Adaptive Filter Reference Constructed From the 2 Noisy Signals To Be Filtered

In one situation the two clean signals correlate by +1 and the noise in the 2 signals correlate by negative 1.

A clean reference, therefore, can be derived by adding one noisy signal to some factor times the other noisy signal.

You have the voltage signal between 2 inductors and the first derivative of current signal.

The driving voltage is between the known inductor and ground and the noise voltage is between the unknown inductor and ground.

If you want to determine the unknown inductance by taking the quotient of V/(di/dt) then the noise will be worse in the quotient than the noise in the worst signal.

The reference allows for match filtering of the signals, however.

This is new in at least one application. The question is if it is new for _any_ application.

Bret Cahill

Bret,

I will try to translate the essence of your question for my own clarity:

You have, in concept, S1 and S2, the two "clean" signals. You have, in concept, N1 and N2, the two "noises". You have in realitiy, S1 + N1 and S2 + N2 ... is that right?

You have asserted that the correlation between S1 and S2 is +1.0

You have also asserted that the correlation between N1 and N2 is - 1.0.

But, you don't mention anything about time or periodicity. Why bother? Well, I can imagine that a periodic signal will have both +1 and -1 correlation depending on the time shift in the correlation. (My assumption is that when you say "+1" that you mean the correlation

*peak* is +1).

So, if the correlation between N1 and N2 peaks at -1 then maybe it's important to know when this happens in comparison to the correlation of S1 and S2. What if the N1 and N2 peak occurs when S1 and S2 have a negative peak? Then a suitably scaled addition would result in suppression of S1 and S2 as well - although maybe not 100%.

I don't get the inductor example at all because the terms are so loose as to not make much sense. You say: "You have the vootage signal between 2 inductors and the first derivative of current signal"

I don't know what "between" means here. If they are connected in series then the voltage "between" them is zero, eh?

You say: "The driving voltage is between the known inductor and ground and the noise voltage is between the unknown inductor and ground."

I don't know the difference between a "driving voltage" and a "noise voltage".

I'm sorry to be picky but I'm trying to understand the question and the example.

Fred

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This is a pretty system so we can cut right to the chase.

Transducer 1 puts out S1 + N1 and, by the way the system behaves, transducer 2 puts out mS1 - m1N1.

where

m1 =3D known constant m =3D unknown const. to be determined.

For noise free signals just take the quotient of the signal from transducer 2 divided by transducer 1. If this is done in real time then there may be zero crossings issues. If both signals are rectified and integrated, however, you get a nice average of m over just a fraction of a cycle.

Adding noise to the signals, however, introduces an error to m. The when the noise in transducer 2 causes the numerator to err high the noise in transducer 1 causes the denominator to err low. The noise is therefore magnified in the quotient by a greater % than in either raw signal alone.

The noise is in the same band as the signal so some kind of adaptive filtering is desired.

A noise free reference is readily available simply by multiplying the signal from transducer 1 by m1 and then adding that to the output from transducer 2.

reference =3D m1(S1 + N1) + mS1 - m1N1 =3D S1(m1+ m)

There may be a phase angle between the signals which isn't an issue with match filtering.

The signals from the transducers do not need to be sinusoidal or even periodic.

The SNR is pretty high anyway, 4 - 20, so the filtering only needs to reduce the noise by a factor of 5 - 20 in most cases for 99.5% accuracy.

Bret Cahill

OK. Thanks for clarifying.

Other than frequency and phase considerations, this looks a lot like an adaptive noise canceller with a single coefficient to be adjusted.

To keep things more or less standard, I'd not add noise one place and subtract it another as long as there's a coefficient to deal with it. I'd use S + N in all cases.

So S + N1 and mS + m1N1

You put mS1 - m1N1 into the direct input (i.e. the input to the adaptive filter). mS + m1N1

The adaptive filter single weight adapts to m1. Then, the output of the adaptive filter is:

-m1( S1 + N1)

This is subtracted from the direct input:

[mS + m1N1] - [m1(S1 +N1)] = (m-m1)S1

So, I think one of us got a sign wrong here. It's a bit bothersome that S1 is multiplied by a difference but if m1 is relatively negative in comparison to m as you've suggested then it's better.

Fred

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It may have gotten lost somewhere but both noisy signals from both transducers are filtered the same way with the same reference.

After that and then rectification and smoothing, the quotient is taken.

If those are the two noisy signals from the 2 transducers, then the + sign on one of the noise terms needs to be negative.

Also, are we dropping the subscript to N? S as well as N don't really need one.

That's just to filter the numerator. (It looks like we're using my notation above again)

For the denominator the input is S1 + N1.

If that's the output to transducer 2 then that + or the + in the other transducer would need to be negative for the -1 correlation for noise.

er.

m and m1 are just two unrelated positive constants with the same units.

Bret Cahill

...

Correlated constants?

Jerry

--
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The "noise cancellation" takes place in the creation of the reference. The reference is the only unique thing about the filter. After that it's no different than any other reference based filtering, match filtering or phase sensitive rectification.

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We know that the noise in each signal correlates by -1 so the signals must be added after one signal is first multiplied by a factor to create a noise free reference.

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Well, I guess that's what got me. Normally variables can be positive or negative. So, why not negative m1 and S+N type notation?

Denominator? Where'd that come from?

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m and m1 are constants.

In the circuit problem -- which is probably academic but serves to illustrate how this can be used for filtering -- the goal is to measure an unknown inductance.

Inductance is always positive. It's best to keep everything kosher. Going to a negative inductance may work in some cases but it could introduce problems down the road.

The only purpose is to get an accurate measurement of inductance. One sensor measures voltage and the other current.

Taking the quotient of voltage / di/dt =3D inductance.

where:

di/dt =3D the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance =3D (voltage * ref)/((di/dt) * ref)

where * represents match filtering (multiplication in the frequency domain) or phase sensitive rectification.

Any scalars in the ref cancel out in the quotient so there's no reason to worry about the magnitude of the ref.

It's important to note that this is a new filtering approach only with respect to how the reference is created/derived.

Bret Cahill

Whatever .... I wasn't addressing the inductor example because I hadn't got that far yet. I was awaiting better description - as mentioned earlier. So this comes as a change in the subject. I don't think that limiting constants to positive values is particularly useful if it gets in the way of clear understanding.

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You mean a more "general" statement?

Supposing it doesn't exist?

There is at least one more real world application where the constant is always positive.

If anyone can come up with more applications, it would be most interesting.

I've never been able to find anything like it myself.

Bret Cahill

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Well, I was viewing it as a coefficient in an equation. If the coefficient has to be positive then so be it. But the math doesn't require it. I still prefer S + N as a general form which can be extended to mS1 + m1N1. m and m1 are coefficients which might be positive or negative.

I did ask very specific questions about the inductor model and didn't get any answers. I'm still unclear what the schematic / circuit diagram is / was intended to be. So I still can't comment....

Fred

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Very simple. Two inductors in series. An unknown signal driving voltage is between the known inductance, L1 and ground.

And an unknown noise generating voltage is between the unknown inductor, L, and ground.

Both voltages fluctuate aperiodically with some overlap in band width.

The 2 transducers measure,

1. voltage, V, between the node between the two inductors, and,
2. current, i, in the circuit.

To determine the unknown inductance just divide V by the 1st derivative of current, di/dt.

To filter the noise create the reference, V + L1 * ( di/dt) which equals the driving voltage and then use it to match filter or PSR both signals.

Bret Cahill

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Which end of the inductor? Is the other end grounded?

If the noises aren't from the same source, they don't correlate at all.

Jerry

--
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The reference can be calculated in either the time or frequency domain.

Since you have to take the Fourier transform of each signal anyway when match filtering it saves some time to calculate the reference in the frequency domain, 2 FTs instead of 3.

Bret Cahill

...

What does "A voltage is applied between L1 and ground." mean? What are the ends of L1 connected to?

Hmm. The two inductors are in series. What does the circuit look like? I can guess, but You haven't told me yet, so I don't know. And what is a noise-generating voltage, anyhow?

But only one of them generates (induces?) noise.

A voltage between the node and what? Voltmeters have two leads.

No. V is RMS voltage. You want the instantaneous voltage, v. What is the second inductor for?

What noise? v = L*di/dt whether v is noisy or not. The only noises that van disturb the measurement are the sensor noises. There is no hope that they will be correlated.

Jerry

--
Engineering is the art of making what you want from things you can get.

The circuit is a simple loop:

Ground -- Vs(t) -- L1 -- L -- Vn(t) -- Ground

Vs(t) is the unknown clean signal.

Vn(t) is unknown uncorrelated noise.

L(1) is the known inductor

L is the unknown inductor to be determined.

Vm(t) is the voltage measured at the node between L1 -- L and ground. (Not shown)

i is the current in the loop.

If you know

1. the voltmeter voltage Vm(t) measured between ground and the node between the inductors.

1. the current i through the loop

2. the noise, Vn(t) = 0

then it's easy to determine L:

L = Vm(t)/(di/dt)

(except near crossings)

If Vn(t) is significant and in the same band as Vs(t) then the noise from Vn(t) can be filtered by calculating Vs(t) as a noise free reference:

Vs(t) = Vm(t) + L1(di/dt) = reference

For phase sensitive rectification,

Integral [Vm(t) * (Vm(t) + L1(di/dt))] / Integral [(di/dt) * (Vm(t) + L1(di/dt))] => L

Bret Cahill

I read that as ground -- L1 -- L -- ground, with Vs(t) and Vn(t) referenced to ground. Where are their other ends connected?

Not correlated to what?

Whet use does it have?

Is Vm(t) the same as Vn(t)? If not, where is it in your scheme?

Since your circuit is a loop with two nodes (one of them ground), there is only one place to measure any voltage. The same voltage is across both L1 an L. What causes it?

How do you measure or compute di/dt?

I wouldn't presume to tell you that you don't know what you're talking about. I can say with confidence that I don't know what you're talking about.

Jerry

--
Engineering is the art of making what you want from things you can get.

Feel free to start another thread if you want a circuit w/o voltage or current sources.

The circuit on this thread is:

Ground -- Vs(t) -- L1 -- L -- Vn(t) -- Ground

You get 3 guesses and the 1st 2 don't count.

1. Some circuits have to have it.

1. Without it then Vm(t) would =3D Vs(t) which may be more expensive and less accurate to measure than with L1.

2. When Vm(t) =3D Vs(t) then Vm(t) does not need to be filtered and it can be used a reference to filter di/dt but this isn't as interesting as with L1 in the circuit.

Not as long as L1 is between the 2 voltages.

Vm(t) is measured between ground and the node between the two inductors

You think the entire circuit is at 1 voltage?

Do you mean the same voltage _drop_? The voltage drop over each inductors will generally be different.

It's plugged into something.

Analog or digital?

Bret Cahill