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.
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