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Minimum Energy to Send $k$ Bits Through the Gaussian Channel With and Without Feedback

机译：通过和不通过反馈通过高斯通道发送$ k $位的最小能量

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摘要

The minimum achievable energy per bit over memoryless Gaussian channels has been previously addressed in the limit when the number of information bits goes to infinity, in which case it is known that the availability of noiseless feedback does not lower the minimum energy per bit, which is ${-}1.59$ dB below the noise level. This paper analyzes the behavior of the minimum energy per bit for memoryless Gaussian channels as a function of $k$, the number of information bits. It is demonstrated that in this nonasymptotic regime, noiseless feedback leads to significantly better energy efficiency. In particular, without feedback achieving energy per bit of ${-}1.57$ dB requires coding over at least $k=10^6$ information bits, while we construct a feedback scheme that transmits a single information bit with energy ${-}1.59$ dB and zero error. We also show that unless $k$ is very small, approaching the minimal energy per bit does not require using the feedback link except to signal that transmission should stop.
机译：当信息位数达到无穷大时，先前已在无内存高斯信道中达到了每位最小可实现的最小能量，在这种情况下，已知无噪声反馈的可用性不会降低每位最小的能量，即比噪声水平低$ {-} 1.59 $ dB。本文分析了无记忆高斯通道每比特最小能量随$ k $（信息位数）的变化情况。结果表明，在这种非渐近状态下，无噪声反馈可显着提高能量效率。特别是，在没有反馈的情况下，要达到$ {-} 1.57 $ dB的每比特能量，就需要对至少$ k = 10 ^ 6 $个信息比特进行编码，而我们构造了一个反馈方案，该机制必须以能量$ {-}传输单个信息比特1.59 $ dB和零误差。我们还表明，除非$ k $非常小，否则接近最低位能量不需要使用反馈链接，只是发出信号表示传输应该停止。

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