Phase Difference of Sampled Waves

Date: 05/27/2002 at 09:21:31
From: Sean Goddard
Subject: Extracting Phase information

Hi Dr. Math,

OK, I'm no kid, and my maths is way too rusty to be of use to 
anyone (me included).

My problem is that I have two SIN waves which I have sampled into 
my PC, and I need to find the LAG/LEAD of the second with respect 
to the first (reference).

What I'm trying to do is return the result in the form of a 
phasor, so I obviously need to be looking in using an I and Q 
system.

I'm trying to give a number in the range of -90 to +90 for the 
phase shift, with an accuracy of +/- 0.1 degrees.

I'm REALLY struggling, and have derived something that sort of 
works, but not very well.

Please help!

Date: 05/29/2002 at 14:01:15
From: Doctor Douglas
Subject: Re: Extracting Phase information

Hi, Sean,

It's a little difficult to answer this fully without knowing more
about your system.  However, if we know that the two sine waves
are of the same frequency (they'd better be, otherwise their
phase difference will be changing with time), then you can do
the following:

  1.  Find the period T (in say, sampling intervals) of 
      both waves. You can use zero-crossings to do this.

  2.  Determine a zero-crossing time Z of the first wave.  

  3.  Starting from that time Z, step forward until the next
      event of the SECOND wave crossing zero with the same
      polarity (i.e. both crossings are positive-going or 
      both are negative-going).

      Call this time Y.  Note that T, Z, and Y are all 
      measured in number of sampling intervals and that 
      Y >= Z.

  4.  The number
       
         P = [360] * [(Y-Z)/T]       (degrees)

      will be the phase measurement that you require, where 
      360 is the number of degrees in a full period, and the
      numbers Y, Z, and T are derived from your measurements.  
      
      If the number P is greater than 180, then you can 
      subtract 360 from it to get a number in the range 
      -180 to +180.  I think you want to consider this 
      range rather than just the range from -90 to +90.

To get the accuracy that you require, the numbers Y, Z, and
T will need to be of sufficient accuracy.  Roughly speaking, 
you'll want T to be at least 3600 sampling intervals (360/0.1).  
More is better (longer period sine waves or faster sampling).

I don't think that there's any need to make things more 
complicated with a in-phase and quadrature (if that's what you 
meant by I-Q) analysis.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/