الفهرس 
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Abstract
Compressive sensing is a sampling technique that exploits the sparsity of signals to operate at sampling rates below the Nyquist rate. Theoretical results ensure that, if a signal is sparse or has a sparse representation in some domain, then it can be reconstructed accurately from a small number of measurements obtained at much lower rates than the Nyquist rate using simple optimization algorithms. It has therefore become a research area of increasing interest over the past decade in applications that require processing a large amount of data in short periods of time. Such is the case in wideband communication systems and cognitive radio, in which sampling the wideband spectrum at the Nyquist rate or higher is both time consuming and expensive. Our research focuses mainly on applications related to cognitive radio, and particularly spectrum sensing, which enables unlicensed users to opportunistically access the spectrum when licensed user are inactive by using ecient sensing methods to identify unoccupied portions of the spectrum. There are three main design aspects in compressive sensing; the design of the basis or sparsifying matrix, the sensing (measurement) matrix and the reconstruction algorithm. The basis matrix depends on the domain in which the signal of interest is known to be sparse; such as the Fourier domain or the Wavelet domain. The sensing matrix is what de nes the compression ratio and how the compressed measurements are obtained. The sensing matrix, together with the basis matrix must satisfy the restricted isometry property, a property that guarantees that the matrices preserve the length of sparse vectors to which they are multiplied, which in turn guarantees successful reconstruction. Certain random constructions have been proven to satisfy this condition with high probability, however, deterministic sensing matrices have gained increasing interest in research since deterministic matrices are much easier to implement, generate. More importantly, it is much easier to verify the restricted isometry property for deterministic matrices. In this dissertation, the di erent constructions of sensing matrices are explored; including constructions based on chaotic codes, algebraic curves, polynomials over nite elds, linear and nonlinear codes. . . ) as well as the possibility of utilizing the properties of certain constructions to improve the overall performance of the system. Chaotic codes are studied and applied to collaborative cognitive radio networks and have been proven to improve the performance, complexity and security of the system. The theory of random matrices is also studied, and its application in spectrum sensing in cognitive radio is investigated. This dissertation focuses on the part of random matrix theory that studies the limiting distributions of the eigenvalues of large random matrices. Since pure noise signals are characterized by their randomness, the matrices arising from noise in spectrum sensing applications follow the distributions dictated by the random matrix theory, and any deviation from the threshold and limits dictated by it indicate the presence of a deterministic signal, which means that it can be used to di erentiate between the signali present case and the noise-only case in cognitive radio. Such spectrum sensing methods are known as eigenvalue-based detection methods. Deterministic matrices are also proposed for compressive eigenvalue-based spectrum sensing as part of this research, particularly ones based on Golay-paired Hadamard matrices. In this dissertation, complementary matrix sets are designed from Golay-paired Hadamard matrices to be used as sensing matrices in collaborative eigenvalue-based compressive spectrum sensing in a way the overall system performance equivalent to the performance of the non-compressive framework.