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In this paper, decoding of LDPC convolutional codes with rational parity-check matrices (LDPC-CC-RPCM) is investigated. We show that Tanner graph of every LDPC-CC-RPCM can always be transformed into an equivalent one with enlarged girth and finite memory order suitable for practical pipeline decoder. Based on the transformed graph, a dynamic scheduling-aided decoding scheme with the enhancement of signal perturbation and error cancellation is presented to improve the convergence speed and bit-error-rate performance in both of the waterfall and error-floor regions. Simulation results also reveal that LDPC-CC-RPCM may outperform ordinary LDPC-CC with polynomial parity-check matrices in some cases under the same code rate and decoding complexity.