Re: [Discuss-gnuradio] Using DSP for precise zero crossing, measurement?

[Bob McGwier]

Lee:

The marginal distributions for the parameters never contain as much 
information as the joint probability distributions except those 
conditions where the underlying problem is truly separable.  It is 
almost always better from an information theoretic and probabilistic 
point of view to use the joint conditional distributions for the 
parameters given the observations and do the estimation jointly.

In this case however, we are actually making measurements on two 
independent signals,  and the observations are r1,r2 as you say.  It 
would be better in this case,  since they are truly separated,  to 
measure each independently.   The joint observation process would arise 
if John measured the difference signal through a mixer or time interval 
counter, etc.

Achilleas identified incorrect parameters given John's statement of the 
problem.  Achilleas used amplitude and phase as the parameters and the 
original statement of John's problem has constant 1V peak to peak as the 
amplitude and the unknowns are frequency and phase.   Your formulation 
of the problem is correct, but it is more general that John's statement 
of the problem since you include A1,A2 as (possibly) different when 
John's statement of the problem is that A1=A2=1.

r(t)  = sin(wt+phi)+n(t)

and determining w and phi in the presence of noise n(t) is just about 
the oldest problem in town.   Let us consider John's original problem 
given the system he claims he has.   Since John's statement is that he 
is doing the measurements on each separately using a coherent system,  
he can repeatedly estimate w and phi using FFT's and downsampling.    
One way to reduce the impact of the noise given a fixed size FFT, is to 
use the coherence as stated and to do long term autocorrelations,  where 
the autocorrelations are computed using  FFT's and then simply added, 
complex bin by bin.  This coherent addition of the correlations will 
produce a very long term autocorrelation where accuracy of the estimates 
from this process goes up like N where N is the number of FFT's added.  
THIS ASSUMES THE SYSTEM IS REALLY COHERENT FOR THAT N * FFTsize SAMPLES 
and THE NOISE REALLY IS ZERO MEAN GAUSSIAN.   Phase noise, drift, 
non-Gaussian noise, etc.  will destroy this coherence assumption and the 
Gaussian properties we use in the analysis.  He can reduce the ultimate 
computational complexity by mixing, downsampling  and doing the FFT 
again and then mixing, downsampling and doing the FFT again, etc.  until 
the final bin traps his w to sufficient accuracy for his needs and then 
phi is simply read off from the FFT coefficient.  The mixing and 
downsampling would be a usable approach.   Careful bookkeeping must be 
done on the phase group delay through the downsampling filters should 
this approach be used or phi will be in error by the neglected phase 
group delay.

This is  one approach that I believe John can take and it is pretty 
simple to put together even if it is not necessarily the most 
computationally benign.    He can grab tons of samples and do this in 
Octave on his favorite Linux computer.  In the case the signals are not 
actually 1V pk-pk,  this will also yield the amplitude since the power 
of the sinusoid as measured by the FFT's in use above will yield the 
result for the amplitude.  If this is to be done real time,  then a 
cascade of power of 2 reductions in sample rate and frequency offset can 
be done until the parameters are trapped sufficiently for the exercise.

Bob


Lee Patton wrote:
> On Mon, 2006-09-18 at 12:32 -0400, Achilleas Anastasopoulos wrote:
>   
> > John,
> >
> > If want to measure the time difference between two sine waves in noise 
> > and accuracy is your primary objective, then you should start with the
> > "optimal" solution to the problem and not with an ad-hoc technique
> > such as measuring the zero crossings.
> >
> > In the simplest scenario, if your model looks like:
> > s(t) = s(t;A1,A2,t1,t2) = A1 sin(w (t-t1)) + A2 sin(w (t-t2))
> > r(t)=s(t)+n(t)
> >     
>
> John, 
>
> Does your model look like the one Achilleas described above, or is it
> like the following? 
>
> s1(t) = A1*sin(w1(t-t1)+phi1); r1(t) = s1(t) + n1(t)
> s2(t) = A2*sin(w2(t-t2)+phi2); r2(t) = s2(t) + n2(t)
>
> In this model, you have separate observations of each sinusoid.  i.e.,
> r1(t) and r2(t) respectively. 
>
>
> Bob, Achilleas -
>
> On an intuitive level, it seems to me that (ML) estimating the
> parameters of s1(t) and s2(t) from r1(t) and r2(t) respectively, instead
> of jointly from r(t) = s1(t)+s2(t)+n(t), would result in better
> accuracy.  Do you agree?  At the very least, I think an adaptive
> technique would converge faster if only three parameters needed to be
> estimated instead of six. (Obviously, you would estimate both signals in
> parallel to get dphi = phi1-phi2).
>
> -Lee
>
>
>
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>   


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Robert W. McGwier, Ph.D.
Center for Communications Research
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