Given an AWGN channel, we look at the problem of designing a source controlled binary antipodal signaling system for transmitting blocks of binary symbols generated either by a Markov chain source (MC) or by a hidden Markov model source (HMM). The goal is to minimize the average SNR required for any given block error rate. The particular case where the binary source is memoryless with nonuniform symbol probabilities has been studied in I. Korn et al. (2003) by optimally allocating the signal energies. In this paper we extend the previous work to include the important case of sources with memory. The proposed system integrates the block sorting burrows wheeler transform (BWT) with an optimal energy allocation scheme based on the first order probabilities of the transformed symbols. This method is compared with standard communication techniques for the transmission of sources with memory.