Abstract: The basic theory of reconstruction signals is that the signals are sparse or approximate sparse in a transform domain. Based on the approximate sparsity of speech signal in the DCT domain, compressed sensing theory is applied to reconstruct speech signal. The iterative hard thresholding (IHT) algorithm with good performances is widely used to reconstruct signals, however, its convergence speed is too slow. How to improve the convergence speed of the iterative hard thresholding algorithm has been a hot topic.The accelerated Landweber iterative hard thresholding (ALIHT) algorithm in the DCT domain for speech reconstruction is proposed to solve the problem that the convergence speed is too slow when the iterative hard thresholding algorithm is applied to the compressed sensing. The accelerated Landweber iterative hard thresholding algorithm firstly makes the original speech signal to its DCT domain, and then speeds up the convergence speed by discomposing the each one step of the Landweber iteration in the iterative hard thresholding algorithm into two steps as the matrix computation and solution in the DCT domain, and modifying the matrix computation step. The experimental simulations show that the accelerated Landweber iterative hard thresholding algorithm increases convergence speed and reduces the calculation measures.