Electromyogram (EMG) signal decomposition is a special case of convolutive blind source separation (CBSS) problem. Specific characteristics of EMG signals and their corresponding channel matrix leads us to proposing family of algorithms using multi-domain sparsity of EMG signals and low rank structure of channel matrix. In this thesis we propose family of algorithms for CBSS that exploits sparse source signals in multiple domains. Our approach is to jointly estimate the sparse vector of concatenated sources and mixing matrix. We propose CBSS algorithms which are composed of two steps for multi domain sparse vector recovery and mixing matrix estimation. In particular, we propose two efficient algorithms coined multi domain adaptive thresholding (MDAT) for both under-determined and over-determined cases and multi domain blind approximate message passing (MDBAMP) for multi-channel convolutive blind sparse source separation for under-determined cases, when the source signals are sparse in multiple domains. In multichannel source separation, channels are usually correlated. Therefore, in this thesis, we propose a CBSS algorithm that exploits both low rank channel matrix and sparse source signals in the time domain. In sparse vector recovery and low rank matrix estimation algorithms, the mixing matrix and the linear operator of low rank matrix are known. Here, we propose a CBSS algorithm which is composed of two steps for sparse vector recovery and low rank matrix estimation when the mixing matrix and the corresponding operator of low rank channel matrix are unknown. In particular, we propose an efficient and guaranteed algorithm coined SMiRE for multi-channel convolutive blind sparse source separation, where the channel matrix is low rank. All three proposed algorithms are based on the sparsity of source signals and they are solved using compressed sensing (CS) methods. Checking the restricted isometry property (RIP) of measurement or mixing matrices is a promising approach to guarantee the unique solution in compressed sensing methods. In convolutive source separation, mixing matrices have banded block Toeplitz structure. Therefore, the analytical RIP bound for banded block Toeplitz matrices is calculated in this thesis as an initial step prior to proposing sparsity based algorithms.