Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 190

°/m. For L1, d 3 R / dt 3 = (98 m/s 3 ) × (360°/cycle) × (1,575.42 × 10 6 cycles/s)/ c =

185,398°/s 3 where c 299,792,458 m/s is the speed of light. For L2, d 3 R / dt 3

= = 98 ×

144,666°/s 3 .
Using (5.15), the 3-sigma stress error for an 18-Hz third-order PLL is 15.35º for
L1 and 11.96º for L2. These are well below the 45º 3-sigma rule-of-thumb levels.

10 6 / c 360 × 1,227.60 × =

5.6.1.5 Reference Oscillator Acceleration Stress Error
The PLL cannot tell the difference between the dynamic stress induced by real
dynamics and the dynamic stress caused by changes in frequency in the reference
oscillator due to acceleration sensitivity of the oscillator. The oscillator change in
frequency due to dynamic stress is:

() ()

∆ f = 360 S f G t ° s (5.16)

g g L

where:

S g = g-sensitivity of the oscillator ( ∆ f / f per g )

f L = L-band input frequency (Hz)

= 1,575.42 MHz for L1

= 1,227.76 MHz for L2

G ( t ) = acceleration stress in g as a function of time

For the components of G ( t ) due to acceleration ( g ), the units of f g are °/s, a
velocity error as sensed by the loop filter. For an unaided second-order carrier track-
ing loop, this acceleration-induced oscillator error can be ignored because it is insen-
sitive to velocity stress. For the components of G ( t ) due to jerk stress ( g /s), the units
of f g are °/s 2 , an acceleration error as sensed by the loop filter. For an unaided
third-order carrier tracking loop, this jerk-induced oscillator error can be ignored
because it is insensitive to acceleration stress. In reality, there will always be some
level of dynamic stress that will adversely affect tracking loop regardless of the loop
filter order because there are always higher order components of dynamic stress
when the host vehicle is subjected to dynamics. Nothing can be done about this for
an unaided tracking loop except to align the least sensitive S g axis of the reference
oscillator along the direction of the anticipated maximum dynamic stress, but this is
often impractical. For an externally aided tracking loop where the LOS dynamic
stress can be measured and S g is known, it is prudent to model this acceleration stress
sensitivity and apply the correction to the aiding. Note that, like all oscilla-
tor-induced errors, the error is common mode to all receiver tracking channels, so
one correction applies to all aided channels.

5.6.1.6 Total PLL Tracking Loop Measurement Errors and Thresholds
Figure 5.24 illustrates the total PLL jitter as a function of ( C / N 0 ) dB for a third-order
PLL, including all effects described in (5.5), (5.6), (5.9), (5.13), and (5.15). Equation
(5.5) can be rearranged to solve for the dynamic stress error, and this can be solved