  The fundamental task of a digital receiver is to decide the transmitted
symbols in the best possible way, i.e., with respect to an appropriately
defined performance metric. Examples of usual performance metrics are the
probability of error and the Mean Square Error (MSE) of a symbol estimator. In
a coherent receiver, the symbol decisions are made based on the use of a
channel estimate. This paper focuses on examining the optimality of usual
estimators such as the minimum variance unbiased (MVU) and the minimum mean
square error (MMSE) estimators for these metrics and on proposing better
estimators whenever it is necessary. For illustration purposes, this study is
performed on a toy channel model, namely a single input single output (SISO)
flat fading channel with additive white Gaussian noise (AWGN). In this way,
this paper highlights the design dependencies of channel estimators on target
performance metrics.
