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

Autocorrelation and Kalman filtering are a couple of examples.

Cheers

Phil Hobbs

-- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics

hobbs at electrooptical dot net

I don't know what you mean by "2 noisy signals to be filtered".

Are you suggesting that there are 2 signals of interest that will each be filtered using different adaptive filters? That would be one interpretation in which case asking about 1 signal will do fine.

Or are you suggesting that there are 2 signals and you want to filter one of them and might use the other as a reference?

Both solutions are common.

The first might be an adaptive line enhancer or ALE in which there is 1 signal in and 1 filtered signal out. There is no "reference" really, just a delayed version of the input so that there's no correlation of the noise from the direct input and the delayed input. Then one or the other is adaptively filtered to minimize the difference between the direct input and the delayed/filterd version of it. The output of the adaptive filter (ahead of the differencer) is the output. This tends to create a comblike set of bandpasses in the adaptive filter that passes the periodic parts of the input.

The second might be an adaptive noise canceller (ANC) where there are two inputs:

- one is the "signal of interest that's perturbed by sinusoidal noise ... "interference".

- the other is the best capture of the perturbing sinusoidal noise (the interference) that's possible to get. This is called the "reference".

Then, the reference is filtered so that the difference between that and the signal of interest is minimized. If it's minimized then the best possible job of removing the interference.

Fred

Same reference and same filter for both signals.

The noise in one correlates by negative 1 to the noise in the other so the sum of one plus some factor times the second signal yields the reference.

In one situation one signal is pretty clean and the other noisy so the noisy signal can be filtered with the clean signal alone.

For uniformity or balance the clean signal should be filtered with itself.

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I'll check it out.

Bret Cahill

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You need a "noise-alone" signal for it to work in an ordinary adaptive filter. For speech this usually means a voice-activity detector. Failing that you might try Independent Component Analysis if you know the PDF of the signals.

Hardy

Something similar:

Seems to be able to work on any kind of noise, not just sinusoidal, with some kind of correlation of the noise in both signals.

If you knew the signal correlation was +1 and the noise correlation was -1, however, you should be able to exploit the 2nd fact for the best filter. Just add one signal to a known factor times the other signal. That reference could then be used to match filter or otherwise process both signals.

Bret Cahill

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I forgot to add that both clean signals correlate by +1.

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This filtering situation is possible in at least one [probably academic] electronics situation.

Determining the inductance of an inductor with a fluctuating voltage by dividing that voltage by the 1st derivative of current.

The unknown inductor is wired to an inductor with a known inductance and the voltage is measured between the two inductors.

An unknown randomly fluctuating voltage source, in the circuit between ground and the unknown inductor, is the noise that appears in both signals.

The noise in the signal voltage goes up when the noise in the derivative of the signal current goes down so the noise in one signal correlates by -1 to the noise in the other signal.

The noise in the quotient, of course, is worse than the noise in either signal.

But if you add the noisy voltage to some constant times the derivative of the current then you get a clean reference for both signals.

Bret Cahill

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It would be pretty nifty to get a reference named after me.

On the other hand references may be like tornadoes and not get names.

Bret Cahill

Well, I don't think that "it works on any kind of noise, not just sinusoidal" unless you make some rash assumptions that don't hold well in a number of practical situations. I'm not saying that it *never* happens but:

A reasonable model is that the "noise" is made up of broadband components and spectral "lines" or sinusoids. It's easy for the sinusoidal components to correlate. It's not so easy for the broadband parts to correlate unless there is very low time delay between the reference and the signal to be cleaned up.

Noise cancelling headphones work because there is very low time delay between the "noise" and the headphone active output. In that way, broadband noise can be subtracted because it's, if you will, highly correlated. And such implementations aren't even "adaptive" as such.

But, in many other practical situations where adaptation is warranted, there is quite a delay between the reference and the signal. In this case there is no hope of reducing broadband noise because uncorrelated broadband noise cannot subtract from other broadband noise - it only adds.

So, in the paper you reference, I think what they mean by "correlated" noise is that there are sinusoidal parts - although I don't see that they mention that - and the rest is likely broadband AND uncorrelated.

Interesting that the paper you cite has as first reference the paper by Widrow et al. I worked with McCool, Hearn and Zeidler when their paper was published and got some first hand insights at the time.

Fred

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But Fred, your example is for active noise cancellation (ANC) and its physical limitations, not limitations of the algorithm (IIRC ANC uses Filtered-X not the configuration of the paper). I have put delays in the reference path to ensure correlation (causality) many times using broadband noise.

Maurice

Maury,

I can't imagine that we disagree. The original question is about active noise cancellation. I don't know of any other kind of *adaptive* noise cancellation or perhaps I have some terms off?

I don't get what you mean by "physical limitations" and not "limitations of the algorithm". As far as I'm concerned, "the algorithm" is all about how one adapts the filter and not much about how the overall system works. I wasn't addressing the algorithm at all.

The simple block diagrams in my head pretty much agree with the student paper that was referenced. In Figure 1, they show an ANC. (I don't know what your "X" is in "filtered-X") The point of that diagram is that there is: S + No ... a signal of interest plus some noise and N1 ... a version of No that would ideally be high "SNR" for the noise and not perturbed (have components of) S.

In this case the stuff that I'm familiar with did not have the luxury of having the broadband parts of No be correlated with the broadband parts of N1. So the adaptive filter "shuts off" in frequency bands where that's all there is. So, it effectively becomes a set of bandpasses for the sinusoidal components in N1 whose amplitudes and phase are adjusted by the adapted filter to do the best possible job of cancelling their presence in No.

But, if there were good correlation between the broadband parts of No and N1 then a simple delay and scaling might be just the thing - just like in the noise cancelling headphone case. In that case I'm sure that there *is* an "adaptation" of sorts:

- first you start out with an inversion

- then you scale to match the headphone characteristics. OK - it's one-time and manual but that works. And, of course, just as needed for the Figure 1 diagram to work well, the signal components in N1 are either zero or very low.

Did I miss something?

Fred

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The confusion here is probably due some less than clear terminology, i.e., using one word for two different things.

In the OP it is suggested that a clean reference for use in, say, match filtering, can be made in some situations by adding one noisy signal to another noisy signal times some factor.

We know the clean signals correlate by +1 and the noise in the signals by -1. It's easy to create a nice clean reference from such signals on SPICE by adding the voltage to the first derivative of current times an inductance.

Any "noise cancellation" -- if that's what you want to call it -- is only in cobbling together the reference.

The noisy signals are then sent with that reference to an adaptive filter to reduce the noise but I wouldn't call that step of the process "noise cancellation."

Cancellation of noise in a reference is not really the same thing as cancelling the noise in the signals, although it leads to the same overall goal

Bret Cahill

The mind invents new things more easily than new words to describe those things. That's why so many misleading terms appear in language.

-- Tocqueville (pointing out that the U. S. constitution isn't for a federal, but an "incomplete national government.")

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Fred, It's probably all trems. ACTIVE noise cancellation refers the the generation of a physical anti-noise (using speakers, transducers, etc.) to effectively cancel out the unwanted noise.This is what is used in headsets, also around machinary, and so forth. ADAPTIVE noise cancellation is the reduction of noise on a signal. With active noise cancellation there can't be much delay because the anti-noise must be generated *soon enough* to be an effective cancellation signal

What I was refering to was your statement

With adaptive noise cancellation (which is waht thw Widrow paper is addressing) broadband noise can be cancelled, even if the delay is long, by putting in a delay register to force the signals to be within the adaptation filter range.

So when you said *adaptation* and referred to the Widrow paper, ADAPTIVE noise cancellation is what came to mind.

Maurice

Well OK but that requires some pretty big constraints on the signals.

In my experience, in adaptive noise cancellation with a reference, there is NO correlation between the broadband noises. So you can delay all you want and on either channel and no improvement.

And, you will notice that there is no "delay register" in the ANC block diagram.

So there must be something about the noise in your reference and the noise in your signal that allows that delay to have a positive impact. What is it? I'm curious.

Fred

Let's say I have a noisy signal, but I have a bit of the noise available. I can use the noise sample as the reference, the noisy signal as the input, and cancel the noise. I just need to make sure the noise reference doesn't get to the summer before the output of the adaptive filter. Thus the delay register in the reference. The delay can be as long as what is needed to ensure causality.

I have used this to actually reduce correlated signals to measure the broadband noise (see patents 7603258, 6725705 for details). In this case, the noise was the seismic rumbling from trucks, cars,clicking from womens' high-heeled shoes, etc., and the broadband seismic noise from underground sources was the desired. The adaptive filter produced an estimate of the noise. Instead of using the error signal (which is normally the case) , use the adaptive filter output.

There is no constraint on Widrow's LMS filter for the signals to be correlated signals. The constraint is that the signal and the noise can't be correlated. The trick is to determine what is the input, what is the reference, and how do I get the reference.

Maurice

How does the adaptive filter give an estimate of the noise? Does it require a "noise" reference?

Then it would be wasting information when processing signals that are correlated.

In the case where the correlation of the signals is +1 and the noise is -1, the reference depends on what is known.

Bret Cahill

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Bret, What started this particular part of your thread (sorry, your thread got a bit hijacked :)) is the idea that you can't have long delays with broadband noise in an adaptive filter. In the last example I gave, the desired signal is not the correlated stuff, but the noise. We weren't exactly addressing the particular configuration you started with, but it leads back to it.

If I have a boadband noise signal currupted with correlated signals, when I remove the correlated signals, I have the wanted noise remaining. In the case of *adaptive noise cancellation* (what Widrow calls adaptive interference cancellation) the signal is actually the reference, and the noise is the input to the adaptive filter. This is the model you showed in the paper. A good analysis of this is in Widrow's book adaptive signal processing, chapter 11 or 12 IIRC (I think it's also in the paper he wrote with McCool, et. al.). Widrow calls this adaptive interference cancellation, rather than adaptive noise cancellation.

With respect to long delays, with the model you referenced, if the signal source, S + No gets to the filter before the noise source, N1, then the filter will not adapt (non-causal). The remedy is to place delays in the signal line to force causality. It doesn't matter if the delay is long.

In the paper you referenced, I believe the S(n) + No(n) label should be after the summation in the signal line. The signal is corrupted with noise from N1(n), but the noise N1(n) gets convolved with some unknown impulse response to become No(n). The goal is to estimate this unknown impulse response so that the sample of noise available, N1(n), can be used to reduce the noise level on S(n). This is precisely the problem I had. I had a seismic sensor corrupted with noise. The noise was the seismic rumbling of trucks, cars, etc. The problem was that the rumbling was convolved with the soil impulse response before it got to the sensor. This is exactly the model in your reference paper, only S(n) was broadband seismic noise, and N1(n) was correlated noise. To further complicate things, the seismic sensors needed to be placed a distance from the noise reference, which meant the noise was on the input of the adaptive filter before the reference. I needed to place a large delay in line with the reference to make the system appear causal. Most people are used to seeing the sample noise as the reference, rather than the input to the adaptive filter.

Hope I'm clearing this up rather than making it more obscure.

Maurice

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So the noise reference still had _some_ signal in it, it just had a much lower SNR?

That approach may be the best thing to do in what seems like a bad situation.

I was determining the quotient of two noisy signals, SNR > 4 - 20, in a few seconds or one or a few cycles with an accuracy of 99.5% half the time.

The 2 signals w/o the noise correlate 100% and the noise in one signal correlates by negative 1 to the noise in the other signal.

The factor to multiply one signal so that that signal has the exact negative of the noise in the other signal is known so it's easy to create a noise free reference by adding one signal to the other signal times that factor.

Then you can use match filtering or, if you have the time, phase sensitive rectification to get the magnitudes.

Bret Cahill

That's an ANC .. the noise that's removed is correlated. What's different? If there's a delay then one has to deal with it .. so that's fine. I'm missing something here it seems.

It seems that I've said that one can't remove uncorrelated noise and you've said that one can remove correlated noise. We're both correct.

Fred