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

> > only S(n) was broadband seismic noise, and N1(n) was correlated noise.
>
> 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.

Not that that is settled it is time to get back to NCIFR, noise
cancellation in fabricating a reference.


Bret Cahill

> Is this situation/solution common? �

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.

> There is at least one example in
> electronics.

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

On 9/16/2011 12:08 PM, Bret Cahill wrote:
>> Is this situation/solution common?
>
> 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.
>
>> There is at least one example in
>> electronics.
>
> 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

> >> Is this situation/solution common?
>
> > 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.
>
> >> There is at least one example in
> >> electronics.
>
> > 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. =A0The question is if it is ne=
w
> > 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?

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

On 9/18/2011 1:27 PM, Bret Cahill wrote:
>>>> Is this situation/solution common?
>>
>>> 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.
>>
>>>> There is at least one example in
>>>> electronics.
>>
>>> 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?
>
> 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 = known constant
> m = 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 = m1(S1 + N1) + mS1 - m1N1 = 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

> >>>> Is this situation/solution common?
>
> >>> 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.
>
> >>>> There is at least one example in
> >>>> electronics.
>
> >>> 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 quotien=
t
> >>> 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. =A0The 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 clarit=
y:
>
> >> 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?
>
> > 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. =A0If this is done in real time
> > then there may be zero crossings issues. =A0If 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. =A0The
> > when the noise in transducer 2 causes the numerator to err high the
> > noise in transducer 1 causes the denominator to err low. =A0The noise i=
s
> > 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. =A0Thanks for clarifying.

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.

> 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

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.

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

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.

> mS + m1N1

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.

> 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)] =3D (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 bett=
er.

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


Bret Cahill

On 9/18/2011 6:36 PM, Bret Cahill wrote:

   ...

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

Correlated constants?

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

> > >>>> Is this situation/solution common?
>
> > >>> In one situation the two clean signals correlate by +1 and the nois=
e
> > >>> 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.
>
> > >>>> There is at least one example in
> > >>>> electronics.
>
> > >>> 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 th=
e
> > >>> noise voltage is between the unknown inductor and ground.
>
> > >>> If you want to determine the unknown inductance by taking the quoti=
ent
> > >>> 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. =A0The question is if it i=
s new
> > >>> for _any_ application.
>
> > >>> Bret Cahill
>
> > >> Bret,
>
> > >> I will try to translate the essence of your question for my own clar=
ity:
>
> > >> 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?
>
> > > 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. =A0If this is done in real time
> > > then there may be zero crossings issues. =A0If 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. =A0Th=
e
> > > when the noise in transducer 2 causes the numerator to err high the
> > > noise in transducer 1 causes the denominator to err low. =A0The 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 fro=
m
> > > 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. =A0Thanks for clarifying.
>
> 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.
>
> > Other than frequency and phase considerations, this looks a lot like an
> > adaptive noise canceller with a single coefficient to be adjusted.

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.

> > 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
>
> 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? =A0S as well as N don't really
> need one.
>
> > You put mS1 - m1N1 into the direct input (i.e. the input to the adaptiv=
e
> > filter).
>
> That's just to filter the numerator. =A0 (It looks like we're using my
> notation above again)
>
> For the denominator the input is S1 + N1.
>
> > mS + m1N1
>
> 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.
>
> > 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:

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.

> > [mS + m1N1] - [m1(S1 +N1)] =3D (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 i=
s
> > relatively negative in comparison to m as you've suggested then it's be=
tter.
>
> m and m1 are just two unrelated positive constants with the same
> units.
>
> Bret Cahill- Hide quoted text -
>
> - Show quoted text -

On 9/18/2011 3:36 PM, Bret Cahill wrote:
>>>>>> Is this situation/solution common?
>>
>>>>> 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.
>>
>>>>>> There is at least one example in
>>>>>> electronics.
>>
>>>>> 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?
>>
>>> 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 = known constant
>>> m = 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 = m1(S1 + N1) + mS1 - m1N1 = 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.
>
> 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.
>
>> 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
>
> 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.
>
>> You put mS1 - m1N1 into the direct input (i.e. the input to the adaptive
>> filter).
>
> 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.
>
>> mS + m1N1
>
> 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.
>
>> 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.
>
> m and m1 are just two unrelated positive constants with the same
> units.
>
>
> Bret Cahill
>
>

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?

fred

> >>>>>> Is this situation/solution common?
>
> >>>>> In one situation the two clean signals correlate by +1 and the nois=
e
> >>>>> 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.
>
> >>>>>> There is at least one example in
> >>>>>> electronics.
>
> >>>>> 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 th=
e
> >>>>> noise voltage is between the unknown inductor and ground.
>
> >>>>> If you want to determine the unknown inductance by taking the quoti=
ent
> >>>>> 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. =A0The question is if it i=
s new
> >>>>> for _any_ application.
>
> >>>>> Bret Cahill
>
> >>>> Bret,
>
> >>>> I will try to translate the essence of your question for my own clar=
ity:
>
> >>>> 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?
>
> >>> 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. =A0If this is done in real time
> >>> then there may be zero crossings issues. =A0If 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. =A0Th=
e
> >>> when the noise in transducer 2 causes the numerator to err high the
> >>> noise in transducer 1 causes the denominator to err low. =A0The 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 fro=
m
> >>> 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. =A0Thanks for clarifying.
>
> > 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.
>
> >> Other than frequency and phase considerations, this looks a lot like a=
n
> >> 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
>
> > 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? =A0S as well as N don't reall=
y
> > need one.
>
> >> You put mS1 - m1N1 into the direct input (i.e. the input to the adapti=
ve
> >> filter).
>
> > That's just to filter the numerator. =A0 (It looks like we're using my
> > notation above again)
>
> > For the denominator the input is S1 + N1.
>
> >> mS + m1N1
>
> > 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.
>
> >> 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)] =3D (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 b=
etter.
>
> > m and m1 are just two unrelated positive constants with the same
> > units.
>
> > Bret Cahill
>
> Well, I guess that's what got me. =A0Normally variables can be positive o=
r
> negative. =A0

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.

> So, why not negative m1 and S+N type notation?

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.

> Denominator? =A0Where'd that come from?

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