A cyclostationary process is a signal having statistical properties that vary cyclically with time.
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.
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.
An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the autocorrelation function).
The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series.
and autocorrelation function: where the star denotes complex conjugation, is said to be wide-sense cyclostationary with period
, helps analyze cyclostationary signals, which exhibit periodic statistical properties.
A signal that is just a function of time and not a sample path of a stochastic process can exhibit cyclostationarity properties in the framework of the fraction-of-time point of view.
This way, the cyclic autocorrelation function can be defined by:[2] If the time-series is a sample path of a stochastic process it is
If the signal is further cycloergodic,[3] all sample paths exhibit the same cyclic time-averages with probability equal to 1 and thus
For a Gaussian cyclostationary process, its rate distortion function can be expressed in terms of its cyclic spectrum.
is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the expected value of the product of the output of a one-sided bandpass filter with center frequency
and the conjugate of the output of another one-sided bandpass filter with center frequency
[5] For time series, the reason the cyclic spectral density function is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the average over all time of the product of the output of a one-sided bandpass filter with center frequency
and the conjugate of the output of another one-sided bandpass filter with center frequency
The cyclic spectrum is: Typical raised-cosine pulses adopted in digital communications have thus only
This same result can be obtained for the non-stochastic time series model of linearly modulated digital signals in which expectation is replaced with infinite time average, but this requires a somewhat modified mathematical method as originally observed and proved in.
[6] It is possible to generalise the class of autoregressive moving average models to incorporate cyclostationary behaviour.
His work follows a number of other studies of cyclostationary processes within the field of time series analysis.
[8][9] In practice, signals exhibiting cyclicity with more than one incommensurate period arise and require a generalization of the theory of cyclostationarity.
Such signals arise frequently in radio communications due to multiple transmissions with differing sine-wave carrier frequencies and digital symbol rates.
The theory was introduced in [10] for stochastic processes and further developed in [6] for non-stochastic time series.
The encyclopedic book [11] comprehensively teaches all of this and provides a scholarly treatment of the originating publications by Gardner and contributions thereafter by others.
Mechanical signals produced by rotating or reciprocating machines are remarkably well modelled as cyclostationary processes.
This happens to be the case for noise and vibration produced by gear mechanisms, bearings, internal combustion engines, turbofans, pumps, propellers, etc.
The explicit modelling of mechanical signals as cyclostationary processes has been found useful in several applications, such as in noise, vibration, and harshness (NVH) and in condition monitoring.
[15] In the latter field, cyclostationarity has been found to generalize the envelope spectrum, a popular analysis technique used in the diagnostics of bearing faults.
Processes whose angle-time autocorrelation function exhibit a component periodic in angle, i.e. such that
The double Fourier transform of the angle-time autocorrelation function defines the order-frequency spectral correlation, where
It is not even a model for time-warped cyclostationarity, although it can be a useful approximation for sufficiently slow changes in speed of rotation.