  Covert communication over an additive white Gaussian noise (AWGN) channel
with finite block length is investigated in this paper. The attention is on the
covert criterion, which has not been considered in finite block length
circumstance. As an accurate quantity metric of discrimination, the variation
distance with given finite block length n and signal-noise ratio (snr) is
obtained. We give both its analytic solution and expansions which can be easily
evaluated. It is shown that K-L distance, which is frequently adopted as the
metric of discrimination at the adversary in asymptotic regime, is not
convincing in finite block length regime compared with the total variation
distance. Moreover, the convergence rate of the total variation with different
snr is analyzed when the block length tends to infinity. The results will be
very helpful for understanding the behavior of the total variation distance and
practical covert communication.
