In this paper, we consider communication over an energy-harvesting (EH) channel which has an input alphabet , an output alphabet and an infinite-sized battery that stores energy harvested from the environment. The channel law of the EH channel is characterized by a conditional distribution where and denote the channel input and output respectively. A source node wants to transmit a message to a destination node through the EH channel. Let be a cost function associated with the EH channel, where represents the amount of energy used for transmitting . At each discrete time , a random amount of energy arrives at the battery buffer and the source transmits a symbol such that
This implies that the total harvested energy must be no smaller than the “energy” of the codeword at every discrete time for transmission to take place successfully. The destination receives from the channel output in time slot for each , where is distributed according to the channel law such that for all . We assume that are independent and identically distributed (i.i.d.), where is a non-negative random variable. To simplify notation, we write if there is no ambiguity. Throughout the paper, we let , the expected value of , denote the battery recharge rate, and we assume that . All results presented in this paper depend on the random variable
only through its first and second moments rather than its distribution.
This paper focuses on the additive white Gaussian noise (AWGN) model where , and . Under the AWGN model, the received symbol at time can be expressed as
for each time where is a standard normal random variable which is independent of and the random variables are independent. Reference  has shown that the capacity of this channel is and proposed two capacity-achieving schemes, namely save-and-transmit and best-effort.
The save-and-transmit scheme consists of an initial saving phase and a subsequent transmission phase. The transmitter remains silent in the saving phase so that energy accumulates in the battery. In the transmission phase, the transmitter sends the symbols of a random Gaussian codeword with varianceas long as the battery has sufficient energy where denotes some small offset from .
The best-effort scheme has a simpler design than the save-and-transmit scheme as it does not have an initial saving phase. As long as the transmitter has sufficient energy to output the symbols of a random Gaussian codeword with variance for some , information gets transmitted.
Following reference , a number of non-asymptotic achievable rates for the save-and-transmit scheme have been presented in references [2, 3, 4]. By contrast, no non-asymptotic achievable rate exists for the best-effort scheme except for a special discrete memoryless EH channel with infinite battery studied in  and a special discrete memoryless EH channel with no battery studied in . A main goal of this paper is to provide a non-asymptotic achievable rate for save-and-transmit with a saving phase of arbitrary length, which will immediately imply a non-asymptotic achievable rate for best-effort.
I-a Related Work
The channel capacity of the AWGN EH channel was characterized in , which showed that the capacity of the AWGN channel with an infinite-sized battery subject to EH constraints is equal to the capacity of the same channel under an average power constraint where the average power equals the average recharge rate of the battery. In particular, save-and-transmit [1, Sec. IV] and best-effort [1, Sec. V] were proposed as capacity-achieving strategies.
For a fixed tolerable error probability , reference.  has performed a finite blocklength analysis of save-and-transmit proposed in  and obtained a non-asymptotic achievable rate for the AWGN EH channel. The first-, second- and third-order terms of the non-asymptotic achievable rate presented in [2, Th. 1] are equal to the capacity, and respectively where the big- notation is used for a positive term which involves the blocklength and which approaches zero at a rate no slower than the argument of the notation as approaches infinity. The formal definition of the big- notation can be found in Section I-D. Subsequently, reference  has refined the analysis in  and improved the second-order term to . Reference  has further improved the second-order term to if where denotes the cumulative density function (cdf) of the standard normal random variable. All the second-order terms obtained by the above studies and the current study are inferior (more negative) to the following second-order term corresponding to the non-EH AWGN channel where all energy is available to the transmitter at the onset and (1) is replaced with the conventional power constraint [10, Th. 54]: .
For the block energy arrival model where the length of each energy block grows sublinearly in [11, 12, 4], reference  has proved that save-and-transmit achieves the second-order term if . In addition, a non-asymptotic upper bound on the second-order term has been proved in  for a general coding scheme, implying that save-and-transmit achieves the optimal second-order scaling if .
I-B Main Contributions
In this paper, we analyze a save-and-transmit scheme with a saving phase of arbitrary length (including zero, which corresponds to the best-effort scheme) and derive a non-asymptotic achievable rate. The derivation involves designing the transmit power to be strictly less than the battery recharge rate (unlike the design in [2, 4] which sets the transmit power equal to ) so that we can effectively bound the number of mismatched positions between the desired transmitted codeword and the actual transmitted codeword subject to a fixed blocklength. The aforementioned non-asymptotic achievable rate is extended to the block energy arrival model where the length of each energy block grows sublinearly in [11, 12, 4]. Our analyzed best-effort and save-and-transmit achieve the second-order scalings and respectively. If , the second-order term for a general coding scheme has been proved to be bounded above by as explained in the previous subsection, which implies that both analyzed schemes achieve the optimal second-order scaling if grows faster than .
In order to compare our results with the existing ones, we focus on the i.i.d. energy arrival case (i.e., ) in the remainder of this subsection. This work provides the first finite blocklength analysis of the best-effort scheme for the AWGN EH channel and presents a non-asymptotic achievable rate. It shows that the first- and second-order terms of the asymptotic achievable rate are equal to the capacity and respectively. This second-order scaling significantly improves the state-of-the-art result in  which did not derive a bound on the vanishing rate for the second-order term. In addition, this work obtains a new non-asymptotic achievable rate for save-and-transmit which outperforms the state-of-the-art result for save-and-transmit with transmit power equal to [4, Th. 1].
I-C Paper Outline
This paper is organized as follows. The notation of this paper is presented in the next subsection. Section II presents the model of the AWGN EH channel. Section III describes the save-and-transmit scheme, states the corresponding preliminary results, and presents the main result — a new non-asymptotic achievable rate for save-and-transmit with a saving phase of arbitrary length. A non-asymptotic achievable rate for best-effort is then obtained by setting the length of the saving phase to be zero. Section IV generalizes the non-asymptotic results in Section III to the block energy arrival model. Section V presents the proof of the new non-asymptotic achievable rate for save-and-transmit for the block energy arrival model which subsumes the proof for the i.i.d. energy arrival model. Section VI contains numerical results which demonstrate the performance advantage of allowing the transmit power for a save-and-transmit to back off from the battery recharge rate in the high battery recharge rate regime for both i.i.d. and block energy arrivals. Section VII concludes the paper.
We use , , and to denote standard asymptotic Bachmann-Landau notations that involve the blocklength variable except our convention that they must be positive. Therefore, we have , , , and . The sets of natural numbers, real numbers and non-negative real numbers are denoted by , and respectively. All logarithms are taken to base throughout the paper.
We use to represent the probability of an event , and we let be the indicator function of . Random variables are denoted by capital letters (e.g., ), and the realization and the alphabet of a random variable are denoted by the corresponding small letter (e.g., ) and calligraphic font (e.g., ) respectively. We use to denote a random tuple , where all of the elements have the same alphabet . We let and
denote the probability distribution of
and the conditional probability distribution ofgiven respectively for random variables and . We let
denote the joint distribution of, i.e., for all and . For random variable and any real-valued function whose domain includes , we let denote for any real constant . For any function whose domain contains , we use to denote the expectation of where is distributed according to . For simplicity, we omit the subscript of a notation when there is no ambiguity. The Euclidean norm of a tuple is denoted by . The distribution of a Gaussian random variable whose mean and variance are and respectively is denoted by .
Ii The AWGN EH Channel
Ii-a Problem Formulation
The AWGN EH channel, as illustrated in Figure 1, consists of one transmitter and one receiver. Energy harvesting and communication occur in time slots, i.e., channel uses. In each time slot, a random amount of energy with alphabet is harvested where
The energy-harvesting process is characterized by independent copies of denoted by . Prior to communication, the transmitter chooses a message . For each , the transmitter consumes units of energy to transmit based on and the receiver observes in time slot . The energy state information is known by the transmitter at time before encoding , but the receiver has no access to . For each , we have:
and are independent, i.e.,
For and every , a transmitted codeword should satisfy
for each .
After the time slots, the receiver declares to be the transmitted based on .
Ii-B Standard Definitions
Formally, we define a code as follows:
An -code consists of the following:
A message set , where is uniform on .
A sequence of encoding functions for each , where is used by the transmitter at time slot for encoding according to .
A decoding function for decoding at the receiver, i.e., .
If the sequence of encoding functions satisfies (5), the code is also called an -EH code.
If an -code does not satisfy the EH constraints (5) during the encoding process (i.e., is a function of alone), then the -EH code can be viewed as an -code for the usual AWGN channel without any cost constraint [13, 14]. The following definition is a formal statement of the channel law (2).
The AWGN EH channel is characterized by a conditional probability distribution such that the following holds for any -code: For each ,
for all and .
For any -code defined on the AWGN EH channel, let be the joint distribution induced by the code. We can factorize as
For an -code defined on the AWGN EH channel, we can calculate according to (6) the average probability of decoding error defined as . We call an -EH code with average probability of decoding error no larger than an -EH code.
Let be a real number. A rate is said to be -achievable for the EH channel if there exists a sequence of -EH codes such that
The -capacity of the AWGN EH channel, denoted by , is defined to be . The capacity of the AWGN EH channel is .
Iii An Achievable Rate for Save-and-Transmit
This section will present a non-asymptotic achievable rate for save-and-transmit. To this end, we first formally describe save-and-transmit in the following subsection.
Iii-a Save-and-Transmit Scheme
Fix a blocklength . Choose a positive real number that may depend on and let
such that . The codebook consists of mutually independent random codewords, which are constructed as follows. For each message , a length- codeword consisting of i.i.d. symbols is constructed where . In other words, the codebook consists of i.i.d. Gaussian codewords where each codeword consists of i.i.d. Gaussian random variables and has average power .
Suppose and , i.e., the transmitter chooses message and the realization of is . Then, the transmitter uses the following save-and-transmit -EH code with encoding functions and decoding function . The save-and-transmit code consists of an initial saving phase and a subsequent transmission phase. Define to be the number of time slots in the initial saving phase during which energy is harvested but not consumed and no information is conveyed. Define in a recursive manner where
For each , let be the symbol transmitted at time . By construction,
for each . Upon receiving where is generated according to
the receiver declares that if is the unique integer in that satisfies
where is the marginal distribution of and is an arbitrary threshold to be carefully chosen later (cf. (60)). Otherwise, the receiver chooses
according to the uniform distribution. The decoding is successful if.
An important quantity that determines the performance of the save-and-transmit -EH code is
which is a random set that specifies the mismatched positions between and during the transmission phase when the chosen message equals . The following lemma presents an upper bound on the probability of seeing more than mismatched positions in the transmission phase. The proof, which is based on analyzing the escape probability of a Markov process, is provided in Appendix A.
Fix any and any such that
and fix a save-and-transmit -EH code with a length- saving phase where
For any , we have
for each .
Lemma 2 (Implied by Shannon’s bound [15, Th. 1])
Let be the probability distribution of a pair of random variables . Suppose , and suppose has the same distribution as and is independent of . Then for each and each , we have
The following lemma is a modification of the Shannon’s bound stated in the previous lemma, and its proof is provided in Appendix B.
Suppose we are given a save-and-transmit -EH code with a length- saving phase as described in Section III-A. Then for each , each and each , we have
Iii-C A Non-Asymptotic Achievable Rate for Save-and-Transmit
Fix an , fix a natural number , fix a non-negative integer , and fix a such that (11) holds. Let . Define , and as in (12), (13) and (14) respectively. Let and let be the marginal distribution of , and let and denote the variance and the third absolute moment of respectively. For any and such that , if and satisfy
then there exists a save-and-transmit -EH code with a length- saving phase which satisfies
In particular, the probability of seeing more than mismatch events can be bounded as
The following corollary is a direct consequence of Theorem 1, and it states a non-asymptotic rate for the save-and-transmit scheme whose second-order term scales as . The proof of Corollary 4 is provided in Appendix C.
Fix an , and fix any and such that . There exists a constant which does not depend on such that for all sufficiently large , we can construct a save-and-transmit -EH code which satisfies
with being defined as
In particular, the probability of seeing a mismatch event in the transmission phase can be bounded as
where each term in the union characterizes the event that the accumulated energy collected during the first time slots is insufficient to output the desired codeword symbols from time to time during the transmission phase.
The parameters and in Corollary 4 have been carefully chosen to achieve the second-order scaling , where the scaling is optimal [4, Th. 1]. Fix any . The best existing lower bound on the second-order term of was derived in [4, Th. 1], which states that there exists a save-and-transmit -EH code that satisfies
for any and such that . The save-and-transmit scheme investigated in  is similar to that described in Section III-A except that is assumed in  while is assumed in this work. Note that the second-order term of the best existing lower bound as stated on the right-hand side (RHS) of (17) decays as as tends to . On the other hand, it follows from (16) in Corollary 4 that the second-order term of our lower bound decays as as tends to . Consequently, the second-order term achievable by the save-and-transmit scheme guaranteed by Corollary 4 is strictly larger (less negative) than the best existing bound for all sufficiently large . In other words, letting be strictly less than instead of equal to achieves a higher rate in the high SNR regime.
Iii-D A Non-Asymptotic Achievable Rate for Best-Effort
We call a save-and-transmit scheme a best-effort scheme if the length of saving phase equals zero, i.e., . By setting , Theorem 1 reduces to the following corollary, which states that the best-effort scheme achieves a non-asymptotic rate whose second-order term scales as . The proof of Corollary 5 is provided in Appendix D.
Fix an , and fix any and such that . Define
There exists a constant which does not depend on such that for all sufficiently large , we can construct a best-effort -EH code with
In particular, the probability of seeing more than
mismatch events can be bounded as
Iv The Block Energy Arrival Model
In this section, we generalize our achievable rates for save-and-transmit and best-effort to the block energy arrival model [11, 12, 4], which is useful for modeling practical scenarios when the energy-arrival process (e.g., solar energy, wind energy, ambient radio-frequency (RF) energy, etc.) evolves at a slower timescale compared to the transmission process.
Iv-a Block Energy Arrivals
We follow the formulation in , which assumes that arrive at the buffer in a block-by-block manner as follows: For each , let
such that is the index of the first channel use within the block of energy arrivals, where denotes the length of each block. The EH random variables that mark the starting positions of the blocks (i.e., ) are assumed to be i.i.d. random variables where satisfies (3). In addition, we assume
for all . In other words, the harvested energy in each channel use within a block remains constant while the harvested energy across different blocks is characterized by a sequence of i.i.d. random variables with mean equal to . By construction, we have the following for each and all ,
The length of each energy-arrival block is assumed to remain constant or grow sublinearly in .
Iv-B Blockwise Save-and-Transmit
Fix a blocklength and choose an . Choose a positive real number and let be as defined in (7) such that . The codebook consists of mutually independent random codewords denoted by , which are constructed as described in Section III-A. Suppose and . Then, the transmitter uses the following blockwise save-and-transmit -EH code with encoding functions and decoding function where . The saving phase consists of blocks of consecutive time slots. Define in a recursive manner where
In other words, the transmitter outputs a block of symbols in the transmission phase during time to if the energy in the battery at time (i.e., ) can support the transmission of the whole block of symbols starting at time . If , the blockwise save-and-transmit scheme defined by (22) reduces to the save-and-transmit scheme presented in Section III-A defined by (8). Let be the symbol transmitted at time for each such that
for each . Upon receiving where is generated according to (9), the receiver declares that if is the unique integer in that satisfies