  Deep learning (DL) has emerged as an effective tool for channel estimation in
wireless communication systems, especially under some imperfect environments.
However, even with such unprecedented success, DL methods are often regarded as
black boxes and are lack of explanations on their internal mechanisms, which
severely limits further improvement and extension. In this paper, we present a
preliminary theoretical analysis on DL based channel estimation for
multiple-antenna systems to understand and interpret its internal mechanism.
Deep neural network (DNN) with rectified linear unit (ReLU) activation function
is mathematically equivalent to a piecewise linear function. Hence, the
corresponding DL estimator can achieve universal approximation to a large
family of functions by making efficient use of piecewise linearity. We
demonstrate that DL based channel estimation does not restrict to any specific
signal model and approaches to the minimum mean-squared error (MMSE) estimation
in various scenarios without requiring any prior knowledge of channel
statistics. Therefore, DL based channel estimation outperforms or is at least
comparable with traditional channel estimation, depending on the types of
channels. Simulation results confirm the accuracy of the proposed
interpretation and demonstrate the effectiveness of DL based channel estimation
under both linear and nonlinear signal models.
