Constrained Codes for Joint Energy and Information Transfer
aa r X i v : . [ c s . I T ] N ov Constrained Codes for Joint Energy andInformation Transfer
Ali Mohammad Fouladgar, Osvaldo Simeone, and Elza Erkip
Abstract
In various wireless systems, such as sensor RFID networks and body area networks with implantabledevices, the transmitted signals are simultaneously used both for information transmission and for energytransfer. In order to satisfy the conflicting requirements on information and energy transfer, this paperproposes the use of constrained run-length limited (RLL) codes in lieu of conventional unconstrained(i.e., random-like) capacity-achieving codes. The receiver’s energy utilization requirements are modeledstochastically, and constraints are imposed on the probabilities of battery underflow and overflow atthe receiver. It is demonstrated that the codewords’ structure afforded by the use of constrained codesenables the transmission strategy to be better adjusted to the receiver’s energy utilization pattern, ascompared to classical unstructured codes. As a result, constrained codes allow a wider range of trade-offs between the rate of information transmission and the performance of energy transfer to be achieved.
I. I
NTRODUCTION
Various modern wireless systems, such as sensor RFID networks [1] and body area networkswith implantable devices [2]-[4], challenge the conventional assumption that the energy receivedfrom an information bearing signal cannot be reused. For instance, implantable devices canbe powered by the received radio signal, hence alleviating the need for a battery and reduc-ing significantly the size of the devices. This realization has motivated a number of research
A.M. Fouladgar and O. Simeone are with the CWCSPR, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail:{af82,osvaldo.simeone}@njit.edu).E. Erkip is with the Department of Electrical Engineering, Polytechnic Institute of New York University, Brooklyn, NY (email:[email protected]).
August 10, 2018 DRAFT groups to investigate the design of wireless systems under joint information and energy transferrequirements .The research activity in this area has focused so far on optimal resource allocation in thepresence of information and energy transfer for various network topologies. Specifically, refer-ence [8] studied a single point-to-point channel, while [9][10] investigated power allocation fora set of parallel point-to-point channels under energy transfer and information rate constraints.The optimization of beamforming strategies under the same criteria was studied in [11]-[14] formultiantenna broadcast channels and for two-user multiantenna interference channels in [15].Optimal resource allocation assuming wireless energy transfer was also investigated in [16] forcellular systems, in [17]-[19] for relay systems, in [20] for two-way interactive channels, andin [21] for graphical multi-hop networks. Considerations on the design of the receiver underthe constraint that, when harvesting energy from the antenna, the receiver is not able to use thesame signal for information decoding, can be found in [22].Unlike all prior work summarized above, this work focuses on the code design for systemswith joint information and energy transfer. We focus on a point-to-point link as shown in Fig. 1,in which the receiver’s energy requirements are modeled as a random process. The statistics ofthis process generally depend on the specific application to be run at the receiver, e.g., sensing orradio transmission. The performance in terms of energy transfer is measured by the probabilitiesof overflow and underflow of the battery at the receiver. The probability of overflow measuresthe efficiency of energy transfer by accounting for the energy wasted at the receiver. Instead, theprobability of underflow is a measure of the fraction of the time in which the application run atthe receiver is in outage due to the lack of energy.Classical codes, which are designed with the only aim of maximizing the information rate, areunstructured (i.e., random-like), see, e.g., [23]. As a result, they do not allow to control the timingof the energy transfer, and hence to optimize the probability of overflow and underflow. Withthis in mind, here it is proposed to adopt constrained run-length limited (RLL) codes [24] in lieuof conventional unconstrained codes. The constraints defining RLL codes ensure that the codedoes not includes bursts of energy either too frequently, thus limiting battery overflow, or tooinfrequently, thus controlling battery underflow. Constrained RLL codes have been traditionally It is worth noting that wireless energy transfer, has a long history [5] and is available commercially (see, e.g., [6][7]).
August 10, 2018 DRAFT studied for applications related to magnetic and optical storage [24]. The application to theproblem at hand of energy transfer has been previously studied in the context of point-to-pointRFID systems in [25], although no analysis of the information-energy trade-off was provided. Incontrast, in this work, a thorough analysis is provided of the interplay between information rate adenergy transfer in terms of probabilities of battery overflow and underflow. The analysis revealsthat, by properly choosing the parameters that define RLL codes depending on the receiver’sutilization requirements, constrained codes allow to greatly improve the system performance interms of simultaneous energy and information transfer.The reminder of this paper is organized as follows. In Sec. II, the system model is intro-duced along with performance criteria. In Sec. III and Sec. IV, we study the performance ofclassical unconstrained codes and of constrained RLL codes, respectively, in terms of energyand information transfer. Sec. V presents numerical results. Finally, some concluding remarkscan be found in Sec. VI. II. S
YSTEM M ODEL
We consider the point-to-point channel illustrated in Fig. 1. We assume that at each discretetime i , the transmitter can either send an “on” symbol X i = 1 , which costs one unit of energy,or an “off” signal X i = 0 , which does not require any energy expenditure. The receiver eitherobtains an energy-carrying signal, which is denoted as Y i = 1 , or receives no useful energy, whichis represented as Y i = 0 . The channel is memoryless, and has transition probabilities as shown inFig. 1. Accordingly, p represents the probability that energy is lost when propagating betweentransmitter and receiver . At the receiver side, upon reception of an energy-carrying signal Y i = 1 ,the energy contained in the signal is harvested. The harvested energy is temporarily held in asupercapacitor and, if not used in the current time interval i , is stored in a battery, whose capacitylimited to B max energy units (see, e.g., [26]).The receiver’s energy utilization is modeled as a stochastic process Z i ∈ { , } , so that Z i = 1 indicates that the receiver requires one unit of energy at time i , while Z i = 0 implies that noenergy is required by the receiver at time i . This process is not known at the transmitter and A more general model would allow also for a non-zero probability p of receiving energy when no energy is transmitted.This could be interpreted as the probability of harvesting energy from the environment (see [20]). We do not consider thisextension in this work. August 10, 2018 DRAFT M ˆ M max B Encoder n X n Y Decoder ReceiverChannelTransmitter p p - Supercapacitor n Z Figure 1. Point-to-point link with information and energy transfer. evolves according to the Markov chain shown in Fig. 2. Note that adopting a Markov modelto account for the time variations of energy usage is a standard practice (see, e.g., [27] andreferences therein). Accordingly, when in state U , there may be bursts of consecutive timeinstants in which no energy is required (i.e., Z i = 0 ); while, when in state U , there may bebursts of consecutive time instants in which energy is required (i.e., Z i = 1 ). The probabilitythat Z i = 0 when in state U is referred to as q and the probability that Z i = 1 in state U isdenoted as q . We observe that the average length of bursts of symbols in which Z i = j in state U j is given by / (1 − q j ) for j ∈ { , } . Also, it is remarked that, when q = 1 − q , the energyusage model becomes a memoryless process with Pr[ Z i = 1] = 1 − q = q .Due to the finite capacity of the battery, there may be battery overflows and underflows. Anoverflow event takes place when energy is received and stored in the supercapacitor (i.e., Y i = 1 ),but is not used by the receiver (i.e., Z i = 0 ) and the battery is full (i.e., B i = B max ), so that theenergy unit is lost; instead, an underflow event occurs when energy is required by the receiver(i.e., Z i = 1 ) but the supercapacitor and the battery are empty (i.e., B i = 0 and Y i = 0 ). In therest of this section we define all the parts of the system in Fig. 1 in detail. A. Transmitter
The transmitter aims at communicating a message M , uniformly distributed in the set (cid:2) nR (cid:3) ,reliably to the decoder, while at the same time guaranteeing desired probabilities of battery August 10, 2018 DRAFT U U q - q q - q Z = Z = Z = Z = Figure 2. Energy utilization model at the receiver. overflow and underflow (see Sec. II-B). Note that n is the codeword length and R represents theinformation rate in bits per channel use, while the constraints on the probabilities of overflowand underflow represents the requirements on energy transfer. As discussed in Sec. I, in thiswork, we investigate the performance in terms of information and energy transfer achievable ofconventional unconstrained codes and of RLL codes. We introduce RLL codes next following[24].The codewords x n ( m ) , with m ∈ (cid:2) nR (cid:3) , of a type- i RLL code satisfy run-length constraintson the number of consecutive symbols i , where i = 0 or i = 1 . To elaborate, let d and k beintegers such that ≤ d ≤ k . We say that a finite length binary sequence x n ( m ) satisfies thetype-0 ( d, k ) -RLL constraint if the following two conditions hold (see Fig. 3): • the runs of 0’s have length at most k , and • the runs of 0’s between successive 1’s have length at least d ; note that the first and lastruns of 0’s are allowed to have lengths smaller than d .Therefore, a type-0 ( d, k ) -RLL code is such that the codewords include sufficiently long stretchesof zero-energy symbols 0, via the selection of d , thus limiting battery overflow, but not tooinfrequently, via k , thus partly controlling also battery underflow. As a result, type-0 ( d, k ) -RLL codes are suitable for overflow-limited regimes in which controlling overflow events ismost critical. An example of a sequence satisfying the type-0 ( d, k ) = (2 , -RLL constraintis x n ( m ) = 00100001001000000010 where n = 20 . Overall, the set of all sequences x n ( m ) Classical RLL codes as discussed in, e.g., [24] are type-0, but here we find it useful to extend the definition to include alsotype-1 RLL codes.
August 10, 2018 DRAFT satisfying a type-0 ( d, k ) -RLL constraint is then described by all the possible n -bit outputs ofthe finite state machine in Fig. , where the outputs are shown by the binary labels of the directededges. Note that finite-state machine consists of k + 1 states (the numbered circles) and the initialstate is arbitrary.A type-1 ( d, k ) -RLL code is defined in the same way, upon substitution of all “0” for “1” andvice versa in the edge labels of Fig. 3. Therefore, type-1 ( d, k ) -RLL codes allow one to controlthe stretches of “1” symbols in the codewords, and are hence well suited for underflow-limitedregimes, in which controlling the probability of underflow is most important.
00 0 d d + k Figure 3. The codewords of a type-0 ( d, k ) -RLL code must be outputs of the shown finite-state machine. A type-1 ( d, k ) -RLLconstrained code is instead obtained by substituting all “0” for “1” and vice versa. B. Receiver
Transmitter and receiver communicate over the binary channel shown in Fig. 1 with theprobability of p of flipping symbol “1” to symbol “0”. As mentioned, this probability can beinterpreted in terms of energy losses across the channel. The received signal Y n is used by thedecoder both to decode the information message M encoded via the constrained code at thetransmitter and to perform energy harvesting. The harvested energy is used to fulfill the energyrequirements of the receiver as dictated by the process Z n , where the requirements are given interms of probability of overflow and underflow. This is discussed next.Let B i denote the number of energy units available in the battery at time i . At the i th timeperiod, the decoder first receives signal Y i , and stores its energy (if Y i = 1 ) temporarily in asupercapacitor (see Fig. 1). Then, if Z i = 1 , the receiver attempts to draw one energy unit fromthe supercapacitor or, if the latter is empty, from the battery. If the energy in the supercapacitoris not used, it is stored in the battery in the next time slot. As a result, the amount of energy in August 10, 2018 DRAFT the battery evolves as B i +1 = min (cid:0) B max , ( B i + Y i − Z i ) + (cid:1) , (1)where ( a ) + = max (0 , a ) .When the receiver harvests a unit of energy, Y i = 1 , no energy is used, Z i = 0 , and the batteryis full, B i = B max , we have an overflow event. To keep track of the overflow events, we definea random process O i such that O i = 1 if the event { B i = B max , Y i = 1 , and Z i = 0 } occursand O i = 0 otherwise. This can be expressed as O i = 1 { B i = B max , Y i = 1 and Z i = 0 } . (2)When the receiver wishes to use a unit of energy, Z i = 1 , and both the supercapacitor and thebattery are empty, Y i = 0 and B i = 0 , we have an underflow event. To describe underflow events,we introduce a random process U i such that U i = 1 if the event { B i = 0 , Y i = 0 and Z i = 1 } takes place and U i = 0 otherwise. This can be expressed as U i = 1 { B i = 0 , Y i = 0 and Z i = 1 } . (3)A sample path of the battery state process B i along with Y i , Z i , U i and O i , is shown in Fig. 4.We define the probability of underflow as Pr {U } = lim sup n →∞ n n X i =1 E [ U i ] , (4)and the probability of overflow as Pr {O} = lim sup n →∞ n n X i =1 E [ O i ] . (5)We note that in (4) and (5), the expectation is taken over the distribution of the message M , ofthe channel and of the receiver’s energy utilization process Z n . C. Performance Criteria and Problem Formulation
The point-to-point link under study will be evaluated in terms of its performance for bothinformation and energy transfer. A triple ( R, P of , P uf ) of information-energy requirements issaid to be achievable by an encoder-decoder pair if the information transfer at rate R is reliable,i.e., if lim sup n →∞ Pr h ˆ M = M i = 0 (6) August 10, 2018 DRAFT max B - i i + max B B a tt e r y S t a t e Time
Underflow Y i + Overflow i + i B Z Y Z Y Z Y Z i Y i Z i Y + i Z + i Y + i Z + i Y + i Z + i + ( 1) i O = ( 1) i U = Figure 4. A sample path of the evolution of the battery B i . Also indicated is the assumed order of energy arrival and departureevents from the battery (i.e., Y i and Z i ) and the overflow and underflow events (where not specified, we have U i = 0 and O i = 0 ). and if the energy transfer fulfill the constraints Pr {O} ≤ P of , (7)and Pr {U } ≤ P uf . (8)We are interested in investigating the set of achievable triples ( R, P of , P uf ) for different classesof codes, namely unconstrained and ( d, k ) -RLL constrained. To obtain further insight, in Sec.V, we will consider the problemminimize max( P of , P uf ) subject to ( R, P of , P uf ) is achievable , (9)where R is fixed and the minimization is done over all codes belonging to a certain class.Problem (9) is appropriate when both underflow and overflow are equally undesirable and onewishes to reduce both equally as much as possible. Alternatively, one could, e.g., minimize eitherone of P of or P uf under a given constraint on the other and on the rate. August 10, 2018 DRAFT
III. U
NCONSTRAINED C ODES
In this section, we study the information-energy transfer performance of classical unconstrainedcodes. To this end, we adopt Shannon’s classical random coding argument. Accordingly, weassume that the codewords x n ( m ) , m ∈ (cid:2) nR (cid:3) , are generated independently as i.i.d. Ber ( p x ) processes and evaluate the corresponding performance on average over the code ensemble. As itis well known (see, e.g., [23]), the maximum information rate R achieved by this code is givenas R = I( X ; Y )= H ( Y ) − H ( Y | X )= H ( p x (1 − p )) − p x H ( p )= H ( p y ) − p y − p H ( p ) , (10)where we have defined the probability p y , Pr[ Y i = 1] = p x (1 − p ) and the binary entropyfunction H( a ) , − a log a − (1 − a ) log (1 − a ) . (11)We now turn to the evolution of the performance in terms of energy transfer. In order tosimplify the analysis and obtain some insight, we first assume the special case for then receiver’senergy utilization model in which the process Z n is i.i.d. and hence q = 1 − q , q . Note that q is the energy usage probability, in that we have q = Pr[ Z i = 1] . The extension to the moregeneral Markov model of Fig. 2 will be discussed in Remark 3. If the process Z n is i.i.d., thebattery state evolves according to the birth-death Markov chain shown in Fig. 5. Using standardconsiderations and recalling (2) and (3), we can then calculate the probability of overflow andunderflow respectively, as Pr {O} = π B max p y (1 − q ) , O ( p y ) , (12)and Pr {U } = π (1 − p y ) q , U ( p y ) , (13)where π i is the steady-state probability of state i ∈ [0 , B max ] for the Markov chain in Fig. 5.This can be easily calculated as π i = A i A + ... + A B max , (14) August 10, 2018 DRAFT0 max B max B - (1 )(1 ) y y p q p q + - - (1 ) y p q - (1 )(1 ) y q p q + - - (1 ) y p q - y q p q - + (1 ) y p q - (1 ) y p q - (1 ) y p q - (1 ) y p q - Figure 5. The birth-death Markov process defining the battery state evolution along the channel uses with unconstrained (i.i.d.)random codes and i.i.d. receiver’s energy usage process Z n (i.e., q = q = 1 − q ). where A = p y (1 − q )(1 − p y ) q . The following lemma summarizes our conclusions so far. Lemma 1.
Given a receiver energy usage i.i.d. process with energy usage probability q , the setof achievable information-energy triples ( R, P of , P uf ) for unconstrained (i.i.d.) codes is givenby ( ( R, P of , P uf ) : ∃ p y ∈ [0 , − p ] such that R ≤ H ( p y ) − p y − p H ( p ) (15) P of ≥ O( p y ) , P uf ≥ U( p y ) ) , (16) where H( p y ) , O( p y ) and U( p y ) are defined in (11), (12) and (13), respectively.Remark . The region (15) is in general not convex, but it can be convexified if one allowsfor time sharing between codes with different values of p y (see, e.g., [28, Ch. 4] for relateddiscussion).In order to get further insight into the performance of unconstrained codes, we now assumethat the channel is noiseless, i.e., p = 0 and, as a result, we have Y i = X i for all i = 1 , ..., n and p y = p x . Moreover, we consider problem (9), which reduces to the following optimizationproblem minimize p x ∈ [0 , max (O( p x ) , U( p x )) subject to : H( p x ) ≥ R, (17) August 10, 2018 DRAFT1 where R is fixed. The solution of problem (17) is summarized in the following lemma. Lemma 2.
The optimal solution p ⋆x of problem (17) is given as q if R ≤ H( q ) (18a) H − ( R ) if R > H( q ) and q ≤ (18b) − H − ( R ) if R > H( q ) and q > (18c) where H − ( R ) is the inverse of the entropy function in the interval [0 , / . Moreover, the optimalvalue max (O ( p ⋆x ) , U ( p ⋆x )) of the problem (17) is given by (1 − q ) qB max + 1 if R ≤ H( q ) (19a) O (cid:0) H − ( R ) (cid:1) if R > H( q ) and q ≤ (19b) U (cid:0) − H − ( R ) (cid:1) if R > H( q ) and q > . (19c) Proof:
A graphical illustration of Lemma 2 is shown in Fig. 6. To interpret the conditions(18a) and (19a), we observe that the underflow probability U ( p x ) is monotonically decreasingwith p x , while the overflow probability O ( p x ) is monotonically increasing with p x . Therefore, inthe absence of the rate constraint, the optimal value of problem (17) is achieved when p x = q ,since, with this choice, we have O( p x ) = U( p x ) . As a result, if R ≤ H ( q ) , and hence the rateconstraint is immaterial for p x = q , we have p ⋆x = q .Instead, if R > H ( q ) , the rate constraint is active and the optimal solution requires R = H( p x ) .In particular, there are two situations to be considered, namely the overflow-limited regime, defined by the condition q ≤ / , and the underflow-limited regime, where we have q > / . Inthe former regime (Fig. 6-(a)), the rate constraint forces p x to be larger than q , which leads to theoptimal solution p ⋆x = H − ( R ) and causes the overflow probability O ( p x ) to be larger than theunderflow probability U ( p x ) , so that max ( O ( p x ) , U ( p x )) = O ( p x ) . In contrast, in the underflow-limited regime (Fig. 6-(b)), the rate constraint forces p x to be smaller than q , which leads to p ⋆x = 1 − H − ( R ) and causes U ( p x ) to dominate O ( p x ) , or max ( O ( p x ) , U ( p x )) = U ( p x ) . Remark . The proof of Lemma 2 suggests that, when the rate is sufficiently small, problem(9) is solved by "matching" the code structure to the receiver’s energy utilization model. Thisis done, under the given i.i.d. assumption on codes and receiver’s energy utilization, by setting
August 10, 2018 DRAFT2 p ⋆x = q . Instead, when the rate constraint is the limiting factor, one is forced to allow for amismatch between code properties and receiver’s energy utilization model (by setting p ⋆x = q ).These ideas will be useful when interpreting the gains achievable by constrained codes discussedin Sec. IV. H ( ) R - q x p * R R R - - q ( ) a ( ) b H( ) q H( ) q x p * Figure 6. Illustration of the optimal solution p ⋆x of problem (17) (Lemma 2): (a) the overflow-limited regime q ≤ / ; (b) theunderflow-limited regime q > / . Remark . The characterization of the achievable information-energy triples ( R, P of , P uf ) ofLemma 1 can be extended to the more general Markov model in Fig. 2 for the receiver’s energyusage. This is done by noting that the battery evolution under this model is described by theMarkov chain shown in Fig. 7, instead of the simpler birth-death Markov process shown in Fig.5. The calculation of the corresponding steady-state probabilities π i,U j , for i ∈ [0 , B max ] and j = { , } , can be done using standard steps (see, e.g., [29]). Lemma 1 then extends to thescenario at hand by calculating the probabilities of overflow and underflow, similar to (12)-(13) August 10, 2018 DRAFT3 as Pr {O} = π B max ,U p y q + π B max ,U p y (1 − q ) , O ( p y ) , (20)and Pr {U } = π ,U (1 − p y )(1 − q ) + π ,U (1 − p y ) q , U ( p y ) . (21) max 0 , B U U U U max 0 B U - q max 1 , B U max 1 B U - U U U (1 ) y p q - y p q y p q q - (1 )(1 ) y p q - - y p q y p q q (1 ) y p q - y p q (1 ) y p q - (1 ) y p q - (1 )(1 ) y p q - - (1 ) y p q - (1 ) y p q - (1 ) y p q - (1 )(1 ) y p q - - (1 )(1 ) y p q - - (1 ) y p q - y p q (1 ) y p q - (1 ) y p q - (1 ) y p q - (1 ) q - y p q (1 )(1 ) y p q - - (1 ) y p q - (1 ) y p q - (1 ) y p q - (1 )(1 ) y p q - - (1 )(1 ) y p q - - (1 ) y p q - Figure 7. The Markov process defining the battery state evolution along the channel uses with unconstrained (i.i.d.) randomcodes and the Markov receiver’s energy usage model of Fig. 2.
IV. C
ONSTRAINED C ODES
In this section, we study the performance of ( d, k ) -RLL codes. To this end, as with uncon-strained codes, we adopt a random coding approach. Specifically, we take the codewords tobe generated independently according to a stationary Markov chain defined on the finite statemachine in Fig. 3. It is known that this choice is optimal in terms of capacity (see, e.g., [24], [25],[30]). A stationary Markov chain on the graph of Fig. 3 is defined by the transition probabilities P = { p d , p d +1 , ..., p k − } on its edges as shown in Fig. 8. We define as C i the state of theconstrained code at time i , prior to the transmission of X i . For example, the state sequence forthe type-0 ( d, k ) = (2 , -RLL corresponding to the codeword x n ( m ) = 00100001001000000010 is c n ( m ) = 01201234012012345670 where n = 20 . Then, the transition probability p j for j = d, ..., k − is equal to Pr[ C i = j + 1 | C i − = j ] , for i > . Barring degenerate choices for P , it is easy to see that the Markov chain is irreducible, and hence one can calculate the uniquesteady-state distribution π j = Pr [ C i = j ] for j ∈ [0 , k ] (see, e.g., [24]). August 10, 2018 DRAFT4
A. Information Rate
In [31, Lemma 5], it was proved that an achievable rate R with ( d, k ) -RLL codes is given as R = I ( C ; Y | C ) . Evaluating this expression for type-0 ( d, k ) -RLL constrained codes leads to R = H ( Y | C ) − H ( Y | C , C )= k − X j = d π j { H ((1 − p j )(1 − p )) − (1 − p j ) H ( p ) } . (22)Instead, for type-1 code the achievable rate becomes R = k − X j = d π j { H ( p j (1 − p )) − p j H ( p ) } . (23) Remark . We note that if the channel is noiseless, i.e., p = 0 , the information rates (22) and(23) equal the entropy rate of the channel input sequence X n , i.e., R = k − X j = d π j H ( p j ) . (24)Moreover, the maximization of the achievable information rate (24) over the transition proba-bilities P , with no regards for energy transfer, leads to the solution sup P k − X j = d π j H ( p j ) = log λ, (25)where λ is the largest absolute value taken by the eigenvalues of adjency matrix A of the graphthat defines the ( d, k ) -RLL code [24]. p = d d + k d p d p + d p - d p + - k p - = k p - p = Figure 8. Transition probabilities P = { p d , p d +1 , ..., p k − } defining the stationary Markov chain used for random coding withtype-0 ( d, k ) -RLL codes. The adjacency matrix A is a k × k matrix such that the ( i, j ) th element equals 1 if state i and state j are connected in thegraph that defines the code (see Fig. 8) and is zero otherwise. August 10, 2018 DRAFT5
B. Energy Transfer
We now address energy transfer by turning to the calculation of the probabilities of batteryunderflow and overflow, namely Pr {U } and Pr {O} in (4) and (5), respectively. To this end, asfor unconstrained codes, we focus at first on the special case in which the energy usage process Z n is i.i.d. with energy usage probability q . We refer to Remark 6 below for a discussion onthe extension to the Markov model in Fig. 2.We use a renewal-reward argument (see, e.g., [29]). We recall that a renewal process is arandom process of inter-renewal intervals I , I , ... that are positive i.i.d. random variables. Forour analysis, it is convenient to define the renewal event as { C i = 0 } , so that a renewal takesplace every time the state of the constrained code C i is equal to 0. This is equivalent to sayingthat, in the channel use before a renewal event, the transmitted signal X i equals 1 for type-0 ( d, k ) -RLL codes and X i = 0 for type-1 ( d, k ) -RLL codes. We refer to Fig. 9 for an illustration.Based on the above, the renewal intervals I i , for i ≥ , are i.i.d. integer random variables withdistribution p I ( i ) that can be calculated, given P , as p I ( i ) = i ≤ d and i > k + 11 − p d i = d + 1(1 − p i − ) i − Q l = d p l d + 1 < i ≤ k k − Q l = d p l i = k + 1 . (26)Moreover, it is useful to define a Markov chain ˜ B i that defines the evolution of the battery asevaluated at the renewal instants (i.e., for values of i for which C i = 0 ), as illustrated in Fig. 9.We refer to the steady-state probability of this Markov chain as ˜ π b with b ∈ [0 , B max ] . Finally,we define as ˜ O b the random variable that counts the number of overflow events in a renewalthat starts with a battery with capacity b ∈ [0 , B max ] , and, similarly, we define as ˜ U b the randomvariable that counts the number of underflow events in a renewal that starts with a battery withcapacity b ∈ [0 , B max ] . We proceed by treating separately the type-0 and type-1 codes.
1) Type-0 Codes:
For type-0 codes, the transition probabilities for the process ˜ B i are reportedin Appendix A (see also Fig. 10 for an illustration), from which the steady state probabilities ˜ π b can be calculated (see, e.g., [29]). The next proposition summarizes the main result of theanalysis. We use the definition p ( n ; i, q ) = (cid:0) in (cid:1) q n (1 − q ) i − n with n = 0 , ..., i , for the probability August 10, 2018 DRAFT6 I I XY == XY == XY == XY == XY == XY == C = B B = ɶ C = B B = ɶ C = B B = ɶ i Figure 9. A sample of renewal events { C i = 0 } and the corresponding evolution of the battery across channel uses for atype-0 ( d, k ) -RLL code. distribution of a binomial random variable with parameters ( i, q ) . max B max B - Figure 10. The birth-death Markov process defining the battery state evolution along the renewal instants, where we have C i = 0 , for ( d, k ) -RLL codes, i.i.d. receiver’s energy usage pattern Z n , and k = B max . Proposition 1.
Given an i.i.d. receiver energy usage process with energy usage probability q , theset of achievable information-energy triples ( R, P of , P uf ) for type-0 ( d, k ) -RLL codes is given August 10, 2018 DRAFT7 by ( ( R, P of , P uf ) : ∃ P = { p d , p d +1 , ..., p k − } ∈ (0 , n such that R ≤ k − X j = d π j { H ((1 − p j )(1 − p )) − (1 − p j ) H ( p ) } , (27) P of ≥ ˜ π B max E h ˜ O B max i E [ I ] , (28) and P uf ≥ B max P b =0 ˜ π b E h ˜ U b i E [ I ] ) , (29) where we have defined E [ I ] = k +1 X i = d +1 i · p I ( i ) , (30) along with E h ˜ U b i = k +1 X i = d +1 p I ( i ) ( (1 − p ) i − b − X l =1 p ( l + b ; i − , q )+ p i − b X l =1 p ( l + b ; i, q ) ) , (31)and E h ˜ O B max i = k +1 X i = d +1 p I ( i )(1 − p ) p (0; i, q ) . (32) Proof:
See Appendix B.
Remark . The right-hand side of (28) evaluates the probability of overflow as the ratio of theaverage numbers of overflow events in a renewal interval over the average length of a renewalinterval. The right-hand side of (29) can be similarly interpreted. Note that, by the given definitionof renewal events, in order to have an overflow, the initial battery state ˜ B i must be in state B max ,whereas underflow events can potentially happen for all states b ∈ { , ..., B max } . This is reflectedby the numerators of (28) and (29). Remark . Similar to the case of unconstrained codes (see Remark 3), the characterization of theachievable information-energy triples ( R, P of , P uf ) of Proposition 1 can be extended to the moregeneral Markov model in Fig. 2 for the receiver’s energy usage. This is done by noting that the August 10, 2018 DRAFT8 max 1 , B U U U U max 1 B U - max 0 , B U U U U max 0 B U - Figure 11. The Markov process defining the battery state evolution along the renewal instants where we have C i = 0 , and theenergy usage state at the receiver, for ( d, k ) -RLL codes, and k = B max with the Markov receiver’s energy usage model of Fig.2. evolution of the battery state along the renewal instants can be still described by a Markov chain,albeit a more complex one. Moreover, in order to extend the analysis one needs to include inthe state of the Markov process not only the battery state ˜ B i but also the state of the receiver’senergy usage (either U or U ). The corresponding Markov chain is sketched in Fig. 11. Thecalculation of the corresponding transition probabilities is straightforward but cumbersome andis not detailed here.
2) Type-1 Codes:
For type-1 codes, the analysis presented above does not easily generalize inthe case in which the channel loss probability p is nonzero. This can be seen by following themain steps of the proof of Proposition 1, which is based on having at most one non-zero receivedsymbol per renewal interval. However, in the special case in which p = 0 , the approach can August 10, 2018 DRAFT9 be generalized, leading to the following result.
Proposition 2.
Given an i.i.d. receiver energy usage process with energy usage probability q , theset of achievable information-energy triples ( R, P of , P uf ) for type-1 ( d, k ) -RLL codes is givenby ( ( R, P of , P uf ) : ∃ P = { p d , p d +1 , ..., p k − } ∈ (0 , n such that R ≤ k − X j = d π j H ( p j ) , (33) P of ≥ B max P b =0 ˜ π b E h ˜ O b i E [ I ] , (34) and P uf ≥ ˜ π E h ˜ U i E [ I ] ) , (35) where we have defined (30) along with E h ˜ U i = k +1 X i = d +1 p I ( i ) p (0; i, − q ) , (36)and E h ˜ O b i = k +1 X i = d +1 p I ( i ) i − b − X l =1 p ( l + b ; i − , − q ) . (37) Proof:
Proposition 2 follows by the same steps as Proposition 1 and is not detailed here.
Remark . Similar to Proposition 1, the right-hand sides of (34), and of (35), evaluate theprobabilities of overflow, and of underflow, via the ratios of the average numbers of overflow,and of underflow, events in a renewal interval over the average length of a renewal interval. Ina dual fashion with respect to Proposition 1, given the definition of renewal events, underflowcan only occur in renewal intervals with initial battery state ˜ B i is zero, whereas overflow eventscan potentially happen for all states b ∈ { , ..., B max } .V. N UMERICAL R ESULTS
In this section, we compare the performance of unconstrained and constrained codes usingproblem (9) as the benchmark. Fig. 12 shows the optimal value of max( P of , P uf ) for a noiselesschannel, i.e., p = 0 in Fig. 1, when R = 0 . and q = 0 versus q (recall Fig. 2). With q = 0 , August 10, 2018 DRAFT0 the energy usage process Z n is such that a single energy request (i.e., Z i = 1 ) is followed byan average of / (1 − q ) instants where no energy is required (i.e., Z i = 0 ). Therefore, as q increases from 0.1 to 0.9, the average length of an interval with no energy usage increases fromaround 1 to 10. Similar to the discussion in Remark 2 for unconstrained codes, when neglectingthe rate constraint, problem (9) is observed to be optimized by matching the code structure to thereceiver’s energy utilization model. When q is sufficiently small, this can be easily accomplishedwith type-0 ( d, k ) -RLL codes with a small k . This is because k defines the maximum possiblenumber of zero symbols X i sent before a symbol X i = 1 . As q increases, and hence the averagelength of the bursts of zeros grows in the process Z i , the value of k must be correspondinglyincreased. This is confirmed by Fig. 12, which shows the significant gain achievable by the useof RLL codes when properly selecting the code parameters. We observe that type-1 ( d, k ) -RLLcodes would provide exactly the same performance in the symmetric case in which we have q = 0 , and hence intervals of energy usage (i.e., Z i = 1 ) are followed by a single instant withno energy usage (i.e., Z i = 0 ).The impact of the information rate R is illustrated in Fig. 13 for q = q = 0 and p = 0 .Following the discussion above, when the rate is small, with q = q = 0 , it is sufficientto choose a type-0 or type-1 ( d, k ) -RLL code with k = 1 , as this matches the energy usageprocess. However, as the rate grows larger, one needs to increase the value of k , while keeping d as small as possible [24]. For instance, with k = 1 and d = 0 , the maximum achievable rate (25)is R = 0 . ; with k = 2 and d = 0 , it is R = 0 . ; with k = 3 and d = 1 , it is R = 0 . ;and with k = 3 and d = 0 , it is R = 0 . [24, Table 3.1]. Accordingly, Fig. 13 shows againthat, by appropriately choosing d and k , RLL codes can provide relevant advantages.Finally, we observe the effect of the loss probability p in Fig. 14 where we set R = 0 . and q = q = 0 . Following the discussion above (see Remark 2), in order to match the receiver’senergy utilization, the unconstrained code should be designed in such a way that p y = 0 . since Pr[ Z i = 1] = 0 . . Given that p y = p x (1 − p ) , this is only possible for p < . , and hence,for p > . , the performance is degraded as seen in Fig. 14. When p = 0 , as demonstratedabove, RLL codes provide significant gains by providing a better matching to the utilizationprocess Z n . As the losses on the channel become more pronounced this gain decreases due tothe reduced control of the received signal afforded by designing the transmitted signal. August 10, 2018 DRAFT1 q (0, 3) RLL - Unconstrained m a x ( P o f , P u f ) -3 -2 -1 (0,10) RLL - Figure 12. Maximum between probability of underflow P uf and overflow P of as per problem (9) for unconstrained and type-0constrained codes versus q with q = 0 (see Fig. 2) and R = 0 . . To simplify the numerical optimization, the curve for k = 10 has been obtained by optimizing only over p , p , p , p and p in P = { p , p , ..., p } and setting p = p = p = ... = p . VI. C
ONCLUSIONS
A host of new applications, including body area networks with implantable devices, is enabledby the possibility to reuse the energy received from information-bearing signals. With theseapplications in mind, we have investigated the use of constrained run-length limited (RLL) codeswith the aim of enhancing the achievable performance in terms of simultaneous information andenergy transfer. We have proposed a framework whereby the performance of energy transferis measured by the probabilities of underflow and overflow at the receiver. The analysis hasdemonstrated that constrained codes enable the transmission strategy to be better adjusted to thereceiver’s energy utilization pattern as compared to classical unstructured codes. This has beenshown to lead to significant performance gains especially at low information rates. Interestingfuture work includes the investigation of non-binary codes and multi-terminal scenarios.
August 10, 2018 DRAFT2 m a x ( P o f , p u f ) Rate -3 -2 -1 (0, 3) RLL - Unconstrained (0,1) RLL - (0, 2) RLL - (1, 3) RLL - Figure 13. Maximum between probability of underflow and overflow as per problem (9) for unconstrained and type-0 constrainedcodes versus the information rate R with q = q = 0 (see Fig. 2). A PPENDIX AT RANSITION P ROBABILITIES FOR T HE M ARKOV C HAIN IN F IG . 10Using the definitions in Sec. IV, we now calculate the transition probability ˜ p m,m − n , Pr[ ˜ B i = m − n | ˜ B i − = m ] , where ˜ B i is the random process that describes the evolution of the battery atthe renewal instants (see Fig. 9). The probabilities ˜ p m,m − n for m ∈ [0 , B max ] and n ∈ [ − , m ] can be calculated as ˜ p m,m − nm =0 , ,B max = k +1 X i = d +1 p I ( i ) · (1 − p ) p (0; i, q ) n = − − p ) q i − P l = n p ( l ; i − , q ) + p i P l = n p ( l ; i, q ) n = m (1 − p ) [ qp ( n ; i − , q )+(1 − q ) p ( n + 1; i − , q )]+ p p ( n ; i, q ) n = 0 , , ..., m − , (38) August 10, 2018 DRAFT3 p m a x ( P o f , P u f ) (0, 3) RLL - Unconstrained -3 -2 -1 -3 -2 -1 Figure 14. Maximum between probability of underflow and overflow as per problem (9) for unconstrained and constrainedcodes versus p for k = 3 , R = 0 . and q = q = 0 . ˜ p m,m − nm =0 = k +1 X i = d +1 p I ( i ) · (1 − p )(1 − q ) i − P l = n +1 p ( l ; i − , q ) n = − − p ) q i − P l = n p ( l ; i − , q )+ p i P l = n p ( l ; i, q ) n = 0 , = k +1 X i = d +1 p I ( i ) · (1 − p )(1 − q ) n = − − p ) q + p n = 0 , (39) ˜ p m,m − nm =1 = k +1 X i = d +1 p I ( i ) · (1 − p ) p (0; i, q ) n = − − p ) (cid:20) qp ( n ; i − , q ) + (1 − q ) i − P l = n +1 p ( l ; i − , q ) (cid:21) + p p (0; i, q ) n = 0(1 − p ) q i − P l = n p ( l ; i − , q ) + p i P l = n p ( l ; i, q ) n = 1 , (40) August 10, 2018 DRAFT4 and ˜ p m,m − nm = B max = k +1 X i = d +1 p I ( i ) · (1 − p ) [ p (0; i, q ) + qp (0; i − , q )+(1 − q ) p (1; i − , q )]+ p p (0; i, q ) n = 0(1 − p ) [ qp ( n ; i − , q )+(1 − q ) p ( n +1; i − , q )]+ p p ( n ; i, q ) n = 1 , , ...m − − p ) (cid:20) qp ( n ; i − , q )+(1 − q ) i − P l = n +1 p ( l ; i − , q ) (cid:21) + p p ( n ; i, q ) n = m − − p ) q i − P l = n p ( n ; i − , q )+ p i P l = n p ( n ; i, q ) n = m . (41)A PPENDIX BP ROOF OF P ROPOSITION ˜ O b and ˜ U b that count the number of overflow and underflow events across the renewal intervals (recall Sec.IV). To this end, we define a random process that counts the number of renewals (i.e., events { C j = 0 } ) up to time i , namely N ( i ) = |{ j ∈ { , ..., i } : C j = 0 }| , (42)where |·| represents the cardinality of its argument. It is also convenient to classify the renewalevents depending on the value of the battery at the beginning of the renewal interval. We canthen define N b ( i ) = |{ j ∈ { , ..., i } : C j = 0 and B j = b }| . (43)The relationship between (42) and (43) is given as N ( i ) = B max X b =0 N b ( i ) . (44)Moreover, the initial time instant of the i th interval corresponding to an initial battery state b ∈ { , ..., B max } can be written as S b,j = min { i : N b ( i ) = j } . (45) August 10, 2018 DRAFT5
Using (42)-(45), we can now obtain the relationship (see also [29, pp. 239-240]) B max P b =0 N b ( i ) P j =1 ˜ U b,j i ≤ i P j =1 U j i ≤ B max P b =0 N b ( i )+1 P j =1 ˜ U b,j i , (46)where ˜ U b,j = S b,j X k = S b,j − U k . (47)Averaging over all battery states b , we also haveE h ˜ U b i = k +1 X i = d +1 p I ( i ) E h ˜ U b,i i . (48)The left hand side of (46) can be separated as B max P b =0 N b ( i ) P j =1 ˜ U b,j i = B max P b =0 Nb ( i ) P j =1 ˜ U b,j N ( i ) N ( i ) i . (49)Therefore, t → ∞ , since we have N ( t ) → ∞ , the strong law of renewal processes can beinvoked on the second term on the right hand side of (49) to conclude that N ( i ) /i → / E [ I ] with probability one [29]. As for the first term, it can be written as B max P b =0 N b ( i ) P j =1 ˜ U b,j N ( i ) = B max X b =0 N b ( i ) P j =1 ˜ U b,jB max P b ′ =0 N b ′ ( i ) (50) = B max X b =0 N b ( i ) P j =1 ˜ U b,j N b ( i ) N b ( i ) B max P b ′ =0 N b ′ ( i ) (51)As a result, if t → ∞ , and hence N b ( i ) → ∞ , by the strong law of large numbers, notingthe fact that the random variables ˜ U b,j for every b ∈ [0 , B max ] are i.i.d. across j , we have P N b ( i ) j =1 ˜ U b,j /N b ( i ) → E [ ˜ U b,j ] with probability one. Finally, by the law of large numbers for ergodicMarkov chains (see, e.g., [24]), we have N b ( i ) / P B max b ′ =0 N b ′ ( i ) → ˜ π b , where we recall that ˜ π b isthe steady-state of the Markov chain ˜ B i , which can be calculated from the transition probabilities August 10, 2018 DRAFT6 detailed in Appendix A. From the discussion above, we conclude that the following limit holdswith probability one lim i →∞ B max P b =0 N b ( i ) P j =1 ˜ U b,j i = B max P b =0 ( E [ ˜ U b ] . ˜ π b ) E [ I ] . (52)The same limit is obtained by applying the approach detailed above to the right-hand side ofthe inequality (46). This concludes the proof of (29) in Proposition 1. The overflow probability(28) is obtained following the same approach.R EFERENCES [1] V. Chawla and S. H. Dong, “An overview of passive RFID,”
IEEE Commun. Magazine , vol. 45, no. 9, pp. 11-17, Sep.2007.[2] F. Zhang, S. A. , X. Liu, H. Chen, R. J. Sclabassi, and M. Sun, “Wireless energy transfer platform for medical sensorsand implantable devices,” in
Proc. Annual Int. Conf. IEEE Engineering in Medicine and Biology Society (EMBC 2009),pp. 1045-1048, Minneapolis, MN, Sep. 2009.[3] A. Yakovlev, K. Sanghoek, A. Poon, "Implantable biomedical devices: Wireless powering and communication,"
IEEECommun. Magazine, vol. 50, no. 4, pp. 152-159, Apr. 2012.[4] P. V. Nikitin, S. Ramamurthy, R. Martinez, and K. V. S. Rao, “Passive tag-to-tag communication,” in
Proc. IEEE Int. Conf.on RFID , Orlando, FL, Apr. 2012.[5] W.C. Brown, "The history of power transmission by radio waves,"
IEEE Trans. Microwave Theory and Techniques,
Proc. IEEE Int. Symp. Inform. Theory (ISIT2008), pp. 1612-1616, Toronto, Canada, Jul. 2008.[9] P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and power transfer,” in
Proc. IEEE Int. Symp. Inform.Theory (ISIT 2010), pp. 2363-2367, Austin, TX, Jun. 2010.[10] L. R. Varshney, "On energy/information cross-layer architectures," in
Proc. IEEE Int. Symp. Inform. Theory (ISIT 2012),pp. 1356-1360, Cambridge, MA, July 2012.[11] R. Zhang and C. Keong Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” in
Proc.IEEE GLOBECOM 2011 , pp. 1-5, Houston, TX, Dec. 2011.[12] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: architecture design and rate-energy tradeoff,”in
Proc. IEEE GLOBECOM 2012, pp. 3982-3987, Anaheim, CA, Dec. 2012.[13] Z. Xiang and M. Tao, “Robust beamforming for wireless information and power transmission,”
IEEE Wireless Communi-cations Letters , vol. 1, no. 4, pp. 372-375, Aug. 2012.[14] K. Huang, E.G. Larsson, “Simultaneous information and power transfer for broadband wireless systems,”http://arxiv.org/abs/1211.6868, 2012.
August 10, 2018 DRAFT7 [15] J. Park, B. Clerckx, “Joint wireless information and energy transfer in a two-user MIMO interference channel,”http://arxiv.org/abs/1303.1693v2, 2013.[16] K. Huang and V. K. N. Lau, “Enabling wireless power transfer in cellular networks: architecture, modeling and deployment,”http://arxiv.org/abs/1207.5640, 2012.[17] Y. J. Zhang, and K. B. Letaief, “Optimal scheduling and power allocation for two-hop energy harvesting communicationsystems,” http://arxiv.org/abs/1212.5394, 2012.[18] K. Ishibashi, H. Ochiai, and V. Tarokh, “Energy harvesting cooperative communications,” in
Proc. IEEE Int. Symp. onPersonal, Indoor and Mobile Radio Communications , pp. 1819–1823, Sydney, Australia, Sept. 2012.[19] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and informationprocessing,” http://arxiv.org/abs/1212.5406, 2012.[20] P. Popovski, A. M. Fouladgar, O. Simeone, "Interactive joint transfer of energy and information,"
IEEE Trans. Commun. ,vol. 61, no. 5, pp. 2086-2097, May 2013.[21] A. M. Fouladgar, O. Simeone, "Information and energy flows in graphical networks with energy transfer and reuse," toappear in
IEEE Wireless Communications Letters,
IEEE Trans.Wireless Commun. , vol. 12, no. 1, pp. 288-300, Jan. 2013.[23] T. Cover and J. A. Thomas,
Elements of Information Theory , Wiley-Interscience, 2006.[24] B. Marcus, R. Roth, P. Siegel,
Introduction to Coding for Constrained Systems,
Proc. IEEE Information Theory and Applications Workshop , pp. 1-9, San Diego, CA, Feb. 2011.[26] A. Kailas and M. A. Ingram, "A novel routing metric for environmentally-powered sensors with hybrid energy storagesystems," in
Proc. IEEE Wireless VITAE Conference, pp. 1-6, Aalborg, Denmark, May 2009.[27] D. O’Neill, M. Levorato, A. Goldsmith, and U. Mitra. "Residential demand response using reinforcement learning," in
Proc. IEEE Int. Conf. Smart Grid Commun. (SmartGridComm), pp. 409-414, Gaithersburg, MD, Oct. 2010.[28] A. El Gamal and Y.-H. Kim,
Network Information Theory , Cambridge University Press, 2011.[29] R. G. Gallager,
Discrete Stochastic Processes,
Kluwer, Norwell MA, 1996.[30] S. Y. Kofman, “On the capacity of binary and Gaussian channels with run-length-limited inputs,”
IEEE Trans. Commun. vol. 38, no. 5, pp. 584-594, May 1990.[31] E. Zehavi and J. K. Wolf, "On runlength codes,"
IEEE Trans. Inform. Theory , vol. 34, no. 1, pp. 45-54, Jan. 1988., vol. 34, no. 1, pp. 45-54, Jan. 1988.