Estimation of phase processes in oscillatory signals and asymptotic expansions

A tracking control problem is presented in which the signal to be tracked is a sine wave at a known frequency ω
c
with a random phase modeled as a Brownian motion process. The measurement process is the sum of this signal with a corrupting additive white noise. This is an estimation problem for which an extended Kalman filter structure is assumed. This formulation serves as an alternate mechanization for phase-lock loop design from its classical counterpart if the amplitude of the signal is known. Although no additional assumptions are made here, other than those required on the classical PLL, the absence of the low-pass filter allows this alternate loop to operate when the carrier frequency is the same magnitude as the effective "bandwidth" of the phase process. The resulting gain is characterized by a Riccati equation with periodic coefficients. By using multiple time scales, the solution to this Riccati equation can be obtained approximately as an expansion in the frequency ω
c
where
1/\omega_{c}
is used as an expansion parameter. The multiple time scale method for developing an asymptotic expansion solution of this Riccati equation is developed as a useful analysis tool. However, this expansion does require the additional assumption that the carrier frequency is much greater than the effective estimated phase process bandwidth.