Non-Asymptotic Achievable Rates for Gaussian Energy-Harvesting Channels: Best-Effort and Save-and-Transmit

05/08/2018 ∙ by Silas L. Fong, et al. ∙ Penn State University 0

An additive white Gaussian noise (AWGN) energy-harvesting (EH) channel is considered where the transmitter is equipped with an infinite-sized battery which stores energy harvested from the environment. The energy arrival process is modeled as a sequence of independent and identically distributed (i.i.d.) random variables. The capacity of this channel is known and is achievable by the so-called best-effort and save-and-transmit schemes. This paper investigates the best-effort scheme in the finite blocklength regime, and establishes the first non-asymptotic achievable rate for it. The first-order term of the non-asymptotic achievable rate equals the capacity, and the second-order term is proportional to -√( n/n) where n is the blocklength. The proof technique involves analyzing the escape probability of a Markov process. In addition, we use this new proof technique to analyze the save-and-transmit and derive a new non-asymptotic achievable rate for it, whose first-order and second-order terms achieve the capacity and the scaling -O(√(1/n)) respectively. For all sufficiently large signal-to-noise ratios (SNRs), this new achievable rate outperforms the existing ones. The results are also extended to the block energy arrival model where the length of each energy block L grows sublinearly in n. It is shown that best-effort and save-and-transmit achieve the second-order scalings -O(√({ n, L}/n)) and -O(√(L/n)) respectively.

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I Introduction

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

(1)

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

(2)

for each time  where is a standard normal random variable which is independent of  and the random variables are independent. Reference [1] 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 variance 

as 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 [1], 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 [5] and a special discrete memoryless EH channel with no battery studied in [6]. 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.

Note that the results in this paper cease to hold if the size of the battery is finite. The channel capacity for the finite battery case is the subject of recent interests, see [7, 8, 9].

I-a Related Work

The channel capacity of the AWGN EH channel was characterized in [1], 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. [2] has performed a finite blocklength analysis of save-and-transmit proposed in [1] 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 [3] has refined the analysis in [2] and improved the second-order term to . Reference [4] 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 [4] 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 [4] 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 [1] 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.

I-D Notation

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 of

given 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

Fig. 1: The AWGN EH channel

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

(3)

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:

  1. and are independent, i.e.,

    (4)
  2. For and every , a transmitted codeword should satisfy

    (5)

    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:

Definition 1

An -code consists of the following:

  1. A message set , where is uniform on .

  2. A sequence of encoding functions for each , where is used by the transmitter at time slot  for encoding according to .

  3. 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).

Definition 2

The AWGN EH channel is characterized by a conditional probability distribution such that the following holds for any -code: For each ,

where

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

(6)

which follows from the i.i.d. assumption of the EH process in (4), the fact that is a function of (cf. Definition 1) and the memoryless property of the channel described in Definition 2.

Definition 3

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.

Definition 4

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

Definition 5

The -capacity of the AWGN EH channel, denoted by , is defined to be . The capacity of the AWGN EH channel is .

Define the capacity function

for all . It was shown in [1, Sec. III] (see also [2, Remark 1]) that

for all where can be interpreted as the signal-to-noise ratio (SNR) of the AWGN EH channel.

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

(7)

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

(8)

For each , let be the symbol transmitted at time . By construction,

for each . Upon receiving where is generated according to

(9)

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

.

Iii-B Preliminaries

An important quantity that determines the performance of the save-and-transmit -EH code is

(10)

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.

Lemma 1

Fix any  and any such that

(11)

and fix a save-and-transmit -EH code with a length- saving phase where

(12)

Define

(13)

and

(14)

For any , we have

(15)

for each .

Remark 1

In the proof of Lemma 1 which is readily seen in Appendix A by setting , and , an important step is analyzing the escape probability (65) of the Markov process where is the stopping time when the value of the Markov process hits any negative number .

The following lemma [15] is standard for proving achievability results in the finite blocklength regime and its proof can be found in [16, Th. 3.8.1].

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.

Lemma 3

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

The following theorem is the main result of this paper. The proof relies on Lemma 1 and Lemma 3, and it will be presented in Section V.

Theorem 1

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

and

where

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.

Corollary 4

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

(16)

with being defined as

the average transmit power  being defined as in (12), and  being defined as in (13) and (14) respectively, and the length of saving phase  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.

Remark 2

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

(17)

for any and such that . The save-and-transmit scheme investigated in [4] is similar to that described in Section III-A except that is assumed in [4] 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.

Corollary 5

Fix an , and fix any and such that . Define

(18)

and

(19)

There exists a constant which does not depend on  such that for all sufficiently large , we can construct a best-effort -EH code with

and

which satisfies

(20)

In particular, the probability of seeing more than

mismatch events can be bounded as

Remark 3

Although the achievable second-order scaling for best-effort in Corollary 5 is not optimal (the optimal scaling is  [4, Th. 1]), it is a significant improvement compared to the state of the art [1, Sec. V] where the achievable second-order scaling therein for best-effort is .

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 [4], which assumes that arrive at the buffer in a block-by-block manner as follows: For each , let

(21)

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

(22)

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