Phase Noise Reduction and Optimal Operating Conditions for a Pair of Synchronized Oscillators

We investigate the dynamics of a pair of noisy coupled oscillators and derive operating conditions that minimize phase noise. The generic model employed allows for general nonlinear dynamics of the resonant elements of the oscillators, in terms of their amplitude-dependent frequencies, general asymmetric coupling, and both additive and multiplicative noise sources. The model is analyzed using the method of stochastic averaging and the results suggest an optimal operating point for minimizing phase noise in either of the oscillators by varying the operating amplitudes and the nature and strength of the coupling. We also show that the system with asymmetric coupling has the ability to reduce the phase noise of the noisier oscillator beyond the expected result for synchronized oscillators. The analytical predictions, which are validated by Monte-Carlo simulations, show how one can exploit nonlinear behavior to improve phase noise characteristics using a pair of coupled oscillators. The results also point to a means of designing and optimizing larger sets of coupled oscillators to improve phase noise performance.