The minimum energy, and, more generally, the minimum cost, to transmit onebit of information has been recently derived for bursty communication wheninformation is available infrequently at random times at the transmitter.Furthermore, it has been shown that even if the receiver is constrained tosample only a fraction $\rho\in (0,1]$ of the channel outputs, there is nocapacity penalty. That is, for any strictly positive sampling rate $\rho>0$,the asynchronous capacity per unit cost is the same as under full sampling,i.e., when $\rho=1$. Moreover, there is no penalty in terms of decoding delay. The above results are asymptotic in nature, considering the limit as thenumber $B$ of bits to be transmitted tends to infinity, while the sampling rate$\rho$ remains fixed. A natural question is then whether the sampling rate$\rho(B)$ can drop to zero without introducing a capacity (or delay) penaltycompared to full sampling. We answer this question affirmatively. The mainresult of this paper is an essentially tight characterization of the minimumsampling rate. We show that any sampling rate that grows at least as fast as$\omega(1/B)$ is achievable, while any sampling rate smaller than $o(1/B)$yields unreliable communication. The key ingredient in our improvedachievability result is a new, multi-phase adaptive sampling scheme forlocating transient changes, which we believe may be of independent interest forcertain change-point detection problems.
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