It has been recently shown that if a signal can be compressed in some basis, then it can be reconstructed in such basis from a certain number of random projections. By al- lowing additional distortion, this holds even if the projections are corrupted by noise. We extend this result by showing that it is possible to exploit prior knowledge (e.g., if the signal is a realization of a stochastic process,) to significantly improve reconstruc- tion performance. This is done in a fashion resembling standard joint source-channel coding of digital sources. Moreover, the exploitation of such knowledge allows for reconstruction in bases where the signal is not sparse.