Adaptive Filter Reference Constructed From the 2 Noisy Signa

[Bret Cahill]

only S was broadband seismic noise, and N1 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

 

[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

 

[Fred Marshall]

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

 

[Fred Marshall]

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

 

[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

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

 

[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

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

 

[Jerry Avins]

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.

 

[Fred Marshall]

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

 

[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

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.

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?  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:

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.

= (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- Hide quoted text -

- Show quoted text -

 

[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

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.  

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?  Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance = (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

 

[Fred Marshall]

On 9/19/2011 1:48 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.

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? Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance = (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.

Fred

 

[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

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.

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?  Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance =  (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.    

You mean a more "general" statement?

Supposing it doesn't exist?

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.

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

 

[Fred Marshall]

On 9/19/2011 11:03 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.

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? Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance = (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.

You mean a more "general" statement?

Supposing it doesn't exist?

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.

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




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

 

[Jerry Avins]

On 9/20/2011 11:10 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.

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? Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance = (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.

You mean a more "general" statement?

Supposing it doesn't exist?

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.

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

Well, I was viewing it as a coefficient in an equation.

Equations aren't created in a vacuum. They generally come from
somewhere.

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....

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.

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

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.

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

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

 

[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

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.

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?  Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance =  (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.

You mean a more "general" statement?

Supposing it doesn't exist?

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.

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

Well, I was viewing it as a coefficient in an equation.  

Equations aren't created in a vacuum. They generally come from
somewhere.

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....

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

 

[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

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.

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?  Where'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 = inductance.

where:

di/dt = the 1st derivative of current

That's where the denominator comes from.

So filtering the noise in both signals with the ref

inductance =  (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.

You mean a more "general" statement?

Supposing it doesn't exist?

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.

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

Well, I was viewing it as a coefficient in an equation.  

Equations aren't created in a vacuum.  They generally come from
somewhere.

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....

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.

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

 

[Jerry Avins]

On 9/20/2011 11:10 PM, Bret Cahill wrote:

...

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

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

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

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?

Both voltages fluctuate aperiodically with some overlap in band width.

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

The 2 transducers measure,

1. voltage, V, between the node between the two inductors, and,

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

2. current, i, in the circuit.

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

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

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.

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.

 

[Bret Cahill]

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.

2. the current i through the loop

3. 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

 

[Jerry Avins]

On 9/22/2011 1:05 AM, Bret Cahill wrote:

The circuit is a simple loop:

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

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

Vs(t) is the unknown clean signal.

Vn(t) is unknown uncorrelated noise.

Not correlated to what?

L(1) is the known inductor

Whet use does it have?

L is the unknown inductor to be determined.

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

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

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.

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?

2. the current i through the loop

3. 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

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.

 

[Jerry Avins]

On 9/22/2011 5:16 PM, Bret Cahill wrote:

The circuit is a simple loop:

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

I read that as ground -- L1 -- L -- ground,

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

with Vs(t) and Vn(t)
referenced to ground. Where are their other ends connected?

Vs(t) is the unknown clean signal.

Vn(t) is unknown uncorrelated noise.

Not correlated to what?

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

L(1) is the known inductor

Whet use does it have?

1. Some circuits have to have it.

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

3. When Vm(t) = 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.

L is the unknown inductor to be determined.

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

Is Vm(t) the same as Vn(t)?

Not as long as L1 is between the 2 voltages.

Between which two voltages? For a circuit with only two nodes (that one
of them is ground doesn't matter) there is only one voltage to measure.

If not, where is it in your scheme?

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

Agreed. That is the only place a voltage can be measured.

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.

Since your circuit is a loop with two nodes (one of them ground), there
is only one place to measure any voltage.

You think the entire circuit is at 1 voltage?

Yes. Don't you? I said I didn't know what you were talking about. A
picture would help.

The same voltage is across
both L1 an L.

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

Is there a difference between a voltage and a voltage drop?

What causes it?

It's plugged into something.

So there are more than two nodes! I need a picture.

2. the current i through the loop

3. 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

How do you measure or compute di/dt?

Analog or digital?

Either way works. I asked you.

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