Riviello Daniel Gaetano, Alfano Giusi, Garello Roberto
CNR-IEIIT, Istituto di Elettronica e di Ingegneria dell'Informazione e delle Telecomunicazioni, Consiglio Nazionale delle Ricerche, 10129 Turin, Italy.
Department of Electronics and Telecommunications (DET), Politecnico di Torino, 10129 Turin, Italy.
Entropy (Basel). 2025 Jan 12;27(1):63. doi: 10.3390/e27010063.
Quadratic forms with random kernel matrices are ubiquitous in applications of multivariate statistics, ranging from signal processing to time series analysis, biomedical systems design, wireless communications performance analysis, and other fields. Their statistical characterization is crucial to both design guideline formulation and efficient computation of performance indices. To this end, random matrix theory can be successfully exploited. In particular, recent advancements in spectral characterization of finite-dimensional random matrices from the so-called allow for the analysis of several scenarios of interest in wireless communications and signal processing. In this work, we focus on the characterization of quadratic forms in unit-norm vectors, with unitarily invariant random kernel matrices, and we also provide some approximate but numerically accurate results concerning a non-unitarily invariant kernel matrix. Simulations are run with reference to a peculiar application scenario, the so-called spectrum sensing for wireless communications. Closed-form expressions for the moment generating function of the quadratic forms of interest are provided; this will pave the way to an analytical performance analysis of some spectrum sensing schemes, and will potentially assist in the rate analysis of some multi-antenna systems.
具有随机核矩阵的二次型在多元统计应用中无处不在，涵盖从信号处理到时间序列分析、生物医学系统设计、无线通信性能分析以及其他领域。它们的统计特性对于设计准则的制定和性能指标的高效计算都至关重要。为此，随机矩阵理论可以得到成功应用。特别是，来自所谓的有限维随机矩阵谱特征的最新进展使得能够分析无线通信和信号处理中一些感兴趣的场景。在这项工作中，我们专注于具有酉不变随机核矩阵的单位范数向量中二次型的特征描述，并且我们还提供了一些关于非酉不变核矩阵的近似但数值精确的结果。针对一种特殊的应用场景——所谓的无线通信频谱感知进行了仿真。给出了感兴趣的二次型的矩生成函数的闭式表达式；这将为一些频谱感知方案的解析性能分析铺平道路，并可能有助于一些多天线系统的速率分析。
