Advanced Frequency Synthesis by Phase Lock

APPENDIX E

EXCESS PPSD

This development follows that of Cassia et al. [2003].

E.1   DEVELOPMENT OF EQ. (2.4)

We first write the Fourier series for the current pulses from the PFD as

where

is the normalized modulation frequency and the amplitude of the nth harmonic is

Here U(t) is the unit step, I_p is the magnitude of the current during the PFD pulse, and T = L_sequT_ref is the duration of the repeated sequence.

Equation (E.5) represents positive pulses extending from mT_ref to mT_ref + |Δt(m)| when Δt(m) > 0 and negative pulses extending from mT_ref to mT_ref − |Δt(m)| when Δt(m) < 0, thus representing the PFD’s shift in pulse position as a function of phase.

Evaluating (E.5), we obtain

Replacing the last exponential with the first three terms in its expansion, we obtain

We can also represent ...