Spectrum sensing is the most important part in cognitive radios. Wideband spectrum sensing requires high speed and large data samples. It makes sampling process challenging and expensive. In this thesis, we propose wideband spectrum sensing for cognitive radio using compressive sensing to address challenges in sampling and data acquisition during spectrum sensing. Compressive sensing based spectrum sensing for a single network is extended to large frequency overlapping networks and joint reconstruction scheme is developed to enhance the performance at minimal cost. The joint sparsity in large networks is used to improve the compressive sensing reconstruction in large networks. Further, a novel compressive sensing method for binary signal is proposed. Unlike general compressive sensing solution based on optimization process, a simple, reliable and quick compressive sensing method for binary signal using bipartite graph, edge recovery and check-sum method is developed. The proposed models and methods have been verified, proved and compared with existing approaches through numerical analysis and simulations.