What is the optimal source-channel communication system for a given finite block length? The problem of obtaining the vector transformations that optimally map between the m-dimensional source space and the k-dimensional channel space is considered under a given channel power constraint and mean square error distortion measure. Closed form necessary conditions for optimality of the encoder and decoder mappings are derived. The optimal mappings are obtained using an iterative algorithm that updates encoder and decoder mappings according to optimality conditions at each iteration. Such mappings are used in a practical analog joint source channel system that transmits a continuous alphabet discrete time source over a noisy channel. Numerical results are presented for several source-channel distributions and it is shown that the optimal mappings outperform the previous heuristic mappings for both bandwidth expansion and compression.