  In this paper, we focus on the design and analysis of the Analog Fountain
Code (AFC) for short packet communications. We first propose a density
evolution (DE) based framework, which tracks the evolution of the probability
density function of the messages exchanged between variable and check nodes of
AFC in the belief propagation decoder. Using the proposed DE framework, we
formulate an optimisation problem to find the optimal AFC code parameters,
including the weight-set, which minimizes the bit error rate at a given
signal-to-noise ratio (SNR). Our results show the superiority of our code
design in comparison to existing code designs and thus the validity of the
proposed DE framework in the asymptotic block length regime. We then focus on
the selection of the precoder to improve the performance of AFC at short block
lengths. Simulation results show that lower precoder rates obtain better
realised rates over a wide SNR range for short information block length.
