Minimum energy to send k bits with and without feedback

The question of minimum achievable energy per bit over memoryless channels has been previously addressed in the limit of number of information bits going to infinity, in which case it is known that availability of noiseless feedback does not lower the minimum energy per bit. This paper analyzes the behavior of the minimum energy per bit for memoryless Gaussian channels as a function of the number of information bits. It is demonstrated that in this non-asymptotic regime, noiseless feedback leads to significantly better energy efficiency. A feedback coding scheme with zero probability of block error and finite energy per bit is constructed. For both achievability and converse, the feedback coding problem is reduced to a sequential hypothesis testing problem for Brownian motion.