Easiest - Approximate Phase Angle Between Two 4 SNR Signals

[Bret Cahill]

Both signals have mostly the same noise. The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

The phase angle will be less than 5 degrees.

The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

In this particular situation has anyone ever tried integrating both
signals at least once reduce the noise and to eliminate multiple
crossings and then to obtain several phase angles / cycle to average?

The accuracy doesn't need to be better than +/- 20% of the phase
angle. In other words, in absolute terms the phase angle may need to
be good down to 0.1 degrees but the error relative to the phase angle
can be large.


Bret Cahill

 

[John Larkin]

On Wed, 18 Aug 2010 08:37:38 -0700 (PDT), Bret Cahill
<Bret_E_Cahill@yahoo.com> wrote:

Both signals have mostly the same noise. The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

The phase angle will be less than 5 degrees.

The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

Then you have to define "phase angle" before you can measure it. Phase
of the fundamental? Zero crossings? Something else?

In this particular situation has anyone ever tried integrating both
signals at least once reduce the noise and to eliminate multiple
crossings and then to obtain several phase angles / cycle to average?

As noted, lowpass filter it.

The accuracy doesn't need to be better than +/- 20% of the phase
angle. In other words, in absolute terms the phase angle may need to
be good down to 0.1 degrees but the error relative to the phase angle
can be large.

There are all sorts of ways to measure phase differences accurately.

The problem is currently too poorly defined to make a bunch of
suggestions worthwhile.

John

 

[Tim Wescott]

On 08/18/2010 08:37 AM, Bret Cahill wrote:

Both signals have mostly the same noise. The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

The phase angle will be less than 5 degrees.

The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

In this particular situation has anyone ever tried integrating both
signals at least once reduce the noise and to eliminate multiple
crossings and then to obtain several phase angles / cycle to average?

The accuracy doesn't need to be better than +/- 20% of the phase
angle. In other words, in absolute terms the phase angle may need to
be good down to 0.1 degrees but the error relative to the phase angle
can be large.


Bret Cahill


http://en.wikipedia.org/wiki/Crossposting

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Do you need to implement control loops in software?
"Applied Control Theory for Embedded Systems" was written for you.
See details at http://www.wescottdesign.com/actfes/actfes.html

 

[Bret Cahill]

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

Would a simple RC or higher order filter be any better than several
integrations?

The phase angle will be less than 5 degrees.

The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

Then you have to define "phase angle" before you can measure it. Phase
of the fundamental?

That seems to be what works graphically using sin curves instead of
real noise.

Zero crossings? Something else?

In this particular situation has anyone ever tried integrating both
signals at least once reduce the noise and to eliminate multiple
crossings and then to obtain several phase angles / cycle to average?

As noted, lowpass filter it.

The accuracy doesn't need to be better than +/- 20% of the phase
angle.  In other words, in absolute terms the phase angle may need to
be good down to 0.1 degrees but the error relative to the phase angle
can be large.

There are all sorts of ways to measure phase differences accurately.

The problem is currently too poorly defined to make a bunch of
suggestions worthwhile.

Thanks.


Bret Cahill

 

[Bret Cahill]

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

Would a simple RC or higher order filter be any better than several
integrations?

Correction:

Would a simple LC or higher order filter be any better than several
integrations?

The phase angle will be less than 5 degrees.
The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

Then you have to define "phase angle" before you can measure it. Phase
of the fundamental?

That seems to be what works graphically using sin curves instead of
real noise.

Zero crossings? Something else?
In this particular situation has anyone ever tried integrating both
signals at least once reduce the noise and to eliminate multiple
crossings and then to obtain several phase angles / cycle to average?

As noted, lowpass filter it.
The accuracy doesn't need to be better than +/- 20% of the phase
angle.  In other words, in absolute terms the phase angle may need to
be good down to 0.1 degrees but the error relative to the phase angle
can be large.

There are all sorts of ways to measure phase differences accurately.

The problem is currently too poorly defined to make a bunch of
suggestions worthwhile.

Thanks.

Bret Cahill

 

[John Larkin]

On Wed, 18 Aug 2010 09:36:40 -0700 (PDT), Bret Cahill
<Bret_E_Cahill@yahoo.com> wrote:

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

Would a simple RC or higher order filter be any better than several
integrations?

Each integrator will add a -6 dB/octave frequency rolloff, and 90
degrees of phase lag, to your signal. A filter can leave your signal
essentially unchanged, so long as you're not too close to the cutoff
frequency. Integrators also have that little (or enormous) problem
with zero offset, mathematically the "constant of integration."

John

 

[John Larkin]

On Wed, 18 Aug 2010 22:28:43 -0700 (PDT), Bret Cahill
<BretCahill@peoplepc.com> wrote:

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

Would a simple RC or higher order filter be any better than several
integrations?

Each integrator will add a -6 dB/octave frequency rolloff, and 90
degrees of phase lag, to your signal.

Both signals need to be integrated so the lags should cancel out when
determining the phase angle.

Yes.

A filter can leave your signal
essentially unchanged, so long as you're not too close to the cutoff
frequency. Integrators also have that little (or enormous) problem
with zero offset, mathematically the "constant of integration."

Those constants should cancel out as well.

No.

The problem with integration is it kind of like dead reckoning. If
you try to go too far with too many integrations completely blind with
no channel markers you might end up in the wrong ocean.

That won't be an issue if the number of cycles and integrations is low
enough and landing anywhere between San Diego and LA is OK.

Too abstract for me. What are you trying to do?

John

 

[Bret Cahill]

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

Would a simple RC or higher order filter be any better than several
integrations?

Each integrator will add a -6 dB/octave frequency rolloff, and 90
degrees of phase lag, to your signal.

Both signals need to be integrated so the lags should cancel out when
determining the phase angle.

A filter can leave your signal
essentially unchanged, so long as you're not too close to the cutoff
frequency. Integrators also have that little (or enormous) problem
with zero offset, mathematically the "constant of integration."

Those constants should cancel out as well.

The problem with integration is it kind of like dead reckoning. If
you try to go too far with too many integrations completely blind with
no channel markers you might end up in the wrong ocean.

That won't be an issue if the number of cycles and integrations is low
enough and landing anywhere between San Diego and LA is OK.


Bret Cahill

 

[Bret Cahill]

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

Then a lowpass filter will remove the noise.

Would a simple RC or higher order filter be any better than several
integrations?

Each integrator will add a -6 dB/octave frequency rolloff, and 90
degrees of phase lag, to your signal.

Both signals need to be integrated so the lags should cancel out when
determining the phase angle.

Yes.



A filter can leave your signal
essentially unchanged, so long as you're not too close to the cutoff
frequency. Integrators also have that little (or enormous) problem
with zero offset, mathematically the "constant of integration."

Those constants should cancel out as well.

No.

Both can be re centered on the t axis after integration.

The waveforms are only slightly different.

The problem with integration is it kind of like dead reckoning.  If
you try to go too far with too many integrations completely blind with
no channel markers you might end up in the wrong ocean.

That won't be an issue if the number of cycles and integrations is low
enough and landing anywhere between San Diego and LA is OK.

Too abstract for me. What are you trying to do?

Get the phase angle of the principle.


Bret Cahill

 

[whit3rd]

On Aug 18, 8:37 am, Bret Cahill <Bret_E_Cah...@yahoo.com> wrote:

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

The phase angle will be less than 5 degrees.

The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

If I understand correctly, two signals are
A = a1 * sin( w*t + p1) + noise
B = a2 * sin(w*t + p2) + noise2

and what is wanted is a measure of the phase difference, p1-p2, which
is
presumed to be small?

That's relatively easy. First, you need to know 'a1' and 'a2'; this
can be
done by computing the RMS average value of A and B (ignore noise for
this
part).
Then, note that the average of the product

avg( A*B) = avg( a1 * a2 sin(w*t + p1) sin(w*t + p2) + noise terms)

we depend on the average of the noise terms to be zero... and on
the constancy of a1 and a2, so

avg( A * B) = a1 * a2 * avg( sin (w*t + p1) sin(w*t + p2))

now using trig identities

avg( A*B) = a1 * a2 * avg( (sin(w*t) cos(p1) + sin(p1) cos(w*t) )
*(sin(w*t)cos(p2) + sin(p2)cos(w*t))

Expand the product; sines and cosines average to zero, so only two
terms contribute

avg (A * B) = a1 * a2 * avg( sin**2(w*t)cos(p1) cos(p2) + cos**2(w*t)
sin(p1) sin(p2))

now note that the average of sine-squared and cosine-squared are 1/2,
and
the phases p1 and p2 are constants, time-averaging doesn't change them

avg(A*B) = a1 * a2 * 1/2 * (cos(p1) cos(p2) + sin(p1)sin(p2))

now use trig identity

avg(A*B) = a1 * a2 * 1/2 * cos(p1 - p2)

and your answer for the phase difference is

(p1 - p2) = acos( avg(A*B) * 2/(a1* a2))

So, to find out the phase difference is a straightforward matter IF
you can
average (i.e. time-integrate over a known period) A*B as well as
measure
amplitudes (time-integrate over a known period A**2 and B**2).

 

[Bret Cahill]

Both signals have mostly the same noise.  The noise is at least
several times higher in frequency and usually several times lower in
amplitude.

The phase angle will be less than 5 degrees.

The bad news is both the frequency of the fundamental and its waveform
vary somewhat with time.

If I understand correctly, two signals are
A  = a1 *  sin( w*t + p1) + noise
B =  a2 * sin(w*t  + p2) + noise2

and what is wanted is a measure of the phase difference, p1-p2, which
is
presumed to be small?

That's relatively easy.  First, you need to know 'a1' and 'a2'; this
can be
done by computing the RMS average value of A and B (ignore noise for
this
part).
Then, note that the average of the product

avg( A*B) = avg( a1 * a2  sin(w*t + p1) sin(w*t + p2) + noise terms)

we depend on the average of the noise terms to be zero... and on
the constancy of a1 and a2, so

 avg( A * B) = a1 * a2  *  avg( sin (w*t + p1) sin(w*t + p2))

now using trig identities

avg( A*B)  = a1 * a2 * avg( (sin(w*t) cos(p1) + sin(p1) cos(w*t)  )
*(sin(w*t)cos(p2) + sin(p2)cos(w*t))

Expand the product; sines and cosines average to zero, so only two
terms contribute

avg (A * B) = a1 * a2 * avg(  sin**2(w*t)cos(p1) cos(p2) + cos**2(w*t)
sin(p1) sin(p2))

now note that the average of sine-squared and cosine-squared are 1/2,
and
the phases p1 and p2 are constants, time-averaging doesn't change them

avg(A*B) = a1 * a2 * 1/2 * (cos(p1) cos(p2) + sin(p1)sin(p2))

now use  trig identity

avg(A*B) = a1 * a2 * 1/2 * cos(p1 - p2)

and your answer for the phase difference is

(p1 - p2) = acos( avg(A*B) * 2/(a1* a2))

So, to find out the phase difference is a straightforward matter IF
you can
average (i.e. time-integrate over a known period) A*B as well as
measure
amplitudes (time-integrate over a known period A**2 and B**2).

Thanks.

Correcting the amplitudes is the goal but some kind of iterative
approach should home in to give a good phase angle as well as good
amplitudes.

First you get a rough estimate of the amplitudes. Then you get an
estimate of the phase angle and then use that to correct for the
amplitudes and so on.

Actually once or twice will probably be good enough since it is a
small correction, at most a few percent from the original.


Bret Cahill