Does anyone know of a source for the Fourier expansion of a phase comparator output?

I'm hacking out a prelim design where I need to measure the phase difference between two signals to do a doppler phase measurement (to ultimately compute the medium's speed). I'm going to pulse a short burst both upstream and downstream, receive both, and compare the phase difference about the 'no flow' time interval. The receiver will have the gain to saturate the return signal, so it becomes a series ofdigital pulses. This tosses out the amplitude, just giving me the desired time info.

PLL's that do something similar to this have a comparator as simple as an EX-OR gate. I like simple.

I kinda remember there is a simple expansion of such an output, the DC term related to the phase, with the frequency elements high enough so some simple filter can eliminate them.

Anyone know the expansion (or have a nice link)?

Thanks!
Posted on 2003-05-12 12:47:36 by Ernie
Im at work, which unfortunately has nothing to do with electronic's, so i can post any more detailed info. But PLL's (PHASE locked loops) have what your looking for.

You simply wire it such that you dont feed back the signal. PLL's have an open ended feedback pin such that you can do freqency division on the feedback path, and ther neat things.

If you want to do a phase comparison, simply input the ping, and the return into the summing pins of the comparator. The resulting drive voltage for the internal VCO (error) will be the difference in phase. This pin is also produced because it is also used by FM decoding.

You will end up with a linear voltage depending on phase from this pin. Print off a data sheet for a PLL and give it a good look over, you should see what im getting at.

:alright:
NaN
Posted on 2003-05-13 17:11:37 by NaN
NaN,

You're 100% correct in what you say, the input to a PLL is a phase comparator, and could be used in my application.

I'm looking for the fourier expansion so I can provide a mathmatical proof of concept of the device, compute measurement limits, tollerances, ect.

I actually looked for this expansion in some PLL tutorials online, but they all gloss over this point.

Funny, when we did it in school I remember them prooving this point.
Posted on 2003-05-13 22:14:02 by Ernie
Ahhh... found it in a table of trig identities:

1
sin U * sin V = --- [ cos ( U - V } - cos ( U + V) ]
2

If we let:

U = wt

V = wt + d

So:

1
sin wt * sin wt + d = --- [ cos ( wt - (wt + d) } - cos ( wt + (wt + d)) ]
2

1
= --- [ cos ( - d } - cos ( 2wt + d) ]
2

1
= --- [ cos ( d } - cos ( 2wt + d) ]
2

So I have a DC term (1/2 cos (d)) directly related to phase, and some higher frequency components at twice the burst frequency. That should give some wiggle room to insert a lo pass filter.
Posted on 2003-05-13 22:47:59 by Ernie
You saved me from having to dig it up myself ;)
Posted on 2003-05-13 23:31:10 by NaN
Also, after re-reading this, i remember that adding a 2nd order low pass filter will cause a 90' shift in the resulting frequency.

Thus the cos(@) turns to a sin(@) on its output. This has an advantage here being its an ODD function. If you design the filter gain properly, you can produce linear results centered about 0. ( i.e. Keep Max and Min extents within +/- 45 deg, before it starts to round to the max and min of the sine wave curve).

:NaN:
Posted on 2003-05-14 16:40:25 by NaN