Downlink Transmission of Short Packets: Framing and Control Information Revisited

Transcription

1 1 Downin ransmission of Short Pacets: Frag and Contro Information Revisited Kasper Føe riingsgaard, Student Member, IEEE and Petar Popovsi, Feow, IEEE arxiv: v1 cs.i 6 May 2016 Abstract Ceuar wireess systems rey on frame-based transmissions. he frame design is conventionay based on heuristics, consisting of a frame header and a data part. he frame header contains contro information that provides pointers to the messages within the data part. In this paper, we revisit the principes of frame design and show the impact of the new design in scenarios that feature short data pacets which are centra to various 5G and Internet of hings appications. We treat frag for downin transmission in an AWGN broadcast channe with K users, where the sizes of the messages to the users are random variabes. Using approximations from finite bocength information theory, we estabish a framewor in which a message to a given user is not necessariy encoded as a singe pacet, but may be grouped with the messages to other users and benefit from the improved efficiency of onger codes. his requires changes in the way contro information is sent, and it requires that the users need to spend power decoding other messages, thereby increasing the average power consumption. We show that the common heuristic design is ony one point on a curve that represents the trade-off between atency and power consumption. I. INRODUCION Modern high-speed wireess networs heaviy depend on reiabe and efficient transmission of arge data pacets through the use of coding and information theory. he advent of machineto-machine M2M), vehicuar-to-vehicuar V2V), and various streag systems have spawned a renewed interest in deveoping information theoretica bounds and codes for communication of short pacets 123. Additionay, these appications often have tight reiabiity and atency constraints compared to typica wireess systems today. Communication at shorter bocengths introduces severa new chaenges which are not present when considering communication of arger data pacets. For exampe, the overhead caused by contro signas and header data is insignificant if arge data pacets are sent, and hence, this overhead is often negected in the anaysis of protocos. However, more stringent atency requirements ead to shortened bocengths for transmission such that the size of contro information may approach, or even exceed, the size of the data part in the pacet. his is especiay true for mutiuser systems such as broadcast channes, two-way channes, or mutipe access channes, where the contro information must incude information about the pacet structure, security, and user address information for identification purposes. he wor of P. Popovsi and K. F. riingsgaard was supported in part by the European Research Counci ERC Consoidator Grant Nr WILLOW) within the Horizon 2020 Program. K. riingsgaard and P. Popovsi are with the Department of Eetronic Systems, Aaborg University, 9220, Aaborg Øst, Denmar e-mai: {ft,petarp}@es.aau.d). he fundamentas of communication of short pacets have been addressed by Strassen and, recenty, Poyansiy et a. in 4 and 5. It was shown that the maximum coding rate of a -ength code with n channe uses and maximum error probabiity ε over a discrete-time AWGN point-to-point channe has an asymptotic expansion given by R n, ε) = C V n Q 1 ε) + 1 2n og 2 n + O ) 1 n as n. Here, C is the Shannon capacity, V is the channe dispersion, and Q 1 ) denotes the inverse Q-function. In addition to the asymptotic expansion in 1), 5 used nonasymptotic bounds to numericay demonstrate that R n, ε) is tighty approximated by the first three terms of 1). he approximation 1) and simiar ones are important in the design of communication systems because the specifics of code seection can be negected in the optimization of protoco parameters. For exampe, such approximations have been appied in the optimization of pacet scheduing probems 6, hybrid ARQ protocos 7, and coud radio access networs 8. In this paper, we consider downin transmission with a discrete-time AWGN broadcast channe that consists of a transmitter and K users. Downin transmissions are organized in frames, whose structure is the main topic of this paper. In each frame, there is a message from the transmitter to the -th user with a certain probabiity 1 q. If, in a given frame, there is a message for user, then this user is said to be active in that frame. he size of the message to user is denoted by D and is a random variabe itsef. Hence, the transmitter needs to convey information about which users are active, the structure of the transmission, and sizes of the messages. As a resut, the frame duration, which corresponds to the tota transmission time, and the tota power consumption at the users are aso random variabes. An important observation from 1) is that arger data pacets are encoded more efficienty. his introduces an interesting trade-off with two extremes: a) in a broadcast setting one can either encode a messages in one arge pacet, or b) one can encode each message separatey, which is the norm in modern wireess protocos. In a), the average frame duration is imized, which impies that the average atency across the users is imized. However, the downside of a) is that a users need to receive for the whoe period of transmission to be abe to decode their messages, which is undesirabe for devices that are power-constrained. he atter approach b), depicted in Fig. 1, uses codes which are ess efficient, and thus the average frame duration is arger. On the other hand, each user ony needs to decode the information intended for that user. he ey point, however, is that these design considerations enarge the 1)

2 2 space of feasibe protocos and enabe the protoco designer to see a trade-off between frame duration atency) and power consumption at the users. Despite this trade-off, practicay a wireess systems soey use the extreme approach b). Contribution: he purpose of this paper is to revisit the way a downin frame is designed when it contains short pacets. Specificay, it aims at exporing the trade-off between the average frame duration and the average power consumption at the users. Instead of using a traditiona frame structure, we enarge the design space for a frame by doing the foowing: the users are divided into groups that may depend on the reaization of the message sizes and the messages of each group are jointy encoded using optima channe codes. We anayze the probem using asymptotic expansions simiar to 1), and we find a ower bound for the trade-off curve. Next, we introduce three protocos: a) a genie-aided protoco with performance cose to the ower bound, b) protoco with a message that wors for the case in which each message has either the size 0 or α N bits, and c) a protoco with variabe message sizes, where the message sizes are distributed according to a probabiity mass function P D with finite and nonnegative integer support. he protocos b) and c) both convey enough contro information to mae them practicay usabe. Our numerica resuts demonstrate trade-offs which are particuary interesting when the message sizes are sma. Organization: Section II introduces the finite bocength approximations and bounds for optima channe codes whie the system mode is introduced in Section III. Section IV presents a ower bound for the average power at each user expressed as a function of the average frame duration. Section V provides some concrete protoco designs, which are subsequenty compared with the ower bound. Finay, numerica exampes are presented in Section VI and Section VII concudes the paper. Notation: Vectors are denoted by bodface etters e.g., x) whie their entries are denoted by roman etters e.g., x i ). We denote the n-dimensiona a-zero vector and a-one vector by 0 n and 1 n, respectivey. We denote by 0 n i x) the n-dimensiona vector with x in the i-th entry and zeroes in the rest. We et denote the concatenation of two bit string, e.g., for a {0, 1} n and b {0, 1} m, a b is the concatenated bit string. hroughout the paper, the index beongs aways to the set K {1,, K}, athough this is sometimes not expicity mentioned. We define the upper concave enveope of a function f : R + R + as ucef) inf g {g f and g is concave}. Simiary, the ower convex enveope is defined by cef) sup g {g f and g is convex}. Finay, N denotes the set of positive integers, Z + N {0}, and the symbo R indicate the set of rea numbers. II. FINIE BLOCKLENGH BOUNDS AND APPROXIMAIONS In our anaysis, we appy resuts from finite bocength information theory. For the rea) AWGN channe under a shortterm power constraint P, 4 and 5 showed that the maximum coding rate R n, ε) of a code with bocength n and error probabiity ε 0, 1) has the asymptotic expansion given contro information encoded messages Fig. 1. Conventiona approach to downin broadcasting. We denote by M 2, M 6,, M 20 the messages of varying size in bits) destined to the active users. An initia pacet contains contro information that defines the structure of the remaining part of the transmission. Each message is encoded separatey. Ch. uses/information bit N a, ε)/ N, ε)/ N, ε)/ N c, ε)/ 1/C Information bits Fig. 2. Bounds and approximations for N, ε) potted for ε = 10 3 and P = 0 db. he converse N c, ε) and achievabiity bound N a, ε) are potted using the SPECRE toobox. by 1), where the channe capacity C and the channe dispersion V are given by and V C 1 2 og 21 + P ) 2) P P + 2) 2P + 1) 2 og 2exp1)) 2 3) respectivey. One can obtain tight nonasymptotic upper and ower bounds for R n, ε) using the achievabiity and converse bounds in 5, and it was numericay demonstrated that the first three terms of the right-hand side of 1) provide a tight approximation of R n, ε). We define N, ε) {n 0 : nr n, ε) } for 1 and N 0, ε) 0 which is the smaest number of channe uses that aows the encoding of bits with error probabiity ε. We obtain the foowing asymptotic approximation of N, ε) as : N, ε) = V C + C 3 Q 1 ε) 1 2C og 2 + O1). 4) C his can be verified by setting n equa to RHS of 4) and by computing nr n, ε). hen, one finds that nr n, ε) = + O1) from which 4) foows. We define the approximation N, ε) uce C + V C 3 Q 1 ε) 1 2C og 2 C ) 5)

3 3 where uce ) stands for the upper concave enveope. It can be shown that the approximation of N, ε) inside uce ) in 5) 4C is concave for Q 1 ε) 2 V og e 2), impying that N, ε) = 2 N, ε)+o1). Additionay, in a numerica exampes in this 4C paper, we have Q 1 ε) 2 V og e 2) < 1, and hence the upper concave enveope does not affect our numerica resuts. In Fig. 2, we 2 have potted the κ-achievabiity bound N a, ε) and the metaconverse bound N c, ε) from 5 aong with the approximation 5), and N, ε) = n, where n is a soution to: nc nv Q 1 ε) og 2 n =. 6) We observe that N, ε) provides an approximation of N, ε) that matches the converse bound cosey. In the remaining part of this paper, when referring to the bocength of an optima code conveying bits with a probabiity of error not exceeding ε, we consistenty use the approximation N, ε) in pace of N, ε) in a computations and derivations. III. SYSEM MODEL We consider an AWGN broadcast channe with one transmitter and K users. In the t-th time sot, the -th user receive Y,t γ X t + Z,t. 7) where Z,t N 0, 1) and X t R is the channe input. hroughout the paper, we assume that γ = 1. he assumption of equa channe conditions can, to some extend, be justified as foows. Consider a downin broadcast scenario with many users with varying channe conditions. A viabe communication strategy is to first divide the users into severa CSI-groups such that the users assigned to a certain CSI-group have simiar channe conditions. hen, the transmitter serves each CSIgroup sequentiay, and our system mode in 7) modes a singe CSI-group. A sateite-based broadcast system with ine-ofsight to a users and predictabe channe conditions constitute a practica exampe of our system mode. If, however, CSIgrouping is not performed, then the transmitter needs to protect a pacet destined to mutipe users with a code that is strong enough to ensure that even the worst-channe user can decode. he assumption of nonfading channes is mainy introduced for simpicity, but we note that there are resuts in finite bocength information theory for fading channes 9. he message M destined to the -th user is nonempty with probabiity 1 q 0, 1), and we say that the -th user is active if there is a message destined to that user. We assume that the size of the message M in bits) is given by D Z + which is a discrete random variabe distributed independenty according to the probabiity mass function { q if d = 0 P D d) 8) 1 q)p i if d = α i for i {1,, S}. he message M is drawn uniformy randomy from the set {0, 1} D. We use α = α 1, α 2,..., α S ) to denote a S- dimensiona vector of distinct ordered positive integers α i < α s if i < s) that correspond to the possibe message sizes. he frame duration is a random variabe that depends on the message sizes {D }. he transmitter encodes the message {M } into a sequence of channe inputs using the encoder function f t ) such that X t f t {D }, {M }) 9) for t {1,, } and X t = 0 for t { + 1, }. Additionay, we require that 1 E X t P. 10) We define the ON-OFF function g,t : R {e}) t 1 {0, 1} that defines the receiver activity for user : { Y,t, g Ȳ,t,t Ȳ t 1 ) = 1, receiver is ON. 11) e, receiver is OFF he ON-OFF function repaces the t-th channe output with an erasure if the user is OFF at that time. he stopping time represents the time index of the ast nonerased channe output in the sequence Ȳ,t; after the receiver is OFF unti the end of the frame. Formay, inf { n 1 : t > n, g,t Ȳ t 1 ) = 0 } for which we require <. Considering that a user can ony use the channe outputs for which it is ON, we define the decoding function h,t Ȳ t) to estimate the message M based on Ȳ t. he ON-OFF functions are causa in the sense that the decision of whether the users are ON at time t depends on previous channe outputs, Ȳ t 1. Uness an error occurs during decoding, the stopping times are ess than or equa for any practica appications of this mode. We merey define to emphasize that is a random variabe which is not nown by the users, and hence the users need to obtain this information through the sequence Ȳ,t. In a conventiona approach to downin broadcast, as depicted in Fig. 1, contro information in the initia pacet defines the structure of the remaining transmission. Hence, after successfuy decoding the contro information in the initia pacet, the -th user nows and when to be ON and OFF to receive the message intended for that user. he average power consumption of the -th user is given by P E 1 { g,i Ȳ i 1 ) = 1 } 12) where 1{ } is the indicator function, and is detered by the ON-OFF function. Note that EP 1 = EP, for {1,, K}, since the message sizes D are distributed identicay. Finay, the active users need to decode their messages with reiabiity arger than or equa 1 ɛ such that P h, Ȳ ) M D > 0 ɛ 13) for {1,, K} and ɛ 0, 1). he above system mode provides a genera framewor for the probem of downin broadcast frag. For tractabiity, we constrain ourseves to an important and practica cass of protocos described as foows. he transmitter forms L Z + pacets which are encoded using optima codes with error probabiities { ɛ } {1,,L}. Here, L and { ɛ } are random variabes that depend ony on {D }. Let L max be a constant that denotes the maximum number of pacets that the transmitter can

4 4 send, defined as the smaest integer such that L max L for a reaizations of {D }. Let {M C) } Lmax denote the contro information that needs to be conveyed in order to describe how the data for different users is conveyed see the exampe beow). Let {D C) } Lmax denote the sizes in bits) of {M C) } Lmax, i.e., MC) {0, 1} DC) and D C) = 0 for > L. Let {U } Lmax denote disjoint random sets that depend ony on {D } such that L max U = K and such that U = for > L. he -th pacet then consists of the information bits M C) U M which are encoded by an optima code with reiabiity ɛ using N D C) + ) U D, ɛ channe uses. he encoder function f t, ) is defined by sequentiay transmitting the L encoded pacets. he frame duration is given by L N D C) + U D, ɛ ). We assume that the optima code has the foowing property: If j bits are encoded into n channe uses by an optima code with error probabiity ε, then the user needs to receive a n channe uses so as to decode any of the j bits with error probabiity ε. As an iustration, we describe how the genera framewor is instantiated to describe a conventiona downin frame from Fig. 1. Suppose S = 3 such that D {0, α 1, α 2, α 3 }. As there are four possibe engths, the contro information about {D } can be represented by at most 2K information bits which are conveyed in the first pacet, commony referred to as the header. We et D C) 1 = 2K and et M C) 1 be the bitstring of ength 2K representing {D }. Since there is a header pacet and at most K other pacets, we set L max = L = K + 1. We aso set ɛ 1 = ε 1 and ɛ = ε 2 for {2,, L max } where ε 1, ε 2 ) 0, 1 2 are such that ɛ = 1 1 ε 1 )1 ε 2 ). Since a contro information is concentrated in the frame header, we have D C) = 0 for 2. he sets {U } Lmax are defined such that the header has no user data and U 1 =, whie U = { 1} for {2,, L max }. User is ON during the transmission of the first pacet which it decodes with probabiity 1 ε 1. If user successfuy decodes the first pacet, it earns {D }, and thereby it obtains a pointer to the ocation of the +1)-th pacet, which contains the desired message M. After decoding the header, the -th user is OFF for the remaining time except when the +1)- th pacet is transmitted. he + 1)-th pacet is successfuy decoded with probabiity 1 ε 2. he overa probabiity of error for the protoco from the viewpoint of a singe user is given by 1 1 ε 1 )1 ε 2 ) = ɛ as desired. For arge message sizes α s 1 we get the ower bounds: E KED 1 14) C EP 1 ED 1 C. 15) When α s 1, the contro information becomes negigibe, and hence for the conventiona approach both E and EP 1 simutaneousy approach the ower bounds in 14) and 15). Our objective is to expore trade-offs between the competing goas of imizing E and EP 1. IV. LOWER BOUND We estabish a ower bound by assug that the users are provided with contro information from a genie, i.e., {D } are Power ch. uses ) for a > 0 22) with = ) with = ) with = Frame duration ch. uses Fig. 3. Depicts the ower bound in 22) for three different vaues of for P = 0 db, ɛ = 10 4, and α 1 = he bac curve is obtained by evauating 22) for a > 0 and by combining the resuting ower bounds. he dots correspond to five genie-aided protocos described by N 1,, N 4 ) 4, 0, 0, 0), 3, 1, 0, 0), 2, 2, 0, 0), 2, 1, 1, 0), 1, 1, 1, 1) enumerated from top-eft corner to bottom-right corner). nown at a users. In that case, the transmitter and a users can agree on a protoco that ony conveys the messages {M }, i.e., D C) = 0 for {1,, L max }. Hence, the transmitter may encode the messages {M } into at most K separate pacets such that each message is encoded in exacty one of these pacets. Each pacet may contain either no messages at a, a singe message, or mutipe concatenated messages, and they are encoded using optima codes with error probabiities that do not exceed ɛ upon decoding; reca that a users experience the same error probabiity since γ = 1. Any genie-aided protoco can be characterized using K random nonnegative integer vectors N Z S +, for {1,, K}, that depend ony on {D }. he content of the -th pacet is described by N ; the pacet encodes N,1 messages of ength α 1, it encodes N,2 messages of ength α 2, etc. Note that the integer vectors {N } do not uniquey describe which messages are encoded in which pacets. For a genie-aided protoco defined by a set of vectors {N }, we compute the frame duration and average power as foows 1 K = K Nα N, ɛ) 16) K P i = 1 K =1 K 1 S N Nα N, ɛ). 17) Here, and 1 K K =1 P i are random variabes that depend ony on the reaization of {D }. We aim to ower bound E + EP 1 for any > 0 and thereby obtain a ower bound on the average power consumption EP 1 as a function of average frame duration E. Before stating the ower bound, we introduce the technique through an exampe. Suppose K = 4, q = 0, S = 1 such that D 1 = = D 4 = α 1 and the frame duration and average power are deteristic. Since the users now {D } 4 =1 and

5 5 each of the four messages beongs to one encoded pacet, any genie-aided protoco can be described through the four nonnegative integers N 1, N 2, N 3, and N 4 satisfying N 1 + +N 4 = 4. hese integers represent the number of messages encoded in the first, second, third, and fourth pacet, respectivey. For > 0, our objective is to imize P with respect to N 1,, N 4 {0,, 4} subject to N N 4 = 4. For this particuar exampe, one can easiy sove the resuting integer optimization probem. However, we can aso find a ower bound on P through the foowing steps + 4 P 4 = 4 Nα 1 N, ɛ) + 4 n 1,,n 4 {0,,4}: n 1+ +n 4=4 n 1,,n 4 {0,,4}: n 1+ +n 4=4 n 1,,n 4 {0,,4}: n 1+ +n 4=4 4 N Nα 1 N, ɛ) 18) 4 Φ n ) 19) 4 Φ n ) 20) 4 Φ 4 n 4 ) 21) = 4 Φ 1). 22) Here, 19) foows by defining Φ x) Nα 1 x, ɛ) 1 + x ), 23) 4 and by a imization with respect to n 1,, n 4, 20) foows by defining Φ ) as the ower convex enveope of Φ ), and 21) is by convexity of the ower convex enveope of Φ ). Interestingy, the bound in 22) is fairy tight and simpe to compute. We iustrate the bound 22) in Fig. 3, confirg the intuition that when S = 1, one shoud attempt to have an equa number of non-empty messages in each pacet, which for this exampe is 1, 2, or 4 messages in each pacet. For the genera setting with arbitrary S 1 and K, we appy the above ideas in the foowing proposition which enabes us to compute a ower bound on EP 1 for certain E. Proposition 1: For every > 0, we have E + EP 1 E 1 S L 1:S φ L1:S 1 S L 1:S ) 24) where L Z S+1 + is mutinomia distributed with S + 1 categories, K trias, and event probabiities 1 q)p 1,, 1 q)p S, q, L 1:S denotes the first S entries of L, and φ : R S + R + is the ower convex enveope of the function φ x) N α x, ɛ ) S x K ) 25) defined for x R S +. Proof: Fix > 0. he users K can be decomposed into S + 1 disjoint subsets {U s) } s {0,,S} such that U s) { K : D = α s } for s {0,, S}, where we et α 0 0 for notationa convenience. We denote the random) set of active users by U = S U s). Fix a genie-aided protoco. hen, since we assume that the users are provided with contro information by a genie, the protoco must decompose the set of active users U into at most K possiby empty) disjoint subsets {U } {1,,K}. Note that these subsets are random, depend ony on {D }, and are induced by the protoco. Define the random integer vectors N Z S +, for {1,, K} and s {1,, S}, as foows: N,s K 1{ U and D = α s }. 26) he average frame duration and the average power for the genieaided protoco in terms of {N i } are now given by K E = E Nα N, ɛ) 27) 1 K EP 1 = E 1 S N Nα N, ɛ). 28) K Now, we compute a ower bound on E + EP 1 based on 27) and 28): E + EP 1 K = E Nα N, ɛ) + K K = E φ N ) E n 1,,n Z S + : n,s= U s) K 1 S N Nα N, ɛ) 29) 30) φ n ) 31) where 30) is by the definition of φ ) in 25). In 31), the expectation is ony with respect to the random variabes U 1),, U S) and. Next, 31) is ower-bounded by using the ower convex enveope of φ ) and its convexity: E + EP 1 E E = E n 1,,n Z S + : n,s= U s) n 1,,n Z S + : n,s= U s) φ U 1),, U S) φ n i ) 32) ) φ 1 n 33) ). 34) Here, 32) foows because the ower convex enveope φ ) of φ ) is smaer than or equa φ ) and 33) foows from convexity of φ ). he resut foows by noting that the random vector U 1),, U S), U 0) is mutinomia distributed with S + 1 categories, K trias, and event probabiities 1 q)p 1,, 1 q)p S, q. he foowing emma shows that we can use the concavity of N, ɛ) to simpify the computation of φ ).

6 6 Lemma 2: For every > 0, we have φ x) = where we have defined ζ R S : 1 S ζ=1 s:ζ s>0 S ζ s φs) x s/ζ s ) 35) φ s) x) Nα sx, ɛ)1 + x/k) 36) for x 0 and s {1,, S}. Additionay, the optimization probem in 35) is convex. Proof: See Appendix I For the case with message sizes, i.e., when S = 1, Proposition 1 reduces to the foowing coroary. Coroary 3: For every 0, we have E + EP 1 1 q)k 1) φ 1) 37) where φ 1) ) is defined in 36). his readiy foows from L 1:S /1 S L 1:S ) = 1 and because 1 S L 1:S is Binomia distributed with parameters K and 1 q, and hence 1 S L 1:S = 1 q)k. A. Genie-aided protoco We put forth a genie-aided protoco that uses the intuition obtained through Proposition 1 and Lemma 2. Here, genie-aided refers to the fact that the protoco assumes that the nowedge about {D } is avaiabe at a users. Lemma 2 suggests that one shoud group messages of the same sizes together rather than grouping messages of mixed message sizes. he purpose of introducing a genie-aided protoco is to show that it achieves a trade-off cose to that of the ower bound. Moreover, we can compare the non-genie-aided protocos, introduced in Section V, to the genie-aided protoco to show the impact of contro information. Such comparisons are provided in Section VI. First, for a set of users Ū K, V N, and ε 0, 1), we define a Ū, V, ε)-protoco as foows. he users Ū are divided into G Ū /V disjoint sets {Ū} {1,,G} such that { Ū /G + 1, {1,, mod Ū, G)} Ū Ū /G, otherwise. 38) G One can verify that Ū = Ū. Sequentiay, for {1,, G}, the transmitter encodes and conveys a pacet containing M with error probabiity ε using Ū N D, ε) channe uses. Here, denotes the concatenation of messages. Whie the number of channe uses spend at Ū the transmitter is given by G N D, ɛ ), each user ony needs to receive and decode one of the Ūi G pacets. We aso note that a Ū, V, ε)-protoco assumes contro information at a the users Ū, i.e., the users needs to now Ū, {D } Ū, V, and ε. For our genie-aided protoco, we define U s) { K : D = α s }, for s {0,, S}, and fix a vector V K S. Now, sequentiay for each s {1,, S}, the transmitter deivers the messages of the users U s) using a U s), V s, ɛ)-protoco. We denote the average frame duration E and the average power EP 1 by V,ɛ) genie and V,ɛ) genie, respectivey. he vector V is eft to be specified. We can trace of optima trade-off between average frame duration and average power by soving the integer optimization probem for a 0: V,ɛ) V,ɛ) V K S genie + genie 39) V. PROOCOL DESIGN In the foowing, we devise actua protocos that trade-off between average frame duration and average power consumption at the users. In contrast to the genie-aided protoco in Section IV-A, these protocos need to convey contro information. A. Fixed message size We initiate our discussion of protoco design with the case of message size, i.e., S = 1. In this case, the contro information ony consists of which users are active. We divide the set of users K into B K/W disjoint subsets K 1,, K B such that B K i = K and such that { K/B + 1, i {1,, modk, B)} K i = 40) K/B, otherwise. Here, W N is a protoco parameter to be set. he subsets {K i } of K are termed user groups UG). he transmitter forms a pacet that contains ony the number of active users in each UG, i.e., the pacet encodes the vector U K 1, U K 2,, U K B. his vector constitutes a first ayer of contro information and can be uniquey represented by at most 1 = K/W og 2 W bits. We encode the contro information by an optima channe code with error probabiity not exceeding ɛ 1 0, 1) which can be achieved by approximatey N 1, ɛ 1 ) channe uses. After successfuy decoding the first pacet, the users now the number of users in each UG, and thereby the structure of the remaining part of the transmission. he second ayer encodes contro information and messages associated with each UG. Specificay, for the i-th UG, the transmitter needs to inform the users of the i-th UG about which U K i users of K i are active. Hence, the contro information for the i-th UG, can be represented by 2,i og Ki 2 U K i ) bits and is conveyed by using an optima code with error probabiity not exceeding ɛ 2 0, 1), which requires approximatey N 2,i, ɛ 2 ) channe uses. Now, the messages of the active users in the i-th UG U i U K i are conveyed with error probabiity not exceeding ɛ 3 0, 1) using an U i, V, ɛ 3 )-protoco, where V K is another protoco parameter to be set. We emphasize that we can use an U i, V, ɛ 3 )-protoco because the set of active users U i nows U i from the the contro information provided that the first two pacets are successfuy decoded. Based on the description of the protoco above, one can compute E and EP 1 which V,W,ɛ) V,W,ɛ) we denote by and, respectivey. Here, ɛ is the vector ɛ 1, ɛ 2, ɛ 3. he parameters V, W, and ɛ are eft to be specified. We can trace the optima achievabe trade-off of the proposed protoco by soving the foowing optimization probem for a 0: ɛ 0,1 3 : V,W ) K 2 3 =1 1 ɛ )1 ɛ V,W,ɛ) + V,W,ɛ). 41)

7 7 ptr 2 ptr 3 ptr 4 message sizes UG 1 UG 2 UG 3 messages with messages with UG 4 Fig. 4. An exampe of the protoco in Section V-B with S = 2, K = 40, W = 10, and V = 3, 2. Pacets surrounded by bac separators corresponds to an encoded pacet. Grey separators means encoded jointy, e.g., the messages M 12 and M 15 are jointy encoded in one pacet. he red shaded parts of the protoco depicts the pacets that the users 12 and 15 needs to decode. Whie the outer imization is an integer optimization probem which can ony be soved using exhaustive search, the inner imization is convex and can be soved using standard convex optimization agorithms. his is shown in the foowing emma. Lemma 4: he inner optimization probem in 41) is convex in ɛ. Proof: Note that, for V and W, the objective function in 41) depends ony on ɛ through a nonnegative inear combination of Q-functions of ɛ 1, ɛ 2, and ɛ 3, i.e., there exist nonnegative constants a 1, a 2, and a 3 such that V,W,ɛ) + V,W,ɛ) = a 1 Q 1 ɛ 1 ) + a 2 Q 1 ɛ 2 ) + a 3 Q 1 ɛ 3 ). 42) V,W,ɛ) V,W,ɛ) his is because and are evauated using N, ε). o show convexity of the optimization probem 41), we use the substitution ɛ i = 1 expu i ) for u i 0 and i {1, 2, 3}, which yieds the equivaent constraint u 1 +u 2 +u 3 = 3 ) og =1 1 ɛ ) og1 ɛ) which is inear. Consequenty, it is sufficient to show that Q 1 1 expu i )) is convex for i {1, 2, 3}. his foows because the ogarithm of the cumuative distribution function of the Gaussian distribution fx) og1 Qx)) is concave and increasing. hus, its inverse function f 1 x) = Q 1 1 expx)) is convex and increasing. At this point, we have not discussed the possibiity of undetected errors. Approximations ie 1) do not give any guarentee for the probabiity of detecting an error. Using CRCs, the probabiity of undetected error can be made arbitrariy sma, but it is aways positive and ess than or equa ɛ. Suppose that decoding of the first pacet, containing contro information, fais for the - th user. In this case, the subsequent behavior is random, and the -th user wi with high probabiity) not correcty decode the foowing pacets. However, since the pacet sizes are imited by α S, we can compute the worst-case power consumption at the users, say P worst. We then cope with the probem of undetected errors simpy by adding, to the power consumption at each user, the term ɛp worst, which corresponds to the worst-case contribution to the power consumption. B. Variabe message size Next, we consider the case S 2. he users are grouped into B K/W UGs in the same way as for the message size protoco. he UGs are encoded sequentiay after the contro information of the first ayer. he contro information of the first ayer consists of pointers to the time indices of the beginning of each UG. hus, based on the contro information of the first ayer, each user can identify the ocation of its UG. Note that we need ony B 1 pointer because the first UG is transmitted immediatey after the contro information. Each pointer is encoded separatey in a pacet using an optima code with an error probabiity not exceeding ɛ 1. Observe that one can compute the maximum ength in channe uses) of each UG and thereby the number of bits required for each pointer. he contro information of the second ayer for the i-th UG consists of {D } Ki, represented by K i og 2 S + 1) bits. hese bits are transmitted using an optima code with error probabiity not exceeding ɛ 2. Finay, sequentiay for each s {1,, S}, the transmitter encodes the messages of the users U s) i using an U s) i, V s, ɛ 3 )-protoco, where V = V 1,, V S are protoco parameters to be specified. he protoco is iustrated in Fig. 4. V,W,ɛ) variabe V,W,ɛ) variabe We denote E and EP 1 by and, respectivey, and optimize the parameters of the protoco using the optimization probem ɛ 0,1 3 : V,W ) K S+1 3 =1 1 ɛ )1 ɛ V,W,ɛ) variabe + V,W,ɛ) variabe. 43) As for the message size protoco, the inner imization is convex. VI. NUMERICAL RESULS In this section, we pot the ower bound aong with the optima achievabe trade-offs for the proposed protocos. A resuts are for ɛ = 10 4, P = 0 db, q = 0.5. We first present resuts for the case with message size. Fig. 5 and Fig. 6 show the trade-offs for α 1 = 100 and S = 1 for K = 16 and K = 128, respectivey. We pot the ower bound given by Proposition 1 and the trade-offs achievabe by the genie-aided protoco and the message size protoco. For the message size protoco, we aso pot the trade-off for the case where the inner imization in 41) is not performed and ɛ 1, ɛ 2, ɛ 3 are set equay to 1 1 ɛ) 1/3. For the protocos, we pot the ower convex enveopes and note that any point on them can be achieved by time-sharing between two sets of protoco parameters. We observe, as expected, that differences between the genie-aided protocos and the ower bounds are negigibe. Optimizing over ɛ aso improves the trade-off sighty. his happens because the contro information which is destined to many users needs better protection compared to a group of messages destined ony to a group of users. Finay, we observe a significant gap between the genie-aided protoco and the message size protoco which refects the significance of contro information for broadcast of sma messages. Fig. 7 and Fig. 8 shows the trade-offs for α = In this case, we see that the gap between the genie-aided protoco and the message size protoco becomes ess significant. Finay, in Fig. 9 and Fig. 10, we depict the trade-offs for K = 16, p = 0.5, 0.5 and with α = 50, 150 and α = 500, 1500, respectivey. Our observations are simiar to those for the message size protoco.

8 8 Average power ch. uses Protoco no ɛ opt.) Protoco Genie-aided protoco Average power ch. uses Protoco no ɛ opt.) Protoco Genie-aided protoco Average frame duration ch. uses Average frame duration ch. uses 10 4 Fig. 5. rade-off between average transmission time and average power consumption for the case K = 16, P = 1, q = 0.5, α = 100, S = 1, and ɛ = Here, Protoco refers to the message size protoco, whie Protoco no ɛ opt.) refers to the message size protoco with ɛ 1 = ɛ 2 = ɛ 3 = 1 1 ɛ) 1/3 ). Fig. 7. rade-off between average transmission time and average power consumption for the case K = 16, P = 1, q = 0.5, α = 1000, S = 1, and ɛ = Here, Protoco refers to the message size protoco, whie Protoco no ɛ opt.) refers to the message size protoco with ɛ 1 = ɛ 2 = ɛ 3 = 1 1 ɛ) 1/3 ). Average power ch. uses Protoco no ɛ opt.) Protoco Genie-aided protoco Average power ch. uses Protoco no ɛ opt.) Protoco Genie-aided protoco Average frame duration ch. uses Average frame duration ch. uses 10 5 Fig. 6. rade-off between average transmission time and average power consumption for the case K = 128, P = 1, q = 0.5, α = 100, S = 1, and ɛ = Here, Protoco refers to the message size protoco, whie Protoco no ɛ opt.) refers to the message size protoco with ɛ 1 = ɛ 2 = ɛ 3 = 1 1 ɛ) 1/3 ). Fig. 8. rade-off between average transmission time and average power consumption for the case K = 128, P = 1, q = 0.5, α = 1000, S = 1, and ɛ = Here, Protoco refers to the message size protoco, whie Protoco no ɛ opt.) refers to the message size protoco with ɛ 1 = ɛ 2 = ɛ 3 = 1 1 ɛ) 1/3 ). VII. CONCLUSION In this paper, we considered the AWGN broadcast channe with K users with symmetric channe conditions. he downin transmission is organized in frames. In each frame, a message of random size in bits) is destined to each of the users in such a way that the message sizes are unnown to the users. he message can aso be of size zero, which means the user shoud not receive data in that frame. A user, however, sti needs to decode a certain amount of information from the frame in order to earn that there is no data destined to her in this particuar frame. Hence, in addition to the messages, a protoco needs to convey contro information that describes the structure of the transmission and the sizes of the messages. We used approximations of the maximum coding rate for the AWGN channe from finite bocength information theory to show that jointy encoding different groupings of the messages enabe the protoco designer to trade-off between average frame duration and the average power consumption at the users. Specificay, we derived a ower bound for the trade-off curve which assumed that contro information was avaiabe at the users, a genie-aided protoco, and two practica protocos. Our numerica resuts showed that the genie-aided protoco achieved a trade-off curve

9 9 Average power ch. uses Protoco S 2) Genie-aided protoco Average frame duration ch. uses Fig. 9. rade-off between average transmission time and average power consumption for the case K = 16, P = 1, q = 0.5, p = 0.5, 0.5, α = 50, 150, S = 2, and ɛ = Average power ch. uses Protoco S 2) Genie-aided protoco S a i,s 1 φ 0 S ) ) s ) S a i,s = 1 φ s) ) 1. 48) 2000 In 46), 0 S s x) denotes an S-dimensiona vector with x in the s-th entry and zeroes in the rest, 47) foows by Jensen s inequaity concave) appied to φ x) on the simpex A 1, and ) is by the definition of φ s) ) in 36). bwe can now owerbound 44) as Average frame duration ch. uses 10 4 S I ν i a i,s Fig. 10. rade-off between average transmission time and average power φ x) consumption for the case K = 16, P = 1, q = 0.5, p = 0.5, 0.5, 1 φ s) ) 1 49) α = 500, 1500, S = 2, and ɛ = S I ν i a i,s ) 1 φs) 1 50) that cosey matched the ower bound. For both of our practica protocos, the contro information ed to a significanty worse trade-off curves when the messages were sma and when compared to the genie-aided protoco. here are severa directions for future research: 1) In Section III, we significanty restricted our genera system mode to a space of practica and tractabe protocos. A rigorous information-theoretic treatment of our genera system mode might ead to improved protocos and ower bounds. 2) he system mode has two obvious extensions: one can extend the system mode to incude fading, and one can introduce asymmetric channe conditions using resuts from 9. 3) Whie we are abe to quantify the suboptimaity of our protocos by comparison to the ower bound, our protocos are sti heuristic. One interesting idea for future research is to systematicay investigate the design of good protocos that incude contro information. APPENDIX I PROOF OF LEMMA 2 By definition of the ower convex enveope, for every x R S +, there exists a vector ν R S + with 1 S ν = 1 and I points a i R S +, for i {1,, I}, such that and such that φ x) = x = I ν i φ a i ) 44) I ν i a i. 45) Since Nn, ɛ) is concave in n, we have that φ x) is concave on the simpex A κ {x R S + : 1 S x = κ} for every κ R +. Consequenty, for i {1,, I}, we have = φ a i ) = φ S S I ) ν i a i,s 1 S a i,s 0 S ) ) 1 s 1 46) φ s) I ν ia i,s 1 1 S ai I ν ia i,s 1 S ai 51) ζ s φs) x s/ζ s ). 52) Here, 49) is by 44) and 48), 50) is by φ s) s) x) φ x) for x 0, 51) foows by Jensen s inequaity convex) appied to φ s) ), and 52) foows by setting ζ s I ν ia i,s and by 1 S ai using I ν ia i,s = x s by 45). hus, we have shown that the LHS of 35) is arger than or equa the RHS of 35). Next, we estabish the equaity in 35). Suppose, on the contrary, that there exists a positive vector ζ R S such that

10 10 1 S ζ = 1 and such that φ x) > S impies a contradiction: φ x) > = S ζ φs) s x s/ ζ s ). his ζ s φs) x s/ ζ s ) 53) S ζ s φ 0 S s x s /ζ s )) 54) φ x). 55) Here, 55) foows by Jensen s inequaity convex) appied to φ ). We concude that 35) must be satisfied with equaity. Note that it is sufficient to write imum instead of infimum in 35) because we have shown the existence of a feasibe point in 35) that attains the imum. o show convexity of the optimization probem in 35), it is sufficient to show that the function x φ s) y/x) is convex in x > 0 for a constant y > 0 and s {1,, S}, i.e., for every x 1 > x 2 > 0 and α 0, 1, we need to show that 6 S. Xu,.-H. Chang, S.-C. Lin, C. Shen, and G. Zhu, Energy-efficient pacet scheduing with finite bocength codes: Convexity anaysis and efficient agorithms, arxiv, pp. 1 30, Mar Onine. Avaiabe: 7 B. Mai,. Svensson, and M. Zorzi, Finite boc-ength anaysis of the incrementa redundancy harq, IEEE Wireess Commun. Letters., vo. 3, no. 5, pp , S. Khaii and O. Simeone, Upin harq for distributed and coud ran via separation of contro and data panes, arxiv, pp. 1 27, Dec Onine. Avaiabe: 9 W. Yang, G. Durisi,. Koch, and Y. Poyansiy, Quasi-static mutipeantenna fading channes at finite bocength, IEEE rans. Inf. heory, vo. 60, no. 7, pp , αx 1 φs) y/x 1 ) + 1 α)x 2 φs) y/x 2 ) αx α)x 2 ) φ s) y/αx α)x 2 )).56) Fix, without oss of generaity, arbitrary x 1 > x 2 > 0, α 0, 1, and s {1,, S}. Define the function y ) x gx) = 2 x y y x 2 y φs) + x y ) x 1 y y x 1 x 1 x 2 y φs). 57) x 1 x 2 Note that xgy/x) and gx) are affine functions in x > 0 and that gy/x 1 ) = φ s) y/x 1 ) and gy/x 2 ) = φ s) y/x 2 ). hus, since φ s) ) is convex, we have gx) φ s) x) for x y/x 1, y/x 2. o verify 56), we write αx 1 φs) y/x 1 ) + 1 α)x 2 φs) y/x 2 ) = αx α)x 2 )gy/αx α)x 2 )) 58) αx α)x 2 ) φ s) y/αx α)x 2 )) 59) his estabishes the convexity of the optimization probem in 35) because we can redo the above argument for a x 2 > x 1 > 0 and α 0, 1. REFERENCES 1 P. Popovsi, Utra-reiabe communication in 5G wireess systems, in IEEE Int. Conf. 5G for Ubiquitous Connectivity, Levi, Finand, Nov. 2014, pp G. Durisi,. Koch, and P. Popovsi, owards massive, utra-reiabe, and ow-atency wireess: he art of sending short pacets, Proc. IEEE, 2016, to appear. 3 F. Boccardi, R. W. Heath, A. Lozano,. L. Marzetta, and P. Popovsi, Five disruptive technoogy directions for 5G, IEEE Commun. Mag., vo. 52, no. 2, pp , V. Strassen, Asymptotische abschätzungen in Shannon s informationstheorie, in rans. 3rd Prague Conf. Int. heory, Prague, Czech Repubic, 1962, pp Y. Poyansiy, H. V. Poor, and S. Verdú, Channe coding rate in the finite bocength regime, IEEE rans. Inf. heory, vo. 56, no. 5, pp , 2010.