Tracking GPS Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 167

By applying bilinear transform in Equation (8.32) to Equation (8.23), the result
can be written as,

(ω n t s ) 2 ] + 2 (ω n t s ) 2 z − 1 + [ (ω n t s ) 2 rζω n t s ] z − 2 [4 ζω n + −

H(z) =

(ω n t s ) 2 ] z − 2
(8.38)
By equating the denominator polynomials in the above two equations, C 1 and
C 2 can be found as

[4 + 4 ζω n t s + (ω n t s ) 2 ] + [2 (ω n t s ) 2 − 8] z − 1 + [4 − 4 ζω n t s +

1
k 0 k 1

8 ζω n t s
4 + 4 ζω n t s +

C 1 =

(ω n t s ) 2

4 (ω n t s ) 2 1

k 0 k 1

C 2 = (8.39)

4 + 4 ζω n t s + (ω n t s ) 2

The applications of these equations will be discussed in the next two sections.
In reference 6 a third-order phase-locked loop is also implemented. The filter
is implemented in digital format and the result can be used for phase-locked loop
designs, but it is not included in this topic.

8.6 CARRIER AND CODE TRACKING ( 4 )

Before discussing the usage of the above equations, let us concentrate on the
tracking of GPS signals. The input to a conventional phase-locked loop is usually
a continuous wave (cw) or frequency-modulated signal and the frequency of the
VCO is controlled to follow the frequency of the input signal. In a GPS receiver
the input is the GPS signal and a phase-locked loop must follow (or track) this
signal. However, the GPS signal is a bi-phase coded signal. The carrier and code
frequencies change due to the Doppler effect, which is caused by the motion of
the GPS satellite as well as from the motion of the GPS receiver as discussed in
Chapter 3. In order to track the GPS signal, the C/A code information must be
removed. As a result, it requires two phase-locked loops to track a GPS signal.
One loop is to track the C/A code and the other one is to track the carrier
frequency. These two loops must be coupled together as shown in Figure 8.3.
In Figure 8.3, the C/A code loop generates three outputs: an early code, a
late code, and a prompt code. The prompt code is applied to the digitized input
signal and strips the C/A code from the input signal. Stripping the C/A code
means to multiply the C/A code to the input signal with the proper phase as
shown in Figure 7.1. The output will be a cw signal with phase transition caused
only by the navigation data. This signal is applied to the input of the carrier loop.
The output from the carrier loop is a cw with the carrier frequency of the input
signal. This signal is used to strip the carrier from the digitized input signal,
which means using this signal to multiply the input signal. The output is a signal
with only a C/A code and no carrier frequency, which is applied to the input of
the code loop.