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

[John Ackermann N8UR]

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