A theory of polyspectra for nonstationary stochastic processes
A theory of polyspectra for nonstationary stochastic processes
复制标题
非平稳随机过程的多谱理论
DOI：
10.1109/tsp.2003.810298
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
L. Scharf
中科院分区：
文献类型：
--
作者：
A. Hanssen;L. Scharf
可调和过程是一类重要的非平稳随机过程。我们提出了一个理论的多谱（高阶矩谱）的协调类。我们定义并讨论了四个基本量：n阶矩函数，n阶时频多谱，n阶模糊函数，和n阶频频多谱。后者将传统的多谱推广到非平稳随机过程。这四个函数通过傅立叶变换相互关联。我们证明了时频多谱的频率和时间边缘分别是瞬时的n阶矩和常规的n阶平稳多谱。除了n阶模糊函数之外的所有量都可以用希尔伯特空间内积进行深入的解释。内积图像导致两个新的和非常强大的定义的多相干的非平稳随机过程。多相干性是n阶平稳性的客观度量，可用于构建各种统计检验。最后，我们给出了一些具体的例子，并将理论应用到线性时变系统，这是流行的模型衰落多径通信信道。
Harmonizable processes constitute an important class of nonstationary stochastic processes. We present a theory of polyspectra (higher order moment spectra) for the harmonizable class. We define and discuss four basic quantities: the nth-order moment function, the nth-order time-frequency polyspectrum, the nth-order ambiguity function, and the nth-order frequency-frequency polyspectrum. The latter generalizes the conventional polyspectrum to nonstationary stochastic processes. These four functions are related to one another by Fourier transforms. We show that the frequency and time marginals of the time-frequency polyspectrum are the instantaneous nth-order moment and the conventional nth-order stationary polyspectrum, respectively. All quantities except the nth-order ambiguity function allow for insightful interpretations in terms of Hilbert space inner products. The inner product picture leads to two novel and very powerful definitions of polycoherence for a nonstationary stochastic process. The polycoherences are objective measures of stationarity to order n, which can be used to construct various statistical tests. Finally, we give some specific examples and apply the theory to linear time-varying systems, which are popular models for fading multipath communication channels.