 There is a special correlation theory in the cyclostationary time series. A time series is cyclostationary when and only if there is quadratic time invariant transformation which makes spectral lines. The time series exhibits spectral correlation. There are special properties of characterisation for spectral correlation function is developed.

These properties effects the periodic modulation and time invariant linear filtering. There are relationship between spectral correlation function and radar ambiguity function and the winger ville distribution. The special correlation properties between rice’s presentation of band pass time series is derived.

A normalisation between the wiener function from the spectral density function and the spectral correlation function is now developed. There are also generalisation of periodic sampling and frequency conversion formula for amplitude modulation. From there the spectral density function to the spectral correlation function is developed.

The statistical spectral analysis of empirical time series of periodic phenomena is known as cyclostationary time series. The term indicates time series data for physical phenomenon. Spectral analysis denotes the decomposition of time series in sine wave components. Statistical means averaging that is used to reduce random effects in the data which mask spectral characteristics of phenomenon under the study.

Products of pair of sine wave components are averaged and it produces spectral correlation. Here the comprehensive theory of spectral correlation of the cyclical stationary time series is analysed. Special concepts and special time series analysis methods of periodic phenomena is setup.

Many application of conceptual gap between concept and deterministic theory is presented. It narrow down and make it easier to bridge conceptual gap between practice and the more abstract probabilistic theory.

Here you can obtain probabilistic interpretation of deterministic theory for periodically time variant fraction of time distributions. You can also compare the random data and periodic phenomena on the foundation of cyclic stationary stochastic process. The deterministic applies into single time series whereas the probabilistic theory applies to random samples of time series that is defined on the abstract probabilistic space.

It is analogous to the fact of the great majority of theoretical treatment of random data which is based on probabilistic foundation of stationary stochastic process. The statistical and probabilistic approach are synonymous. Here the average of measured spectra is a statistical spectrum.

There is nothing contradictory in the notion of deterministic or non-probabilistic theory for the statistical spectral analysis. This is common term in engineering. The non-probabilistic theory is defined by infinite limits of the time average. Deterministic and probabilistic theories are functional and stochastic theories.

The term random is used as it means that nothing more than erratic unpredictable behaviour. It is mainly used in probabilistic concepts, it gives rise to random data abound engineering and science. In the concept of mechanical vibration machinery and periodicity arises by the rotation, revolution and reciprocation of gear, belts, chains, shafts.

It has usage in atmospheric science the periodicity of seasons comes by rotation and revolution of earth. The time series has usage in communications, signals, telemetry, multiplexing etc.

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