Advanced Frequency Synthesis by Phase Lock

APPENDIX Q

QUANTIZATION PPSD

The phase power spectral density (PPSD) at the synthesizer output is

In this appendix, we will show that

where S_δN is the PSD of the variation in divide number. The shape of S_φ with a MASH modulator, Eq. (2.3), is obtained by combining Eq. (Q.1) with S_δN, as derived in Appendix M and given there by Eq. (M.4).

The original form of Eq. (F.8.74) [revised as Eq. (2.3)] was obtained by employing the equivalence between a change in N and a frequency change introduced after the VCO,^Q1 as developed in Section F.2.5. However, this equivalence depends on steady state during the time preceding the change in N. While this applies to a single step in N, it is not accurate for modulated N. To obtain Eq. (Q.1) [and, therefore, also Eq. (2.3)] correctly, we must compute how changes in N produce changes in phase at the divider output. We will initially assume a constant synthesized frequency (which is our design goal). If the system were time independent, we could use superposition to add this result to the effects of changes in output frequency (as with feedback or due to a change in ). We will see that this is still a good approximation in most cases.

Q.1   DEVELOPMENT ...