# Cyclostationary process

A cyclostationary process is a signal having statistical properties that vary cyclically with time.[1] 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.

## Definition

There are two differing approaches to the treatment of cyclostationary processes.[2] The probabilistic approach is to view measurements as an instance of a stochastic process. As an alternative, the deterministic approach is to view the measurements as a single time series, from which a probability distribution can be defined as the fraction of time that events occurs over the lifetime of the time series. In both approaches, the process or time series is said to be cyclostationary 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 additive finite-strength sine-wave components is said to exhibit cyclostationarity if there exists some nonlinear transformation of the signal that produces positive-strength additive sine wave components.

## 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 $x(t)$, we define the autocorrelation function as
$R_x(t;\tau) = E \{ x(t - \tau/2) x^*(t + \tau/2) \}.\,$
The signal $x(t)$ is said to be wide-sense cyclostationary with period $T_0$ if $R_x(t;\tau)$ is cyclic in $t$ with cycle $T_0,$ i.e.,
$R_x(t;\tau) = R_x(t+T_0; \tau)\text{ for all }t, \tau.$[2]
• For a deterministic time series $x(t)$, 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 $x(t)$ is said to be wide-sense cyclostationary with period $T_0$ if $\hat{R}_x^\alpha$ is not identically zero for $\alpha = n T_0$ for some integers $n$, but is identically zero for all other values of $\alpha$.[2]
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.

## Cyclostationary models

It is possible to generalise the class of autoregressive moving average models to incorporate cyclostationary behaviour. For example, Troutman[3] treated autoregressions in which the autoregression coefficients and residual variance are no longer constant but vary cyclically with time. His work follows a number of other studies of cyclostationary processes within the field of time series analysis.[4][5]

## References

1. ^ Gardner, William A.; Antonio Napolitano; Luigi Paura (2006). "Cyclostationarity: Half a century of research". Signal Processing (Elsevier) 86 (4): 639–697. doi:10.1016/j.sigpro.2005.06.016.
2. ^ a b c Gardner, William A. (1991). "Two alternative philosophies for estimation of the parameters of time-series". IEEE Trans. Inf. Theory 37 (1): 216–218. doi:10.1109/18.61145.
3. ^ Troutman, B.M. (1979) "Some results in periodic autoregression." Biometrika, 66 (2), 219–228
4. ^ Jones, R.H., Brelsford, W.M. (1967) "Time series with periodic structure." Biometrika, 54, 403–410
5. ^ Pagano, M. (1978) "On periodic and multiple autoregreessions." Ann. Statist., 6, 1310–1317.