Quadratic Forms in Random Matrices with Applications in Spectrum Sensing
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 b...
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MDPI AG
2025-01-01
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| author | Daniel Gaetano Riviello Giusi Alfano Roberto Garello |
| author_facet | Daniel Gaetano Riviello Giusi Alfano Roberto Garello |
| author_sort | Daniel Gaetano Riviello |
| collection | DOAJ |
| description | 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 <i>polynomial ensembles</i> 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. |
| format | Article |
| id | doaj-art-3c35d1c0fafa4060abe99cdee037e775 |
| institution | Kabale University |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj-art-3c35d1c0fafa4060abe99cdee037e7752025-01-24T13:31:52ZengMDPI AGEntropy1099-43002025-01-012716310.3390/e27010063Quadratic Forms in Random Matrices with Applications in Spectrum SensingDaniel Gaetano Riviello0Giusi Alfano1Roberto Garello2CNR-IEIIT, Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni, Consiglio Nazionale delle Ricerche, 10129 Turin, ItalyDepartment of Electronics and Telecommunications (DET), Politecnico di Torino, 10129 Turin, ItalyDepartment of Electronics and Telecommunications (DET), Politecnico di Torino, 10129 Turin, ItalyQuadratic 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 <i>polynomial ensembles</i> 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.https://www.mdpi.com/1099-4300/27/1/63spectrum sensingquadratic formsmulti-antennarandom matrix theorycognitive radios6G |
| spellingShingle | Daniel Gaetano Riviello Giusi Alfano Roberto Garello Quadratic Forms in Random Matrices with Applications in Spectrum Sensing Entropy spectrum sensing quadratic forms multi-antenna random matrix theory cognitive radios 6G |
| title | Quadratic Forms in Random Matrices with Applications in Spectrum Sensing |
| title_full | Quadratic Forms in Random Matrices with Applications in Spectrum Sensing |
| title_fullStr | Quadratic Forms in Random Matrices with Applications in Spectrum Sensing |
| title_full_unstemmed | Quadratic Forms in Random Matrices with Applications in Spectrum Sensing |
| title_short | Quadratic Forms in Random Matrices with Applications in Spectrum Sensing |
| title_sort | quadratic forms in random matrices with applications in spectrum sensing |
| topic | spectrum sensing quadratic forms multi-antenna random matrix theory cognitive radios 6G |
| url | https://www.mdpi.com/1099-4300/27/1/63 |
| work_keys_str_mv | AT danielgaetanoriviello quadraticformsinrandommatriceswithapplicationsinspectrumsensing AT giusialfano quadraticformsinrandommatriceswithapplicationsinspectrumsensing AT robertogarello quadraticformsinrandommatriceswithapplicationsinspectrumsensing |