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 probabilistic approach is to view measurements as an instance of a . As an alternative, the deterministic approach is to view the measurements as a single time series, from which a probability distribution for some event associated with the time series can be defined as 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 deterministic 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 positive-strength additive sine-wave components.

An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the function). These are called wide-sense cyclostationary signals, and are analogous to wide-sense stationary processes. The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series.

A stochastic process ${\displaystyle x(t)}$ of mean ${\displaystyle \operatorname {E} [x(t)]}$ and autocorrelation function:

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