Statistical analysis of frequency modulated signals

In the transmission of signals, one of the common models used for transmission is the frequency modulated model. Suppose the original signal is of the form
f_{t} = \beta \sin(\thetat+\psi)
, then the contaminated frequency Modulated signal is given by
y_{t} = A \cos(\theta_{c}t + \phi + \beta\sin(\thetat + \psi)) + z_{t} (t = 1,2 ....)
, where θ
c
is the carrier frequency, θ is the frequency of the original signal, φ and ψ are random phases, each having a uniform distribution over the interval [-π,π], β is a modulation index. Assuming z
t
is a second order stationary process, we consider the spectral analysis of the process y
t
. Suppose we have a sample y
1
,y
2
.....y
n
, we consider the (maximum likelihood) estimation of the frequencies θ
c
, θ, β and A on the basis of the sample, under the assumptions (i) {z
t
} is a Gaussian white noise; (ii) {z
t
} is a second order stationary Gaussian process.