Could you elaborate on this statement?
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).
Are you saying that theta_hat(t) should be quantized, and therefore not set
directly to arg(r(t))? Thanks.
-William
On Fri, Oct 7, 2011 at 10:08 AM, Achilleas Anastasopoulos <anas...@umich.edu
> wrote:
>
> 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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>> WWW:
>> http://www.commsys.isy.liu.se/**en/staff/mikael<http://www.commsys.isy.liu.se/en/staff/mikael>
>> Phone: +46 - (0)13 - 28 1343
>> Telefax: +46 - (0)13 - 13 9282
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>> -----------
>>
>>
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