling the signal+noise by a reference sine wave, and averaging the result, is mathematically equivalent to doing the exactly-best complex discrete Fourier transform and taking the in-phase component as the result. You can't do better than that.

A lot of very smart people have thought about this for a very long time, in situations where a small increase in s/n ratio is worth gigabucks.

John

This isn't some epidemiology study on the public health effects of eating broccoli.

This can be determined right here right now by first principles.

And?

Again?

Bret Cahill

"The superman is the lightening out of the dark cloud man."

-- Nietzsche

Only in situations so limited they aren't worth considering.

OK, how much money is at stake? I can put the original idea on the back burner for awhile if necessary.

Previously I just wanted to plug something into a net book and then call it a day.

Bret Cahill

He is talking from personal delusions. He has never done the tests.

So you have not done the experiments or the theory.

You means you have not done any study on this.

Yes?

So why do your signal to noise conversion instead of doing it correctly?

Well, you should at least try to follow the laws of physics. You have been wanting to ignroe them.

Yes, just hope and pray instead of actually working.

Agreed, except in most circumstances the minimum requirement is to send the input signal to *two* separate multipliers, and multiply by the reference in sine and cosine phases, then use the recovered DC values to compute magnitude. Otherwise, with a single reference multiply, you get huge errors due to signal phase changes... including getting no output at all if the signal has undergone a 90 degree phase shift.

Best regards,

Bob Masta DAQARTA v4.51 Data AcQuisition And Real-Time Analysis

Quotient sync requires knowing when the reference is near zero.

Other than that the phase angle between the signal and the ref are treated in a manner equivalent to any conventional lock in.

That drawback seems minor even in the case of low SNR as the sampling angle can be adjusted to an optimum. A larger sample angle extracts more information from the signal but increases the noise monitored when the denominator is near zero. A smaller sample angle decreases the noise when the ref is near zero but it also decreases the signal sample.

The big advantage with quotient sync is in the case of high SNR where the amplitude of the signal needs to be known to a high degree of precision.

Only the noise needs to be smoothed so the "aquisition" time is much shorther.

So if you don't know the SNR or if the SNR varies wildly, quotient sync could save a lot of time out in the field.

Bret Cahill

This is absolutely true but the traditional lockin has one multiplier and a phase shift on the reference so that you could tune for maximum output. This worked since generally the signal came from a forcing signal such as an optical chopper or a modulation signal of some sort. Now that the multiplication can be done in software, life is a lot easier.

Limiting the fraction of the time you look at a signal means a larger noise bandwidth and thus less efficiency. Then there are those unfortuate infinities you have to deal with.

Well, no, the laws of physics do not depend on the SNR.

You really have done no signal processing or analysis.

Except, of course, that it is a silly way to do anything.

So? whats that got to do with the price of fish?

I agree in principle, but saying a thing isn't the thing.

Well, for the lim x-->1 f(x) =3D (xx-1)/(x-1) .... that limit is 2... but the strange thing is at a glance one would assume undefinable.

its potential simplifies simple roots, as a sum of two composites constructed along two signal processing line {split by edge transition (instantaneous switch -slight sink difference)}. or at least so it seems in principle to me. and it eliminates the transition -of course it assume a relative constant value for transition - average where the sink difference >

transition max

was thinking (applications of two times rising-edges & falling-edge is all) separated by transition.

if you think so, but that aint it to me. The spark is the lightning.

...

Mathematically, the original multiply/integrate action of a lock in amplifier is an inner product, and produces the full value of the in-frequency signal with zero out-of-band noise. The action of a 'divide' does produce high noise sensitivity in out-of-band regions, and the proposed 'only outside a certain angle' will reduce noise BUT ALSO reduce signal.

The phase-locked amplifier with reference multiplication is better both on signal recovery (gain) and noise rejection.

Other phase-locked amplifiers use a square-wave reference, and capture a few harmonics of the fundamental; those (so-called boxcar averagers) are using a reference that goes from +1 to -1, so the 'multiply' versus 'divide' is a distinction without a difference, and they DO work, just not quite the same as sinewave multiplying units.

There's a real way to improve signal * reference type phase-locked amps, and it is to add a statistical weight function, i.e. signal * reference * W(phase, time, temperature) where W is a positive definite function that is proportional to the inverse of the expected squared amplitude of the signal measurement error (1/sigma-squared). That weight function would track any conditions that reduce the expected signal measurement error (and would, in the case of the 'divide' algorithm, pretty much undo the division, and for completely sound mathematical reasons).

If you were looking at starlight, the W function would turn your experiment off during daylight hours when lots of scattered sunlight was dominant. It's what an experimenter would do 'by hand' in that simple case.

us

A very minor loss in almost all applications.

A lot of conventional lock in applications only sample part of _one_ cycle _anyway_.

The "aquisition time" of a conventional lock in is much too long in many situations.

Apparently they aren't too concerned about losing some signal info.

I may get back into signal recovery in a few months if there's any money in it, but I don't have any noise problems at the moment.

Bret Cahill

Or, it's important to reject one phase, the 90-degree-from-reference component, and other issues are less important. The stability of a boxcar averager is quite good.

many situations.

What does that mean? The output filter of a conventional lockin is configurable. You choose your own time constants...

Huh? All the lock-ins I have seen use the full cycle... that's the referene frequency. The time constant controls the number of cycles used.

I'm beginning to wonder if maybe you are confusing two different aspects of lock-in amps. "Acquisition time" includes the time for the phase-locked loop in the lock-in to lock to the signal. The time constant controls the S/N improvement.

PLL lock time is totally unneeded if you already have the reference signal in sin/cos phases. I don't know the current situation, but for years lock-in makers seemed wedded to the idea of only using the PLL to generate the reference. Even when they provided a voltage-controlled reference, it was a simple oscillator that fed into the PLL to get sin/cos, instead of just controlling the PLL VCO directly. Duh!

So, if you are dismayed by long lock-in acquisition times from a lock-in simulator, maybe things aren't as bad as you think. Maybe they are just dutifully including PLL lock time in the simulation.

Best regards,

Bob Masta DAQARTA v4.51 Data AcQuisition And Real-Time Analysis

the

Yup, Except for some signals that last only for a short time compared to the repetition time. Then a boxcar averager (gated integrator) may be better. I never hear anything about boxcar averagers any more. I assume this is all done in software these days.

George H.

We may be using one word for two different things.

These kinds of "misunderstandings" don't happen in math.

Consider a hypothetical problem:

Two AC signals just happen to have the exact same shape and just happen to always be in sync, phase angle = 0. Only the amplitudes are different. Both have the same noise, maybe 20% the amplitude of the the first signal.

The first signal must be divided by the other for the output so both signals stand on equal basis. There is no "ref."

Without any noise the output should ideally always be positive DC, changing only as the ratio of signal amplitudes change.

In reality there is noise and dividing AC signals won't work because the denominator will approach zero with noise in the numerator. The quotient then becomes undefined, +/- infinity. In fact, the quotient would become undefined at 0 / 0 even without any noise.

Instead of sampling away from the zero intersections, as initially suggested as a solution, rectify both signals along with their noise components _then_ divide.

Again, this eliminates the humps from multiplying two in sync signals so only the noise needs to be smoothed. The time constant is lower, much much lower at low noise levels.

At high noise levels the time constant would increase but it still remain below lock in.

Rectification quotient sync is superior to lock in for every application.

Bret Cahill

It sounds like you are looking for a single constant value from two signals, the proportionality constant. In that case, instead of the mean of the quotient, best approximation of the value is the weighted average

integral( W * X/Y)/integral(W)

where W is a positive definite weight function equal to 1/sigma- squared of the X/Y value; in the case of white noise, X +/- dX and Y +/- dY this comes to W =3D ( X**2 Y**2 dX**2 + X**4 dY**2)/ Y**6

I'm not aware of any important uses of a lock-in amplifier that use less than one cycle of the reference signal; all the filtering theorems apply only to single or multiple cycles of the reference. You can track/hold at every downgoing zero crossing if you want to eliminate the 2* F ripple, but an integrator will swamp it after a few hundred cycles regardless.