A general fraction-of-time probability framework for chirp cyclostationary signals

The fraction-of-time (FOT) framework, a pseudo-probability framework, is proposed to promote the stochastic signal processing theory into practical applications. Technically, the ensemble averaging that requires infinite realizations is replaced by the time averaging of a single realization. This skill is further generalized for cyclostationary signals and accordingly the periodic FOT framework is obtained. However, neither the FOT nor the periodic FOT is proper for the chirp cyclostationary (CCS) signals which are utilized to model most of the nonstationary signals in mobile communications, radar/sonar systems. In this paper, a novel general FOT framework is proposed, which is obtained by constructing an isomorphic map associated with fractional Fourier transform (FrFT). Specifically, a CCS signal and its fractional shift form a Hilbert space, which is isomorphic to the Hilbert space formed by the CCS stochastic processes. The chirp-period FOT framework for CCS signals with a single chirp period is obtained directly, which is further extended for CCS signals with incommensurate chirp periods. Then the statistics of the CCS signals are defined and their properties are introduced in this framework. Finally, their estimators are investigated.