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.

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.

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."

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

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.

If I understand correctly, two signals are A =3D a1 * sin( w*t + p1) + noise B =3D 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

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).

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.

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