The response of a phase-locked loop to a sinusoid plus noise

The phase-locked loop is a practical device for separating a sinusoidal signal from additive noise. In this device the incoming signal-plus-noise is multiplied by a noise-free sinusoid generated by a voltage-controlled oscillator (vco). The filtered product is used to lock the phase of the vco output to that of the incoming signal, thus producing a relatively clean version of the incoming signal in which the noise manifests itself as a small phase modulation. Analysis of this noise-produced phase modulation is complicated by the presence of the multiplier at the input to the loop. This paper presents a perturbation method which reduces this inherently nonlinear servo analysis problem to the analysis of a series of linear systems, the first of which is related to the linear model used by previous authors. The perturbation technique permits the phase modulation resulting from an arbitrary noise input to be computed to any desired accuracy. This analysis is particularly useful in predicting loop performance when it is used as a narrow-band receiver in a phase-comparison angle-measuring system,