Cyclostationary                                        From Wikipedia, the free encyclopedia                                                                                                Jump to: navigation, search
                                                A signal having statistical properties that vary cyclically with time is called a cyclostationary process.[1] A cyclostationary process can be viewed as multiple interleaved stationary processes.For example, the maximum daily temperature in New York City can bemodeled as a cyclostationary process: the maximum temperature on July21 is statistically different from the temperature on December 20;however, the temperature on December 20 of different years has(arguably) identical statistics. Thus, we can view the random processcomposed of daily maximum temperatures as 365 interleaved stationaryprocesses, each of which takes on a new value once per year.

Contents [hide] 1 Definition 2 Wide-sense cyclostationarity 3 References 4 External links

[edit] DefinitionThere 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 oftime that events occurs over the lifetime of the time series. In bothapproaches, the process or time series is said to be cyclostationary ifits associated probability distributions vary periodically with time.However, in the deterministic time-series approach, there is analternative but equivalent definition: A time series that contains noadditive finite-strength sine-wave components is said to exhibitcyclostationarity if there exists some nonlinear transformation of thesignal that produces positive-strength additive sine wave components.
[edit] Wide-sense cyclostationarityAn important special case of cyclostationary signals are those whichexhibit cyclostationarity in second-order statistics (e.g., the autocorrelation function). These are called wide-sense cyclostationary signals, and are analogous to wide-sense stationaryprocesses. The exact definition differs depending on whether the signalis treated as a stochastic process or as a deterministic time series.

• For a stochastic process x(t), we define the autocorrelation function as

.The signal x(t) is said to be wide-sense cyclostationary with period T0 if Rx(t;τ) is cyclic in t with cycle T0, i.e.,    for all t,τ.[2]

• For a deterministic time series x(t), we define the cyclic autocorrelation function as

.The time series x(t) is said to be wide-sense cyclostationary with period T0 if is not identically zero for α = nT0 for some integers n, but is identically zero for all other values of α.[2]Equivalently, we may say that a time series having nofinite-strength sine-wave components is wide-sense stationary if thereexists a quadratic transformation of the time series that producesfinite-strength sine-wave components.