Cyclostationary Process - Wide-sense Cyclostationarity

Wide-sense Cyclostationarity

An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the autocorrelation 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.

For a stochastic process, we define the autocorrelation function as

The signal is said to be wide-sense cyclostationary with period if is cyclic in with cycle i.e.,

For a deterministic time series, we define the cyclic autocorrelation function as

$\hat{R}_x^\alpha(\tau) = \lim_{T \rightarrow \infty} \frac{1}{T} \int_{-T/2}^{T/2} x(u+\tau/2) x^*(u-\tau/2) e^{-i 2\pi \alpha u}\, du.$

The time series is said to be wide-sense cyclostationary with period if is not identically zero for for some integers, but is identically zero for all other values of .

Equivalently, we may say that a time series having no finite-strength sine-wave components is wide-sense stationary if there exists a quadratic transformation of the time series that produces finite-strength sine-wave components.

Read more about this topic: Cyclostationary Process