Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 185

θ A = Allan variance-induced oscillator jitter in degrees

5.6.1.1 PLL Thermal Noise
Often the PLL thermal noise is treated as the only source of carrier tracking error,
since the other sources of PLL jitter may be either transient or negligible. The ther-
mal noise jitter for an arctangent PLL is computed as follows:

B
CN

360
2

1

σ =

n

1 + (degrees) (5.6)

PLLt

π 2 TCN

0 0

λ B

CN

1

()

L n

σ = 1 + m (5.7)

PLLt

2 π 2 TCN

0 0

where:

B n = carrier loop noise bandwidth (Hz)

C / N 0 = carrier to noise power expressed as a ratio (Hz)

(/ CN dB )

0
10

= 10 for ( C / N 0 ) dB expressed in dB-Hz

T = predetection integration time (seconds)

λ L = GPS L-band carrier wavelength (m)

= (299,792,458 m/s)/(1,575.42 MHz) = 0.1903 m/cycle for L1

= (299,792,458 m/s)/(1,227.6 MHz) = 0.2442 m/cycle for L2

Note that (5.6) and (5.7) do not include factors relating to C/A code or P(Y)
code or the loop filter order. Also note that (5.6) is independent of carrier frequency
when the error is expressed in units of degrees. The carrier thermal noise error stan-
dard deviation is strictly dependent on the carrier-to-noise power ratio, C / N 0 , the
noise bandwidth, B n , and the predetection integration time, T . The carrier-to-noise
power ratio, C / N 0 , is an important factor in many GPS receiver performance mea-
sures. It is computed as the ratio of recovered power, C , (in W) from the desired sig-
nal to the noise density N 0 (in W/Hz). Methods for determining C / N 0 with and
without external interference are described in Section 6.2.2. If C / N 0 increases (e.g.,
the recovered signal power is increased or the noise level is decreased), the standard
deviation decreases. Decreasing the noise bandwidth reduces the standard devia-
tion. The part of the equation involving the predetection integration time, T ,is
called the squaring loss. Increasing the predetection integration time reduces the
squaring loss, which in turn decreases the standard deviation.
It is a common misconception that the GPS P(Y) (precision) codes always pro-
duce more accurate measurements than the C/A (coarse) codes. While the P(Y) code
signal provides the potential for better code tracking accuracy, the PLL thermal
noise error is the same for either code for the same C / N 0 , since PLL processing uses
quantities after the spreading code has been stripped off. Further, since the C/A code
is 3 dB stronger than P(Y) code, the carrier thermal noise error is smaller for C/A