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

[Bob McGwier]

John,

Achilleas implies how to compute the "score" for the parameters A1, A2, 
t1, and t2 and that the problem is to minimize the distance between r 
and s,  but he does not say how to reestimate or compute the best set of 
parameters.      That is all the meat to the problem (the minimization 
of this L^2 norm).    Since you are the smartest lawyer with no 
engineering training I have ever known in so far as your engineering 
abilities are concerned  you can probably do this  but I suspect some 
further guidance will make this easier.

73's
Bob  (N4HY)



John Ackermann N8UR wrote:
> 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)
> > and if you assume Gaussian noise n(t), then
> > if your observation is t in (0,T)
> > the (maximum likelihood) estimate should minimize
> > the squared Euclidean distance
> > int_0^T |r(t)-s(t)|^2 dt
> > or in discrete time (with appropriate sampling Ts)
> > sum_k |r(k Ts)-s(k Ts)|^2
> >
> > There are several simplifications (eg, if you know
> > apriori that the 2 amplitudes are the same, or if you know one
> > of t1 or t2, etc) you can try (depending on the required accuracy) 
> > but this is the safest way to approach the problem.
> >
> > Achilleas
>
> Achilleas, thanks for the answer, which it will take me a little while 
> to digest as I'm not a mathematician -- more accurately, I'm a 
> mathematical anti-prodigy!  However, I see generally what you're 
> saying and measuring zero crossings isn't the way it "has" to be done, 
> it's the way it's been done in the past for this application.
>
> One factor is that we will *not* know that the two amplitudes are the 
> same; while we can aim for that, in some cases they may be several dB 
> apart because of the nature of the device under test and the reference 
> signal.  And, we don't want to add additional amplifiers, etc. into 
> the system unless we really have to because they could contribute 
> their own noise to the result.
>
> My posting a few minutes ago describes in more detail just what we're 
> trying to accomplish.  Hopefully that will help explain the goal.
>
> Thanks!
>
> John
>
>
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Robert W. McGwier, Ph.D.
Center for Communications Research
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