  This paper investigates covert communication over an additive white Gaussian
noise (AWGN) channel in finite block length regime on the assumption of
Gaussian codebooks. We first review some achievability and converse bounds on
the throughput under maximal power constraint. From these bounds and the
analysis of TVD at the adversary, the first and second asymptotics of covert
communication are investigated by the help of some divergences inequalities.
Furthermore, the analytic solution of TVD, and approximation expansions which
can be easily evaluated with given snr (signal noise ratio) are presented. In
this way, the proper power level for covert communication can be approximated
with given covert constraint of TVD, which leads to more accurate estimation of
the power compared with preceding bounds. Further elaboration on the effect of
such asymptotic characteristics on the primary channels's throughput in finite
block regime is also provided. The results will be very helpful for
understanding the behavior of the total variation distance and practical covert
communication.
