Symbol timing recovery design

Question

I am trying to design the Symbol timing recovery(#STR) block of #DVBS Receiver. These are the specifications

• Symbol rate ($f_s$) = $2\textrm{ MHz}$
• Acquisition range = $10\%$ of Symbol rate
• Acquisition time = better than $50\textrm{ ms}$
• Interpolation rate = $4$
• Roll off factor = $0.5$
• Symbol time ($T_s$) = 1/$f_s$
• Sample time ($T$) = $T_s$/Interpolation rate = $T_s$/4

The main thing I need for the design is to find out the loop filter coefficients $\mathbf{K1}$ and $\mathbf{K2}$ of PI filter which depends on $\zeta$, $B_n$, $T_s$(symbol time), $K_d$(TED gain) and $\mathbf{K0}$(VCO gain).

I am referring to the book Digital Communications: A Discrete-time Approach by Michael Rice, Appendix C, for the design of the loop filter.

From the acquisition range specs, I found:

• The term $B_nT_s \geq 0.01125$.
• Acquisition range $\approx$ $2\pi B_n\sqrt 2 \zeta$.

Besides I also figured out the following values

• $K_d :(\text{I found out from S-curve})= 0.1751$ [Gardner TED]
• $\mathbf{K0} =1$
• $\zeta = 1$

I calculated $\mathbf{K1}$ and $\mathbf{K2}$ from these, but when I am simulating the timing recovery design for noise-free channel, it is not working. I found out through trial that $B_nT_s$ factor should be in the range $0.005$ for retrieved symbols bit amplitude close to the original symbol bit amplitude, not $0.01125$ which I got from my calculation. I am not understanding what possible mistake I could have done with calculation. Or I am missing any important factor in the calculation of $\mathbf{K1}$ and $\mathbf{K2}$.

This is the equation for loop filter coefficient $\mathbf{K1}$ $$ \mathbf{K0}\cdot K_{d}\cdot \mathbf{K1} = \frac{4\zeta\left (\frac{B_{n}T}{\zeta+ \frac{1}{4\zeta}}\right )}{1 + 2\zeta\left ( \frac{B_{n}T}{\zeta + \frac{1}{4\zeta}} \right ) + \left ( \frac{B_{n}T}{\zeta + \frac{1}{4\zeta}} \right )^{2}} $$

Answer 1

The reason the OP is not getting the result expected could be that a factor of $T$ was missing, where $T$ is the loop update period (reciprocal of the update rate). When computing factors in time using seconds, the integrator coefficient should be multiplied by T in the implementation. The z transform equivalent of $1/s$ in discrete time is $z/(z-1)$ when the x axis is unit samples, but is $Tz/(z-1)$ when the x axis is actual time. Therefore in the modeling we would see the accumulator path implemented as $Kz/(z-1)$, but in the implementation we need to multiply the gain coefficient by $T$ to be $KTz/(z-1)$.