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f
L(f)
Each frequency multiplication
by 10 increases the sidebands
by 20 dB.
10X
1X
Vibration-Induced Sidebands
After Frequency Multiplication
   Upon frequency multiplication by a factor N, the vibrational frequency fv is unaffected, as it is an external influence.  The peak frequency change due to vibration, Df, however, becomes
f = (A) Nf0.
   The modulation index  is therefore increased by the factor N.  Expressed in decibels, frequency multiplication by a factor N increases the phase noise by 20 log N.  When exposed to the same vibration, the relationship between the vibration-induced phase noise of two oscillators with the same vibration sensitivity and different carrier frequencies is
 LB(f) = LA(f) +20 log (fB/fA),
where LA(f) is the sideband level, in dBc/Hz (or dBc for sinusoidal vibration), of the oscillator at frequency fA and LB(f) is the sideband level of the oscillator at frequency fB.  For the same acceleration sensitivity, vibration frequency and output frequency, the sidebands are identical, whether the output frequency is obtained by multiplication from a lower frequency or by direct generation at the higher frequency.  For example, when a 2 x 10-9/g sensitivity 5 MHz oscillator’s frequency is multiplied by a factor of 315 to generate a frequency of 1575 MHz, its output will contain vibration-induced sidebands which are identical to those of a 1575 MHz SAW oscillator that has the same 2 x 10-9/g sensitivity.


R. L. Filler, "The Acceleration Sensitivity of Quartz Crystal Oscillators:  A Review," IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 35, No. 3, pp. 297-305, May 1988.

J. R. Vig, C. Audoin, L. S. Cutler, M. M. Driscoll, E. P. EerNisse, R. L. Filler, R. M. Garvey, W. L. Riley, R. C. Smythe, and R. D. Weglein, "Acceleration, Vibration and Shock Effects - IEEE Standards Project P1193," Proc. 1992 IEEE Frequency Control Symposium, 763-781, 1992; also, The Effects of Acceleration on Precision Frequency Sources, U. S. Army Laboratory Command Research and Development Technical Report SLCET-TR-91-3, March 1991, AD-A235470.