In this paper, we develop the adaptive algorithm for system identification where the model is sparse. The classical configurations of adaptive filtering are system identification, prediction, noise cancellation. The low-complexity adaptive filtering algorithms are developed which exploit the sparsity of signals and systems are designed. We design and develop the adaptive algorithm which we term least mean square (LMS), normalized least mean square and zero attractors normalized LMS, These algorithms are analysed and applied to the identification of sparse systems. The reweighted ZA-NLMS (RZA-NLMS) are developed to improve the filtering performance. In common sensing, the L_1 relaxation is applied to improve the performance of adaptive LMS type filtering. The ZA-LMS is developed by combining the quadratic LMS cost function and a L_1 norm penalty, which helps to generate a zero attractor in the LMS algorithm. This results in two new algorithms, the Zero-Attracting LMS (ZA-LMS) and the Reweighted Zero-Attracting LMS (RZA-LMS). During the filtering process, this zero attractor proposed sparsity in taps and therefore the speed of convergence increased in the sparse system identification process.