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

[Lee Patton]

Thanks, Bob -

I think this problem is a little trickier than previously assumed
because:

1) I believe John wants to measure frequency drift over time (i.e.,
drift of one clock with respect to a reference)  So, the clock signals
cannot be assumed to be 100% coherent.  Otherwise, there would be no
drift to measure.  The problem here is that there is no underlying model
for how this drift occurs. (It's what he's trying to find out!)  The
signals are probably very close to coherent over some finite window, but
the problem becomes one of determining a window size that helps
integrate out noise but doesn't do too much averaging of the drifting
signal to obtain parameter estimates.

2) John wants to measure to the best possible accuracy.  So, while we
assume A1=A2=1, this might not be the best assumption.  Even if he
measures the amplitude at the output of his clocks, there is no
guarantee that both digitizing channels are completely calibrated.  They
may be "good enough," but that depends on his application.

3) Relating to (1), I think John needs estimates of instantaneous
frequency over time.  A sliding window method similar to the one
outlined below would work. Again though, how is the window size chosen
if the nature of the drift isn't known?  Similarly, there are adaptive
techniques that could be used.  They should track to some degree, but
again, at what point do they break lock, and how do we know?

I think this may become a "how do you test the test equipment" problem.

Lee

On Tue, 2006-09-19 at 12:21 -0400, Bob McGwier wrote:
> 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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> 
>