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Luby-Transform (LT) codes are a class of Fountain codes which can approach the capacity of the erasure channels. In this paper, we investigate the performance of the LT codes over AWGN channels with BPSK modulation. First, the ensemble weight distribution of an LT code is derived. Secondly, we use a refined union bound to analyze the performance of LT codes under the maximum likelihood decoding assumption. Finally, simulation results using the iterative sum-product decoding algorithm are given for the comparison with the computational bounds.