Advanced Frequency Synthesis by Phase Lock

APPENDIX N

SAMPLED NOISE

Consider a train of pulses of width T and frequency f_ref multiplying a noise waveform in the time domain. In the frequency domain, the noise power spectrum will be convolved with the Fourier power spectrum of the pulse train (Fig. N.1). See Section N.5 for clarification. The original noise spectrum at B will be replicated, each new replica being centered on one of the lines in the spectrum at A. We will consider three cases that differ in width of the noise spectrum compared to the width of the pulse spectrum and its repetition frequency f_ref.

N.1   CASE 1: W_n« f_ref

The noise power spectrum is narrow, for example, flicker noise that falls off with frequency. Unlike the flat spectrum shown at B in Fig. N.1, this spectrum is high in the center and falls off rapidly. Then, the other spectrums (C, D, etc.), resulting from convolution with the spectral lines at A, will have the same peaked shape and will not overlap significantly, and the noise seen near 0 will mainly just be the spectrum at B.

N.2   CASE 2: 1/T» W_n» f_ref

This may represent phase noise that affects the switching of a logic signal only during the rise time of the signal. The pulses are very narrow, so the pulse spectrum is spread out and near the center looks like a flat impulse train. If the width of the noise spectrum is narrow (the frequencies are low) relative to the width of the pulse spectrum (1/T), the total power near 0 will depend on how many spectral lines are in the width of the noise ...