Re: [Discuss-gnuradio] Phase demodulation

[Achilleas Anastasopoulos]

Just to make a note on the more theoretical aspect of this problem:

If you want to avoid all complications arising from the discontinuous 
arctan, you can (and technically/theoretically you should)
work with the actual incoming signal r(t)=exp(j theta(t)) + n(t)
and its hypothesized estimate s_hat(t) = exp(j theta_hat(t)),
and minimize their Euclidean distance (over a horizon T) under the 
constraints (bandwidth) of your signal theta_hat(t).
What this boils down to is:

find theta_hat(t) satisfying the BW constraints that maximizes
the quantity sum_{t=1}^T |r(t)| cos[arg(r(t))-theta_hat(t)].

BTW, the fine point above is that you cannot set theta_hat(t)=arg(r(t)),
because this assignment does NOT satisfy the BW constraints of your 
initial signal theta(t) or theta_hat(t).

This maximization problem is solved APPROXIMATELY using a PLL which is 
implemented in gnuradio in various forms.

Also the algorithms suggested earlier (evaluate arctan() and smooth it)
can be thought of as approximate solutions of the above problem.

Achilleas





On 10/7/2011 5:40 AM, Mikael Olofsson wrote:
> Achilleas Anastasopoulos wrote:
> > arctan (imag(out)/real(out)) plus some filtering to remove out of band
> > noise (it would be nice also to make sure you avoid discontinuities
> > inherent in the arctan function)...
>
> I would like to add that you should use four quadrant arctan, i.e. add
> or subtract pi to the above if the real part is negative.
>
> Regards
> /Mikael Olofsson
> Universitetslektor (Associate Professor)
> Linköpings universitet
>
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