Variable-length Feedback Codes with Several Decoding Times for the Gaussian Channel

Abstract

We investigate variable-length feedback (VLF) codes for the Gaussian point-to-point channel under maximal power, average error probability, and average decoding time constraints. Our proposed strategy chooses K < ∞ decoding times n1, n2, . . . , nK rather than allowing decoding at any time n = 0, 1, 2, . . . . We consider stop-feedback, which is one-bit feedback transmitted from the receiver to the transmitter at times n1, n2, . . . only to inform her whether to stop. We prove an achievability bound for VLF codes with the asymptotic1− , where ln (K) (•) denotes the K-fold nested logarithm function, N is the average decoding time, and C(P ) and V (P ) are the capacity and dispersion of the Gaussian channel, respectively. Our achievability bound evaluates a non-asymptotic bound and optimizes the decoding times n1, . . . , nK within our code architecture. Index Terms-Variable-length stop-feedback codes, Gaussian channel, second-order achievability bound.

Variable-Length Stop-Feedback Codes With Finite Optimal Decoding Times for BI-AWGN Channels

“…Next, we follow [18] in describing a VLSF code with m decoding times for the BI-AWGN channel. Due to BPSK, we omit the power constraint from the definition.…”

“…In [18], Yavas et al proved an achievability bound for an (l, n m 1 , M, ) VLSF code for the AWGN channel. With a slight modification, this result is directly applicable to the BI-AWGN channel.…”

“…Remark 2. In [18] (and its full version [23]), Yavas et al obtained Theorem 3 by constructing a random VLSF code according to distribution (18) and applying an information density threshold decoder that favors the largest message index whose cumulative information density exceeds γ for the first time among any other message indices at decoding times {n 1 , n 2 , . .…”

“…In [18], Yavas et al developed an achievability bound for VLSF codes with m decoding times for the additive white Gaussian noise channel with capacity C, dispersion V , and maximal power constraint P . The asymptotic expansion of the maximum message size M is given by ln…”

“…We first develop tight approximations on the tail probability of length-n cumulative information density. Building on the result of Yavas et al [18], for a fixed information density threshold γ, we formulate an integer program of minimizing the upper bound on average blocklength over all decoding times n 1 , n 2 , . .…”