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In many application of noise cancellation, the changes in signal characteristics could be quite fast. This requires the utilization of adaptive algorithms, which converge rapidly. Least Mean Squares (LMS) adaptive filters have been used in a wide range of signal processing application. The Recursive Least Squares (RLS) algorithm has established itself as the "ultimate" adaptive filtering algorithm in the sense that it is the adaptive filter exhibiting the best convergence behavior. Unfortunately, practical implementations of the algorithm are often associated with high computational complexity and/or poor numerical properties. Recently adaptive filtering was presented that was based on Matching Pursuits, have a nice tradeoff between complexity and the convergence speed. This paper describes a new approach for noise cancellation in speech enhancement using the new adaptive filtering algorithm named fast affine projection algorithm (FAPA). The simulation results demonstrate the good performance of the FAPA in attenuating the noise.