Implementing PLL/FLL/DLL in GNSS RX

1. Basic scheme (Kaplan 2006)
The given scheme for PLL and DLL in the tracking loop of GNSS RX

In tracking loop filters part, we have presented the common scheme of tracking loop filter for digital receiver. We will compare the PLL/DLL in GNSS RX with the common scheme. The given scheme actually contains 2 blocks (PLL and DLL). In case of no-noise, they work separately. Unfortunately, in reality, 2 blocks must work together. For the sake of simplify, we consider PLL.
Firstly, we have to compute I and Q in EPL channels
I=Ax(n)+noise(n)
Q=Ay(n)+noise(n)
real_error(n)=f(x(n)/y(n))
estimate_error(n)=f(I(n)/Q(n))
DLL affects to the coefficient A. It's clear that A should be maximum value as possible.

Take PLL as an example. We compare PLL and the CTL
1, Discriminator
2, F(z)
3, NCO or D(z)
The updated frequency of NCO is equivalent to sampling frequency Fs=1/Ts. On the other hand, the PLL updated carrier phase every a duration time of integration Td. We will consider the characteristic of NCO in Z-domain:
The output of PLL is often expressed as $\frac{T_d}{2}\frac{d\Phi_k}{dt}$. It's easy to prove that
$$\Phi_{k+1}=\Phi_{k}+T_d/2(\frac{d\Phi_{k+1}}{dt}+\frac{d\Phi_k}{dt})$$.
Therefore:
$$N(z)=\frac{T_d}{2}\frac{z+1}{z-1}$$