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High resolution broadband source direction of arrival (DOA) estimation is a challenge problem in acoustic array signal processing. While the compressive sensing (CS) is a recent theory exploring the signal sparsity representation, which has been received considerable attentions in various research areas. In this paper, we study on the DOA estimation under CS framework (DOA-CS) making use of the spatial sparsity. Two approaches for constructing the sparse basis matrix are reviewed in details. The formulation of the DOA estimation method under CS for broadband signal is presented following the work of. Intensive experiments for speech DOA estimation application have been conducted to evaluate the DOA estimation accuracy, the impact of the parameter selection and SNRs on its performance. Some advantages and disadvantages have been drawn and discussed according to the simulation results. Performance comparison with MUSIC algorithm showed that the DOA-CS gives higher DOA estimation accuracy than MUSIC algorithm does. It can be concluded that DOA-CS is a new and promising approach to provide robust and high DOA estimation accuracy under different noise levels for narrowband or broadband signals. It deserves further research for developing computational efficient DOA-CS estimation algorithms as well as the methods for the optimal parameter choice.