Advanced Frequency Synthesis by Phase Lock

APPENDIX R

REFERENCE FREQUENCY SPURS

Here, we will show the theoretical levels of sidebands at ±F_ref due to three PFD effects, and how they combine, and we will verify the theory with results from simulations.

First, we will perform an analysis of a train of narrow pulses, generated by the PFD, to be used in subsequent computations; we will assume no resampling until Section R.6. Then we will compute the first sideband amplitude due to leakage current (Section R.1). In Section R.2, we will compute the amplitude of sidebands due to an offset in two “canceling” pulses from a PFD. In Section R.3, we will compute the sideband amplitude due to ΣΔ modulation. In Sections R.4 and R.5, we will find the result of combining ΣΔ spurs with pulse offset spurs and with leakage current spurs, respectively, comparing theoretical and simulation results.

The amplitude of the fundamental component from the Fourier series of a train of rectangular pulses is

where A_p is the pulse amplitude and D_p is the pulse duty factor. Because the pulses that produce spurs are narrow compared to the repetition period, this is approximately

The equivalent input peak phase deviation is

R.1   LEAKAGE CURRENT

Leakage ...