Vanassche : The Difference Between Hajimiri (ISF) and Demir (Exact, PPV) Phase Noise Models


🔹 1. Exact vs. Approximate Models: Similar Math, Different Scope

  • Both models reduce oscillator phase dynamics to a 1D equation driven by an input perturbation:

    • Exact (Demir et al.): Includes oscillator phase in its own evolution → nonlinear, more rigorous

    • Approximate (Hajimiri & Lee): Simpler integral form using ISF; easier to apply, but omits feedback

  • For small perturbations and stationary noise, they yield identical phase noise predictions

🔹 2. Stationary Noise: Both Models Succeed Equally

  • When noise input is stationary (e.g., thermal or 1/f noise), the slow-phase behavior can be captured using averaging

  • Averaging filters out high-frequency effects and isolates the core phase dynamics

  • Both models reduce to the same stochastic differential equation, leading to the same phase noise growth (∝ √t)

🔹 3. Injection Locking: Only the Exact Model Captures It

  • When driven by a non-stationary input (e.g., sine wave), only the exact model predicts phase and frequency locking

  • The approximate model mistakenly predicts a continuous phase drift (frequency pulling), failing to show locking behavior

  • This gap becomes critical in coupled oscillator systems or PLL injection scenarios

🔹 4. Simulation Results Confirm Divergence

  • Under stationary white noise, both models yield identical time-varying phase variances

  • Under injected sine wave, only the exact model shows convergent phase locking; the approximate model does not

  • Experimental simulations support the mathematical derivation

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