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We design and evaluate a novel and highly efficient algorithm, named sparsely interleaved estimation and decoding (SIED), for joint iterative channel estimation and decoding for bit-interleaved coded modulation (BICM) over flat correlated fading channels. In SIED, in order to effectively prevent error propagation, the decoding proceeds until the current decoder stage converges, and only then feeds back the best achievable decoding results to refine channel estimates. The on-line convergence is detected through the soft decoding metrics (SDMs). Further, we provide a unified analytic framework for both the choice of filter length and an asymptotic bit error rate (BER) analysis, where the key parameter is the power correlation coefficient between the fading and its estimate. Simulation results verify that our SIED algorithm can successfully break the barrier of the channel estimation error caused by conventional pilot symbol-assisted modulation (PSAM), and lead the BER to approach the ideal asymptotic performance. Compared to the classical iteratively filtered PSAM (IF-PSAM) algorithm, our proposed SIED algorithm has more than 0.5-dB improvement in Eb/N0 for the normalized fade rate fdTs=0.05 with much lower complexity.