Phase-selective filters

Hi all. 

I've been playing with a couple of ideas regarding extractions of 
signals from complex and complicated raw data. The overall experiment 
setup is that two sensors monitor the same environment such that 
both sensors monitor some source of interest, while each sensor monitor
unique spurious sources + unique observation noise. 

Block diagram:


      N1   S1         S         S2  N2    
       |    |         |         |   |
       V    V         |         V   V
     ----------       |       ----------   
    |    R1    |<-----+----->|    R2    |
     ----------               ----------

Each reciever R1 and R2 measures some raw data that can be separated as

   Dn = S+Sn+Nn

where 

  Dn   is the total signal measured by sensor Rn, 
  S    is the "useful" signal to be extracted that is measured by both 
       sensors R1 and R2 
  Sn   is some "signal-like" component that is of no interest, that is 
       only measured on sensor n (i.e. reciever R2 don't measure S1 and 
       vice versa)
  Nn   is random observation noise that is unique to each sensor, i.e 
       the noise is uncorrelated with the signal components Sn and S, 
       and uncorrelated between the sensors. 

The problem is to extract the "mutual component" S from the data, given 
no other information than that the signal is measured by both recievers 
R1 and R2. If S and Sn are known to be narrow-band spectrum lines (with 
no knowledge of frequency, amplitude or phase), I can do that. 

It's some sort of "implicit de-noising" based on the signal model being 
linear. I think the method should be possible to extend to an arbitrary 
linear signal model, i.e that the measured data is expressed as 

          K
   Dn =  sum ank*Ek + noise
         k=1

where Ek is some basis vector, "ank" is the amplitude of component Ek 
as measured by sensor Rn, and the noise is uncorrelated with the signal 
and between sensors. 

Now, in some applications one would like to separate the signals further 
with a constraint on phase, i.e. that one can assume that if the phase 
relationship between the measurement of S made by R1, call it S12, and 
the measurement of S made by R2, call it S21, is 

                                  0 deg
    arg C12=arg E[S12'*S21] = +/-90 deg  
                                180 deg

for various measurements. In some analyses I'm interested only in 
measurements (waves) that exhibit a +/- 90 deg phase relationship, in other 
cases I am intersted in all the other components (waves). 

So, at last, my question: 

Does anyone know whether ther exist filters that discriminate components 
by phase? Say, if the measured signal is in phase, i.e. 0 or 180 degrees
lag with respect to a reference, it's rejected. If the phase relationship 
is +/-90 degrees, it's admitted. Of course, in my application there is the 
complicating matter that the frequency is unknown, though by assumption 
is known to be equal in the two measurements...

Rune

Hi all. 

I've been playing with a couple of ideas regarding extractions of 
signals from complex and complicated raw data. The overall experiment 
setup is that two sensors monitor the same environment such that 
both sensors monitor some source of interest, while each sensor monitor
unique spurious sources + unique observation noise. 

Block diagram:


      N1   S1         S         S2  N2    
       |    |         |         |   |
       V    V         |         V   V
     ----------       |       ----------   
    |    R1    |<-----+----->|    R2    |
     ----------               ----------

Each reciever R1 and R2 measures some raw data that can be separated as

   Dn = S+Sn+Nn

where 

  Dn   is the total signal measured by sensor Rn, 
  S    is the "useful" signal to be extracted that is measured by both 
       sensors R1 and R2 
  Sn   is some "signal-like" component that is of no interest, that is 
       only measured on sensor n (i.e. reciever R2 don't measure S1 and 
       vice versa)
  Nn   is random observation noise that is unique to each sensor, i.e 
       the noise is uncorrelated with the signal components Sn and S, 
       and uncorrelated between the sensors. 

The problem is to extract the "mutual component" S from the data, given 
no other information than that the signal is measured by both recievers 
R1 and R2. If S and Sn are known to be narrow-band spectrum lines (with 
no knowledge of frequency, amplitude or phase), I can do that. 

It's some sort of "implicit de-noising" based on the signal model being 
linear. I think the method should be possible to extend to an arbitrary 
linear signal model, i.e that the measured data is expressed as 

          K
   Dn =  sum ank*Ek + noise
         k=1

where Ek is some basis vector, "ank" is the amplitude of component Ek 
as measured by sensor Rn, and the noise is uncorrelated with the signal 
and between sensors. 

Now, in some applications one would like to separate the signals further 
with a constraint on phase, i.e. that one can assume that if the phase 
relationship between the measurement of S made by R1, call it S12, and 
the measurement of S made by R2, call it S21, is 

                                  0 deg
    arg C12=arg E[S12'*S21] = +/-90 deg  
                                180 deg

for various measurements. In some analyses I'm interested only in 
measurements (waves) that exhibit a +/- 90 deg phase relationship, in other 
cases I am intersted in all the other components (waves). 

So, at last, my question: 

Does anyone know whether ther exist filters that discriminate components 
by phase? Say, if the measured signal is in phase, i.e. 0 or 180 degrees
lag with respect to a reference, it's rejected. If the phase relationship 
is +/-90 degrees, it's admitted. Of course, in my application there is the 
complicating matter that the frequency is unknown, though by assumption 
is known to be equal in the two measurements...

Rune

Rune Allnor wrote:
> 
  ...
> 
> The problem is to extract the "mutual component" S from the data, given
> no other information than that the signal is measured by both recievers
> R1 and R2. If S and Sn are known to be narrow-band spectrum lines (with
> no knowledge of frequency, amplitude or phase), I can do that.
 
How?

  ...
>
> Now, in some applications one would like to separate the signals further
> with a constraint on phase, i.e. that one can assume that if the phase
> relationship between the measurement of S made by R1, call it S12, and
> the measurement of S made by R2, call it S21, is
> 
>                                   0 deg
>     arg C12=arg E[S12'*S21] = +/-90 deg
>                                 180 deg
> 
> for various measurements. In some analyses I'm interested only in
> measurements (waves) that exhibit a +/- 90 deg phase relationship, in other
> cases I am intersted in all the other components (waves).

Sort of like a lock-in amplifier.
> 
> So, at last, my question:
> 
> Does anyone know whether ther exist filters that discriminate components
> by phase? Say, if the measured signal is in phase, i.e. 0 or 180 degrees
> lag with respect to a reference, it's rejected. If the phase relationship
> is +/-90 degrees, it's admitted. Of course, in my application there is the
> complicating matter that the frequency is unknown, though by assumption
> is known to be equal in the two measurements...
> 
> Rune

Phase is strictly defined for a single frequency. It makes sense only in
special circumstances to compare the phases of signals at different
frequencies, and even then, it amounts to hand waving.

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

Jerry Avins <jya@ieee.org> wrote in message news:<3F4E1BD5.6C364545@ieee.org>...
> Rune Allnor wrote:
> > 
>  ...
> > 
> > The problem is to extract the "mutual component" S from the data, given
> > no other information than that the signal is measured by both recievers
> > R1 and R2. If S and Sn are known to be narrow-band spectrum lines (with
> > no knowledge of frequency, amplitude or phase), I can do that.
>  
> How?

It would take way too long to write an explanation here. I devoted a 
section (along with demonstrations on synthetic data) on that subject 
in my PhD thesis. Suffice it to say that all the necessary information 
is embedded in the data covariance matrixes. The trick is to manipulate 
the covariance matrixes before estimating the parameters of the sinusoidal 
model. For those who have my thesis available, this would be section 2.3 
and figure 4.6. I also used this implicit formalism to search for weak 
signals in real data. The data was a two-component seismic signal 
(horizontal and vertical particle displacements). The idea was that no 
weak signal component (spectrum line) is accepted from one data component 
[sorry for not taking more care to discriminate between "signal component" 
(spectrum line) and "data component" (horizontal/vertical particle 
displacements)] unless there is something in the other that corroborated 
a spectrum line very close to, though not exactly on, that frequency. That 
way one can filter out "false" signal components that are due to spurious 
effects in the processing of a single data component. I have a couple of
plots from demonstrations that show that the trick works with real data, 
but haven't published anything. 

I am also able to count the total number of spectrum lines when measured 
across two sensors in the sense that mutual lines are counted only once.
Hey, that could even be useful somewhere...

>   ...
> >
> > Now, in some applications one would like to separate the signals further
> > with a constraint on phase, i.e. that one can assume that if the phase
> > relationship between the measurement of S made by R1, call it S12, and
> > the measurement of S made by R2, call it S21, is
> > 
> >                                   0 deg
> >     arg C12=arg E[S12'*S21] = +/-90 deg
> >                                 180 deg
> > 
> > for various measurements. In some analyses I'm interested only in
> > measurements (waves) that exhibit a +/- 90 deg phase relationship, in other
> > cases I am intersted in all the other components (waves).
> 
> Sort of like a lock-in amplifier.
> > 
> > So, at last, my question:
> > 
> > Does anyone know whether ther exist filters that discriminate components
> > by phase? Say, if the measured signal is in phase, i.e. 0 or 180 degrees
> > lag with respect to a reference, it's rejected. If the phase relationship
> > is +/-90 degrees, it's admitted. Of course, in my application there is the
> > complicating matter that the frequency is unknown, though by assumption
> > is known to be equal in the two measurements...
> > 
> > Rune
> 
> Phase is strictly defined for a single frequency. It makes sense only in
> special circumstances to compare the phases of signals at different
> frequencies, and even then, it amounts to hand waving.

I know that the sines are at the *same* frequency, I just don't know *what* 
frequency... the number to insert for 'f' in the formula is missing, so to 
speak. What I'm after is the argument of the cross correlation coefficient 
at that (unknown) frequency. One way of doing things is to explicitly 
estimate the frequency and do some Maximum Likelihood amplitude processing, 
and get it from there. That approach sounds a bit crude, though. Some people
I have talked to, who have done these types of things for a one-component 
data set, tell me that the Maximum Likelihood estimates aren't very stable.
Apparently phase estimates are very sensitive to errors in the frequency 
estimate. So if I can get there with some other method... but before I 
make any attempts to go there, I would need to know what already has been 
done in this field. Hence my question.

Rune

allnor@tele.ntnu.no (Rune Allnor) wrote in message news:<f56893ae.0308281324.4628f19b@posting.google.com>...
> Jerry Avins <jya@ieee.org> wrote in message news:<3F4E1BD5.6C364545@ieee.org>...
> > Rune Allnor wrote:
> > > 
>  ...
> > > 
> > > The problem is to extract the "mutual component" S from the data, given
> > > no other information than that the signal is measured by both recievers
> > > R1 and R2. If S and Sn are known to be narrow-band spectrum lines (with
> > > no knowledge of frequency, amplitude or phase), I can do that.
> >  
> > How?
> 
> It would take way too long to write an explanation here. ...

Waddayaknow! After having written about this thing yesterday, I woke up 
this morning with an idea on how to solve the cross correlation problem
without going all the way through explicit frequency and amplitude estimates.
I have, of course, no idea whether the idea will succeed, but it's definately 
a starting point that may somehow be useful. 

Jerry, thanks for your invaluable help!

Rune

Rune Allnor wrote:
> 
> ... but before I
> make any attempts to go there, I would need to know what already has been
> done in this field. Hence my question.
> 
> Rune

Very interesting. Thanks for the enlightenment.

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

Rune Allnor wrote:
> 
  ...
> 
> Jerry, thanks for your invaluable help!
> 
  :-) !
                 ҿ�
Jerry            ���

P.S. Yesterday, I turned 71.
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:f56893ae.0308280120.45cb27e4@posting.google.com...
> Hi all.
>
> I've been playing with a couple of ideas regarding extractions of
> signals from complex and complicated raw data. The overall experiment
> setup is that two sensors monitor the same environment such that
> both sensors monitor some source of interest, while each sensor monitor
> unique spurious sources + unique observation noise.
>
> Block diagram:
>
>
>       N1   S1         S         S2  N2
>        |    |         |         |   |
>        V    V         |         V   V
>      ----------       |       ----------
>     |    R1    |<-----+----->|    R2    |
>      ----------               ----------
>
> Each reciever R1 and R2 measures some raw data that can be separated as
>
>    Dn = S+Sn+Nn
>
> where
>
>   Dn   is the total signal measured by sensor Rn,
>   S    is the "useful" signal to be extracted that is measured by both
>        sensors R1 and R2
>   Sn   is some "signal-like" component that is of no interest, that is
>        only measured on sensor n (i.e. reciever R2 don't measure S1 and
>        vice versa)
>   Nn   is random observation noise that is unique to each sensor, i.e
>        the noise is uncorrelated with the signal components Sn and S,
>        and uncorrelated between the sensors.
>
> The problem is to extract the "mutual component" S from the data, given
> no other information than that the signal is measured by both recievers
> R1 and R2. If S and Sn are known to be narrow-band spectrum lines (with
> no knowledge of frequency, amplitude or phase), I can do that.
>
> It's some sort of "implicit de-noising" based on the signal model being
> linear. I think the method should be possible to extend to an arbitrary
> linear signal model, i.e that the measured data is expressed as
>
>           K
>    Dn =  sum ank*Ek + noise
>          k=1
>
> where Ek is some basis vector, "ank" is the amplitude of component Ek
> as measured by sensor Rn, and the noise is uncorrelated with the signal
> and between sensors.
>
> Now, in some applications one would like to separate the signals further
> with a constraint on phase, i.e. that one can assume that if the phase
> relationship between the measurement of S made by R1, call it S12, and
> the measurement of S made by R2, call it S21, is
>
>                                   0 deg
>     arg C12=arg E[S12'*S21] = +/-90 deg
>                                 180 deg
>
> for various measurements. In some analyses I'm interested only in
> measurements (waves) that exhibit a +/- 90 deg phase relationship, in
other
> cases I am intersted in all the other components (waves).
>
> So, at last, my question:
>
> Does anyone know whether ther exist filters that discriminate components
> by phase? Say, if the measured signal is in phase, i.e. 0 or 180 degrees
> lag with respect to a reference, it's rejected. If the phase relationship
> is +/-90 degrees, it's admitted. Of course, in my application there is the
> complicating matter that the frequency is unknown, though by assumption
> is known to be equal in the two measurements...

Rune,

What isn't clear to me in your question is where the phase reference comes
from.  If there's a solid reference that's one thing.  If there is no such
reference but, rather, you have to compare the two outputs, then that's
another matter, right?

Without getting into the math, I would think of a phase detector (so that I
would have a reasonable, simple model to work with).  How would I detect
phase in one channel relative to the other?  It seems that a very high SNR
with a clipper would provide a reference.  One might measure phase of both
channels relative to the other and average the result......  Or, if both
channels are clipped, XOR'd and integrated, you get the relative phase.

This method is used to get angle of arrival.  It has a relation to DIMUS.
But I think you know all that so....

On the other hand, if this model isn't what you have in mind, then I'd be
very concerned that the SNR would not be adequate to support the
measurement.

If the signals are broadband (as distinct from what's described above) and
if you cross-correlate them then you get a similar measure don't you?

I would go back to basics.  Your objective implies superposition rather than
filtering (well, filtering alone).  By this I mean that it appears you want
to subtract parts of one signal from the other if the conditions are right.
From your comments it appears that you already have an adaptive approach in
mind for simpler cases.

The complicating factor here is that one doesn't generally find broadband
Hilbert transformers in nature - at least I don't think so.  In contrast, we
find things that delay signals - so the phase varies with frequency.  So,
the phase relations you're looking for would seem to be strange.  What the
heck is the situation that causes this to be of interest?

Another way to say this:  "phase" really only applies to a single frequency
(at a time) while "delay" can apply to all frequencies in a composite
signal - by definition.

OK - what am I missing here?

Fred

Mike Rosing wrote:
> 
> Jerry Avins wrote:
> > Rune Allnor wrote:
> >
> >   ...
> >
> >>Jerry, thanks for your invaluable help!
> >>
> >>
> >   :-) !
> >                  ҿ�
> > Jerry            "^"
> >
> > P.S. Yesterday, I turned 71.
> >
> 
> Happy birthday!  What a coincidence too, my kids turned 10 yesterday.
> The great thing with their birthday is that I get 2 cakes to eat from :-)
> 
> I can't wait to read about Rune's solution either, seems like an interesting
> problem.
> 
> Patience, persistence, truth,
> Dr. mike
> 
> --
> Mike Rosing
> www.beastrider.com                   BeastRider, LLC
> SHARC debug tools

It's Goethe's birthday too. My mother had a knack. My twin sisters, both
musicians (though only one professionally) were born on Mozart's
birthday.

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

Jerry Avins <jya@ieee.org> wrote in message news:<3F4F6F9B.97A5FEB3@ieee.org>...
> Rune Allnor wrote:
> > 
>  ...
> > 
> > Jerry, thanks for your invaluable help!
> > 
>   :-) !
>                  &#1215;&#4294967295;
> Jerry            ???
> 
> P.S. Yesterday, I turned 71.

Happy birthday! 

And nice cartoon...

Rune