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A new algorithm to reduce the computational complexity of iterative decoding is presented. The algorithm reduces the complexity by removing certain branches from the corresponding trellis diagram. A threshold on a priori values is used as the elimination criterion. To assess the performance, numerical results for serially concatenated convolutional codes transmitted over an additive white Gaussian noise channel is presented. The results show that for the system considered, the algorithm can reduce the complexity by one third with a negligible degradation in error performance.