  This paper investigates the soft covering lemma under both the relative
entropy and the total variation distance as the measures of deviation. The
exact order of the expected deviation of the random i.i.d. code for the soft
covering problem problem, is determined. The proof technique used in this paper
significantly differs from the previous techniques for deriving exact exponent
of the soft covering lemma. The achievability of the exact order follows from
applying the change of measure trick (which has been broadly used in the large
deviation) to the known one-shot bounds in the literature. For the ensemble
converse, some new inequalities of independent interest derived and then the
change of measure trick is applied again. The exact order of the total
variation distance is similar to the exact order of the error probability, thus
it adds another duality between the channel coding and soft covering. Finally,
The results of this paper are valid for any memoryless channels, not only
channels with finite alphabets.
