All-Digital Frequency Synthesizer in Deep-Submicron CMOS

APPENDIX A

SPURS DUE TO DCO SWITCHING

Let's assume a sinusoidal modulating signal,

For frequency modulation (FM), the instantaneous frequency ω_i is the carrier frequency ω_c modified by the modulating signal times the FM constant k_f :

Defining a new constant called the peak frequency deviation,

we can rewrite Eq. A.2 as

The phase of this FM signal is

where

is a dimensionless ratio of the peak frequency deviation to the modulating frequency.

The resulting signal is

Equation A.7 could be rewritten after some trigonometric expansion:

For small values of β (narrowband FM modulation), we can make the following approximations:

Substituting these into Eq. A.8, we obtain an approximate solution ...