Stochastic systems with small noise, analysis and simulation; a phase locked loop example
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This work addresses the problem of noise analysis of phase locked loops (PLLs). The problem is formulated as a stochastic differential equation and is solved in presence of circuit white noise sources yielding the spectrum of the PLL output. Specifically, the effect of loop filter characteristics, phase-frequency detector and phase noise of the open loop voltage controlled oscillator (VCO) on the PLL output spectrum is quantified. These results are derived using a full nonlinear analysis of the VCO in the feedback loop and cannot be predicted using traditional linear analyses or the phase noise analysis of open loop oscillators. The computed spectrum matches well with measured results, specifically, the shape of the output spectrum matches very well with measured PLL output spectra reported in the literature for different kinds of loop filters and phase detectors. The PLL output spectrum computation only requires the phase noise of the VCO, loop filter and phase detector noise, phase detector gain and loop filter transfer function and does not require the transient simulation of the entire PLL which can be very expensive. The noise analysis technique is illustrated with some examples.