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Channels with a long but sparse impulse response arise in a variety of wireless communication applications, such as high definition television (HDTV) terrestrial transmission and underwater acoustic communications. By adopting the ℓ1-norm as the sparsity metric of the channel response, the channel estimation is formulated as a complex-valued convex optimization problem. A fast fixed point iteration algorithm is developed to solve the resultant complex-valued ℓ1-minimization problem. The proposed fast channel estimation algorithm is easy to implement and has a low computational complexity of O (N log N) per iteration with N the signal length. Simulation results are provided to demonstrate the performance of the proposed fixed point algorithm.