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An optimal joint source-channel coding (OJSCC) scheme is developed for encoding memoryless generalized Gaussian distribution (GGD) sources and transmission over noisy channels, including memoryless channels and bursty channels. In particular, we are able to incorporate both transition probability and bit error rate of a bursty channel model into an end-to-end rate-distortion (R-D) function. This R-D function results in an optimum tradeoff between source coding accuracy and channel error protection under a fixed transmission rate. Based on the results of OJSCC, we further proposed a robust joint source-channel coding (RJSCC) scheme based on a combination of OJSCC with allpass filtering source reshaping. Experimental results based on a broad class of GGD sources suggest that the RJSCC scheme consistently outperforms the OJSCC scheme when the shape factor, a parameter of GGD, is less than 2.0. The comparison of RJSCC with the state-of-the-art robust quantization (RQ) [1] shows that RJSCC is comparable with RQ at low bit rate and significantly better than RQ at high bit rate. In addition, an image transmission system using RJSCC demonstrates that the RJSCC system can be efficiently employed in practical applications