Cyclostationary process

A cyclostationary process is a signal having statistical properties that vary cyclically with time. A cyclostationary process can be viewed as multiple interleaved stationary processes. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year. There are two differing approaches to the treatment of cyclostationary processes. The stochastic approach is to view measurements as an instance of an abstract stochastic process model. As an alternative, the more empirical approach is to view the measurements as a single time series of data--that which has actually been measured in practice and, for some parts of theory, conceptually extended from an observed finite time interval to an infinite interval. Both mathematical models lead to probabilistic theories: abstract stochastic probability for the stochastic process model and the more empirical Fraction Of Time (FOT) probability for the alternative model. The FOT probability of some event associated with the time series is defined to be the fraction of time that event occurs over the lifetime of the time series. In both approaches, the process or time series is said to be cyclostationary if and only if its associated probability distributions vary periodically with time. However, in the non-stochastic time-series approach, there is an alternative but equivalent definition: A time series that contains no finite-strength additive sine-wave components is said to exhibit cyclostationarity if and only if there exists some nonlinear time-invariant transformation of the time series that produces finite-strength (non-zero) additive sine-wave components.