Abstract 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.

中文翻译：

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chirp 循环平稳信号的一般时间分数概率框架

摘要 时间分数（FOT）框架是一种伪概率框架，旨在将随机信号处理理论推向实际应用。从技术上讲，需要无限次实现的整体平均被单个实现的时间平均所取代。该技能进一步推广到循环平稳信号，因此获得了周期性 FOT 框架。然而，FOT 和周期性 FOT 都不适用于 chirp 循环平稳 (CCS) 信号，CCS 信号用于模拟移动通信、雷达/声纳系统中的大多数非平稳信号。在本文中，提出了一种新颖的通用 FOT 框架，该框架是通过构建与分数傅立叶变换 (FrFT) 相关的同构映射而获得的。具体来说，一个 CCS 信号及其分数移位形成一个 Hilbert 空间，它与 CCS 随机过程形成的 Hilbert 空间同构。直接获得了具有单个 chirp 周期的 CCS 信号的 chirp-period FOT 框架，这进一步扩展了具有不相称 chirp 周期的 CCS 信号。然后在这个框架中定义了 CCS 信号的统计数据并介绍了它们的特性。最后，调查他们的估计量。