Tracking GPS Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 166

must be changed into a discrete system. This discussion is based on reference 5.
The transfer from the continuous s -domain into the discrete z -domain is through
bilinear transform as

z − 1 2

t s

1 −

s = ( 8 . 32 )

1 + z − 1

where t s is the sampling interval. Substituting this relation in Equation (8.22) the
filter is transformed to

C 1 z − 1

C 2 (C 1 + C 2 ) −

F(z) = C 1 + = ( 8 . 33 )

1 − z − 1 1 − z − 1

where

2 τ 2 − t s

C 1 =

2 τ 1

t s
τ 1

C 2 = (8.34)

This filter is shown in Figure 8.2.
The VCO in the phase-locked loop is replaced by a direct digital frequency
synthesizer and its transfer function N ( z ) can be used to replace the result in
Equation (8.7) with

k 1 z − 1
1

θ f (z)
V o (z) ≡

N(z) = ( 8 . 35 )

− z − 1

Using the same approach as Equation (8.8), the transfer function H ( z ) can be
written as

θ f (z)
θ i (z) =

k 0 F (z)N(z)

H(z) = ( 8 . 36 )

1 + k 0 F (z)N(z)

Substituting the results of Equations (8.33) and (8.35) into the above equation,
the result is

k 0 k 1 (C 1 + C 2 )z − 1 − k 0 k 1 C 1 z − 2

H(z) = ( 8 . 37 )

1 + [ k 0 k 1 (C 1 + C 2 ) − 2] z − 1 + ( 1 − k 0 k 1 C 1 )z − 2

FIGURE 8.2 Loop filter.