An important model in communications is the stochastic frequency modulation (FM) signal st = A cos (tωc + ∑tk=1 mk + θ0), where the message process {mt} is a stochastic process. In this paper, we investigate the linear models and limit distributions of FM signals. Firstly, we show that this non-linear model in the frequency domain can be converted to an ARMA (2,q + 1) model in the time domain when {mt} is a Gaussian MA (q) sequence. The spectral density of {st} can then be solved easily for MA message processes. Also, an error bound is given for an ARMA approximation for more general message processes. Secondly, we show that {st} is asymptotically strictly stationary if {mt} is a Markov chain satisfying a certain condition on its transition kernel. Also, we find the limit distribution of st for some message processes {mt}. These results show that a joint method of probability theory, linear and non-linear time series analysis can yield fruitful results. They also have significance for FM modulation and demodulation in communications.
