Satellite Signal Acquisition, Tracking, and Data Demodulation - Understanding GPS: Principles and Applications - page 187

must be mounted using vibration isolators in order for the GPS receiver to success-
fully operate in PLL. The equation for vibration induced oscillator jitter is:

() ()

f max

Sf Pf
f

360
2

f

m

L

∫

2

σ = df (degrees) (5.8)

v ν m m

π

2

f min m

where:

L-band input frequency in Hz
S v ( f m )

f L =

= oscillator vibration sensitivity of ∆ f / f L per g as a function of f m

f m = random vibration modulation frequency in Hz

P ( f m )

= power curve of the random vibration in g 2 /Hz as a function of f m

g = the acceleration due to gravity ≈ 9.8 m/s 2

If the oscillator vibration sensitivity, S v ( f m ), is not variable over the range of the
random vibration modulation frequency, f m , then (5.8) can be simplified to:

()

f max

Pf
f

360
2

fS

m

σ =

L v

∫

df (degrees) (5.9)

v m

π

2

f min m

As a simple computational example, assume that the random vibration power
curve is flat from 20 Hz to 2,000 Hz with an amplitude of 0.005 g 2 /Hz. If S v =

1 ×

10 −9 parts/ g and f L = L1 = 1,575.42 MHz, then the vibration-induced phase jitter

using (5.9) is:

2000

df
f

1
20

1
200

=

∫

σ ν = 90 265 0005 . .

m

= 90 265 0005 . . − 142 . °

2

20 m

5.6.1.3 Allan Deviation Oscillator Phase Noise
The equations used to determine Allan deviation phase noise are empirical. They
are stated in terms of what the requirements are for the short-term stability of the
reference oscillator as determined by the Allan variance method of stability mea-
surement. (Appendix B contains a description of the Allan variance.) The equation
for short-term Allan deviation for a second-order PLL is [10]:

θ
ωτ

∆

()

στ = 2. ( dimensionless units of ∆ ff ) (5.10)

A

L

where:

∆θ = rms jitter into phase discriminator due to the oscillator (rad)

ω L = L-band input frequency = 2 π f L (rad/s)

τ = short-term stability gate time for Allan variance measurement (seconds).