Wideband spectrum sensing (WSS) is a significant challenge in cognitive radios (CRs) due to requirement of very high-speed analog-to-digital converters (ADCs), operating at or above the Nyquist rate. To alleviate the need for high sampling rates, compressed sensing can be applied to WSS. Compressed sensing is a signal processing technique to recover high dimensional signals from a fewer measurements than conventional sampling theory demands. In compressed sensing, the superior algorithms are the ones that can recover the signal with fewer measurements.In many applications of compressed sensing such as WSS, the signal has a block sparse structure which can be used to reduce the required sampling rate. Here, we propose a family of iterative algorithms for the recovery of block sparse signals, referred to as iterative reweighted 2 / 1 minimization algorithms (IR-2 / 1 ). As an example, we apply the proposed algorithms to WSS. Our simulation and analytical results on the recovery of both ideally and approximately block sparse signals show that the proposed iterative algorithms have significant advantages in terms of accuracy and the number of required measurements over the existing recovery methods at a small cost in computational complexity.The approximate message passing (AMP) algorithm of Donoho et al. has attracted much attention due to its remarkable performance/complexity trade-off. In this thesis, we also propose a weighting/reweighting scheme to improve the performance of AMP algorithm for the recovery of block sparse signals with known borders. For the case of unknown borders, we propose an iterative algorithm which combines a border detection scheme with a recovery algorithm for block sparse signals with known borders. Simulation results, both in noiseless and noisy scenarios, show a considerably better performance/complexity trade-off compared to other state-of-the-art recovery algorithms.Although the recovery methods can be applied to WSS for the spectrum reconstruction to detect the spectrum holes, to further reduce the sampling rate, one can iii find a sufficiently large fraction of spectrum holes without spectrum reconstruction. Here, we first propose a very low-complexity zero-block detection scheme that can detect a large fraction of spectrum holes from the sub-Nyquist samples, even when the undersampling ratio is very small. We then propose an iterative low-complexity scheme for the reliable detection of zero blocks in a block sparse signal. This scheme is based on the application of verification-based (VB) recovery algorithms in compressed sensing to block sparse signals. To apply both schemes to WSS, we devise a block sparse sensing matrix by designing a novel analog-to-information converter (AIC). The AIC, the sensing matrix and the VB algorithms can be optimized such that the largest number of zero blocks for a given number of measurements can be detected. These works introduce a new paradigm in the recovery of block sparse signals, where one is interested in partial detection of the complement ...
