My question is regarding the design of timing recovery using gardner TED, where kp and ko are the gains of gardner TED and VCO respectively.
My question is whether the product term (kpko) will depend or symbol time Ts or not?
The unit of kp is volt/radians. We get it from S-curve.
The unit of ko is radians/second. In the design. there is a phase increase of 2pi per Ts symbol time. So the gain becomes 2pi/Ts. Would this Ts term gets cancelled while multiplied with kp? If not,(no timing parameter in the numerator of kp) the product term will depend on Ts. which will in turn affect the PI Loop filter constants. So for a large variance of Ts(say from 1 s to 0.0000001 s),loop filter constant will vary largely and affect the whole timing recovery design which does not depend on Ts, rather than relative frequency of sample and symbol rate..
This presentation covers one way to address this:
Note that KoKd (or kpko in your notation) has no dependence on Ts, but it does have a dependence on the clock frequency of the NCO (assuming your implementation is all digital). The loop coefficients, however, have a dependence on Rs (the symbol rate, or 1/Ts).
I hope that helps. This is an area that is not covered very well in most texts.
The proportional loop coefficient depends on the whole kokp term as an inverse proportion. So, if (koKp) has no dependence on Ts, then how the loop coefficient will depend on 1/Ts?
Then this (1/Ts) dependence is not coming from (kokp) product term.