Variable-Rate Channel Capacity

Abstract

Abstract-This paper introduces the notions of variable-to-fixed and fixed-to-variable channel capacity, without feedback. For channels that satisfy the strong converse, these notions coincide with the conventional Shannon capacity. For channels that do not behave ergodically, the conventional fixed-rate Shannon capacity only depends on least-favorable channel conditions, while the variable-rate capacity notions are able to capture the whole range of channel states and their likelihood, even in the absence of any side information about channel state at the transmitter. Particular emphasis is placed on memoryless channels that are governed by finitely valued states. We show that (single-user) variable-to-fixed channel capacity is intimately connected to the capacity region of broadcast channels with degraded message sets, and we give an expression for the fixed-to-variable capacity.

Feedback in the Non-Asymptotic Regime

“…Furthermore, our codes with termination have a particularly convenient structure: the encoder uses the feedback link only to choose the time when to stop the transmission (by sending the termination symbol), and otherwise simply sends a fixed message-dependent codeword. The codes with such structure have been called fixed-to-variable (FV), or fountain, codes in [19]. Thus, in short, we have demonstrated that fountain codes can achieve 90% of the capacity of the BSC with crossover probability at average blocklength and with zero probability of error.…”

“…[19], are VLFT codes required to satisfy two additional requirements: is a function of and the encoder is not allowed to use feedback, i.e., (10) holds. The fundamental limit and the -capacity of variable-length codes are given by (11) (12) 4) fixed-to-variable codes, or FV codes, defined in [19] are also required to satisfy (10), while the stopping time is 2 (13) and therefore, such codes are zero-error VLFT codes. Of course, not all zero-error VLFT codes are FV codes, since in general condition (10) does not necessarily hold.…”

“…(12), is (15) The proof is given in the Appendix. In general, it is known [19,Theorem 16] that the VL capacity, , is equal to the conventional fixed-blocklength capacity without feedback, , for any non-anticipatory channel (not necessarily satisfying the strong converse). On the other hand, the capacity of FV codes for state-dependent non-ergodic channels can be larger than [19].…”

Variable-length channel codes with probabilistic delay guarantees

“…Hence, we conclude that log M * VLFT * ( , ε r , ε d ) ≤ C +K √ , for all sufficiently large ∈ Z + , which, in turn, implies (6).…”

“…(i) Recall that an ( , M, ε) variable-length (VL) code [6], a variable-length feedback (VLF) code [2], and a variablelength feedback code with termination (VLFT) [2], subject to average delay constraint are defined in the same way as their counterparts in Definition …”