The maximum likelihood estimates of all the parameters of a PM signal, when the baseband signal is a harmonic process and the PM signal observed is contaminated by an independent Gaussian white noise, are derived. The sampling distribution of the maximum likelihood estimates of the parameters and the demodulated signal are also investigated. Alternative estimates for the carrier frequency and the modulating frequency, together with their sampling properties, are also studied. The methods are illustrated with two examples and in each case the demodulated signal is compared to the true baseband signal. Some comments about the performance of the demodulated signals are also included.

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 formf_{t} = \beta \sin(\thetat+\psi), then the contaminated frequency Modulated signal is given byy_{t} = A \cos(\theta_{c}t + \phi + \beta\sin(\thetat + \psi)) + z_{t} (t = 1,2 ....), where θcis 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 ztis a second order stationary process, we consider the spectral analysis of the process yt. Suppose we have a sample y1,y2.....yn, we consider the (maximum likelihood) estimation of the frequencies θc, θ, β and A on the basis of the sample, under the assumptions (i) {zt} is a Gaussian white noise; (ii) {zt} is a second order stationary Gaussian process.