### Abstract

We address the Primary User (PU) detection (spectrum sensing) problem, relevant to cognitive radio, from a finite random matrix theoretical (RMT) perspective. Specifically, we employ recently-derived closed-form and exact expressions for the distribution of the standard condition number (SCN) of uncorrelated and semi-correlated random dual central Wishart matrices of finite sizes in the design Hypothesis-Testing algorithms to detect the presence of PU signals. In particular, two algorithms are designed, with basis on the SCN distribution in the absence (H _{0}) and in the presence (H _{1}) of PU signals, respectively. Due to an inherent property of the SCN's, the H _{0} test requires no estimation of SNR or any other information on the PU signal, while the H _{1} test requires SNR only. Further attractive advantages of the new techniques are: a) due to the accuracy of the finite SCN distributions, superior performance is achieved under a finite number of samples, compared to asymptotic RMT-based alternatives; b) since expressions to model the SCN statistics both in the absence and presence of PU signal are used, the statistics of the spectrum sensing problem in question is completely characterized; and c) as a consequence of a) and b), accurate and simple analytical expressions for the receiver operating characteristic (ROC) - both in terms of the probability of detection as a function of the probability of false alarm (P _{D} versus P _{F}) and in terms of the probability of acquisition as a function of the probability of miss detection (P _{A} versus P _{M}) -are yielded. It is also shown that the proposed finite RMT-based algorithms outperform all similar alternatives currently known in the literature, at a substantially lower complexity. In the process, several new results on the distributions of eigenvalues and SCNs of random Wishart Matrices are offered, including a closed-form of the Marchenko-Pastur's Cumulative Density Function (CDF) and extensions of the latter, as well as variations of asymptotic the distributions of extreme eigenvalues (Tracy-Widom) and their ratio (Tracy-Widom-Curtiss), which are simpler than those obtained with the "spiked population model".

Original language | English |
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Article number | 6094130 |

Pages (from-to) | 164-175 |

Number of pages | 12 |

Journal | IEEE Transactions on Communications |

Volume | 60 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 Jan 1 |

### Keywords

- Cognitive radio
- Hypothesis test
- Random matrix
- Spectrum sensing
- Standard condition number

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

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## Cite this

*IEEE Transactions on Communications*,

*60*(1), 164-175. [6094130]. https://doi.org/10.1109/TCOMM.2011.112311.100721