Explain The Cumulant Theory of Cyclostationary Time-Series

The development for the theory of non linear process of cyclostationary time series is started. New type of cumulant for the complex valued variables is introduced and generalised. It has temporal and spectral moments and cumulants of input and output of several sign processing operations are determined.

According to Google Scholar William A. Gardner formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived. Estimates for the temporal moments for the cyclic polyspectra is presented and their properties are discussed. These theory has several usage in weak signal detection.

The development of non linear processing of cyclic stationary time series. It applies with the theory of problems of weak signal detection and interference tolerant time delay estimation. The time series is one of the finite strength additive sine waves which can be generated using non linear transformation. It does not contain itself with finite strength additive to sine wave components.

The minimum order of non linearity is required to generate sine waves is called frequencies. A suppressed carrier amplitude modulated time series is second order cyclostationarity because a sine wave frequency is twice the carrier frequency that can be generated by the squarer.

A PAM time series is positive frequency bandwidth that has equality to the half of the pulse rate that is of fourth order because it has no non linearities of order less than four which has to generate a sine wave from such a time series. But a fourth order non linearity can generate sine wave with frequency equal to the pulse rate.

The theory of second order cyclostationaity has been developed over last decade. It has found application to numerous signal processing problems like the weak signals detection. The time delay estimation and the interference removal, system identification and blind adaptive spatial filtering.

The exploitation of second order cyclic stationary signals is beneficial when the time series of interest is heavily corrupted by the noise and interference. This is due to that the second order cyclic stationary parameters are associated with particular frequency. It can be reliably extracted by amount and type of interference. It is not noise nor the interference level which exhibit SOCS with the same frequency.

The last requirement is time series of interest and have unique symbol rate, chip rate, hop rate, carrier frequency or frequency associated with the underlying periodicity. The analysis framework for cyclostationarity is usual expectation operation which is replaced by the wave extraction operation.

To take advancement of property for tolerance noise and interference time series which are cyclostationary are order of n>2. It is necessary for the generalisation of the theory of second order cyclic stationary signals that are theory of higher order cyclostationarity.

The sine wave extraction are for the multiplication of sine waves which are associated with lower order lag products and factors. In these process the moments and cumulants are useful in characterisation of the sine wave components in the nth order nonlinear transformation of cyclostationary time series. Its aim is to characterise sine wave components.

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