Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 195

B 2

fe

() ( )

− ∫

2

BSf sin π f T f

n s c

1

B 2

σ ≅

fe

tDLL

T

2

B 2

c

fe

()

2

() ( )

∫

2 π CN fS f sin π f DT df

0 s c

− B 2

fe

(5.22)

B 2

fe

() ( )

∫

2

Sf cos π fDTdf

s c

− B 2

×+ 1

fe

(chips)

2

B 2

fe

() ( )

TC N

∫

S f cos π fDT df

0 s c

− B 2

fe

where:

B n = code loop noise bandwidth (Hz)

power spectral density of the signal, normalized to unit area over
infinite bandwidth
B fe =

S S ( f ) =

double-sided front-end bandwidth (Hz)

T c = chip period (seconds) = 1/ R c where R c is the chipping rate

For BPSK-R( n ) modulations (see Section 4.2.3) such as P(Y) code ( n = 10) and

C/A 1), and when using a noncoherent early-late power DLL

discriminator, the thermal noise code tracking jitter can be found by substituting
(4.14) into (5.22). The result can be approximated by [12]:

code ( n =

B
CN

2 π R

B

n

D 1 + , D ≥

c

( )

2 TC N 2 − D

0 0 fe

2

BT

B
CN

1 1 fe c

n

+ D −

2 B T π − 1 BT

R
B

π R

B

c c

0 fe fe

σ ≅

c

<< D

c

( chips) (5.23)

tDLL

2

fe fe

×+ 1 ,

( )

TC N 2 − D

0

B
CN

R
B

1 1

n

1 + , D ≤

c

2 B T TCN

c

0 fe 0 fe

The part of the right-hand side of (5.22) and (5.23) in brackets involving the
predetection integration time, T , is called the squaring loss. Hence, increasing T
reduces the squaring loss in noncoherent DLLs. When using a coherent DLL
discriminator, the bracketed term on the right is equal to unity (no squaring loss)
[12]. As seen in (5.23), the DLL jitter is directly proportional to the square root of
the filter noise bandwidth (lower B n results in a lower jitter, which, in turn, results in
a lower C / N 0 threshold). Also, increasing the predetection integration time, T ,