  A new approach for blind channel equalization and decoding, variational
inference, and variational autoencoders (VAEs) in particular, is introduced. We
first consider the reconstruction of uncoded data symbols transmitted over a
noisy linear intersymbol interference (ISI) channel, with an unknown impulse
response, without using pilot symbols. We derive an approximate maximum
likelihood estimate to the channel parameters and reconstruct the transmitted
data. We demonstrate significant and consistent improvements in the error rate
of the reconstructed symbols, compared to existing blind equalization methods
such as constant modulus, thus enabling faster channel acquisition. The VAE
equalizer uses a convolutional neural network with a small number of free
parameters. These results are extended to blind equalization over a noisy
nonlinear ISI channel with unknown parameters. We then consider coded
communication using low-density parity-check (LDPC) codes transmitted over a
noisy linear or nonlinear ISI channel. The goal is to reconstruct the
transmitted message from the channel observations corresponding to a
transmitted codeword, without using pilot symbols. We demonstrate improvements
compared to the expectation maximization (EM) algorithm using turbo
equalization. Furthermore, unlike EM, the computational complexity of our
method does not have exponential dependence on the size of the channel impulse
response.
