[time-nuts] Re: Complex PLL

[Magnus Danielson]

Hi,

On 2021-03-17 17:20, Detlef Schuecker via time-nuts wrote:
> Hi time-nuts,
>
> a PLL takes the phase difference of the incoming signal and the 
> synthesized signal and feeds that in a loop filter. The output of the loop 
> filter is used to steer the local oscillator. 
>
> In my setup I have an incoming complex signal and my local oscillator is 
> generating a complex signal as well. So calculation of the phase 
> difference is just the quotient of the incoming signal and the local 
> oscillator, it is a sampled system. I take the quotient, calculate the 
> angle using the atan function and then I feed it in the loop filter, a PI 
> controller. The output of the loop filter is converted to a complex phase 
> increment for the local oscillator with the sin and cos function. 
>
> Now I have to get rid of the atan, cos and sin functions.
>
> I am looking for a loop filter which takes the quotient of the 
> incoming/synthesized signal as a complex value. The output of this loop 
> filter should be the phase increment for the local oscillator. It should 
> not use the angle of the complex value explicitly, as this will involve 
> the atan/cos/sin functions.
>
> Is someone aware of such a loop filter? I surfed through Gardners' 
> 'Phaselock Techniques' but did not find a hint.

That book is full of hints. Costas loop is one. Actually, you could just
do complex multiplication and only use the real output (and thus remove
half the complex multiplication) and use that output of the
multiplication as input to normal PI-regulator, that will lock up and
achieve everything you want. You can then also remove the sine with a
squarewave. There is some benefits and losses in doing that, which may
or may not be relevant.

There is a richness of complex detectors to be found in GPS literature,
such as that of "Understanding GPS principles and applications" of
Kaplan and Hegarty. You can also look at "Phase-locked loop circuit
design" by Wolaver for additional inspiration. You end up finding that
Garners' book is actually very comprehensive if you only take time to
dwell into it.

Let me know if you need more hints.

Cheers and 73,
Magnus SA0MAD