A Multi-Objective Optimization Framework for URLLC with Decoding Complexity Constraints
aa r X i v : . [ c s . I T ] F e b A Multi-Objective Optimization Framework forURLLC with Decoding Complexity Constraints
Hasan Basri Celebi,
Student Member, IEEE,
Antonios Pitarokoilis,
Member, IEEE, and Mikael Skoglund,
Fellow, IEEE
Abstract —Stringent constraints on both reliability and latencymust be guaranteed in ultra-reliable low-latency communication(URLLC). To fulfill these constraints with computationally con-strained receivers, such as low-budget IoT receivers, optimaltransmission parameters need to be studied in detail. In thispaper, we introduce a multi-objective optimization frameworkfor the optimal design of URLLC in the presence of decod-ing complexity constraints. We consider transmission of short-blocklength codewords that are encoded with linear block en-coders, transmitted over a binary-input AWGN channel, andfinally decoded with order-statistics (OS) decoder. We investigatethe optimal selection of a transmission rate and power pair,while satisfying the constraints. For this purpose, a multi-objective optimization problem (MOOP) is formulated. Basedon the empirical model that accurately quantifies the trade-offbetween the performance of an OS decoder and its computationalcomplexity, the MOOP is solved and the Pareto boundary isderived. In order to assess the overall performance among severalPareto-optimal transmission pairs, two scalarization methods areinvestigated. To exemplify the importance of the MOOP, a casestudy on a battery-powered communication system is provided.It is shown that, compared to the classical fixed rate-powertransmissions, the MOOP provides the optimum usage of thebattery and increases the energy efficiency of the communicationsystem while maintaining the constraints.
Index Terms —URLLC, low-complexity receivers, channel cod-ing, internet-of-things, order statistics decoder, multi-objectiveoptimization.
I. I
NTRODUCTION
The advent of 5G communication technologies will providemore than mobile broadband data transmission by supportingmassive number of machine-type connections and allowingcommunication with stringent latency and reliability con-straints. With these two new communication frameworks,namely the massive machine-type communication (MTC) andultra-reliable low-latency communication (URLLC), 5G willnot only improve the human-centric connection, but also createa new interconnection network for devices, objects, sensors,such as the Internet of Things (IoT). However, compared tothe previous generations, this new network will require signifi-cantly different architecture and operation modes. For instance,the target end-to-end radio latency limit for 4G networks is ms [1]. However, this requirement is further reduced to ms Hasan Basri Celebi and Mikael Skoglund are with the School of ElectricalEngineering and Computer Science, KTH Royal Institute of Technology,Stockholm, Sweden (e-mail: [email protected], [email protected]).Antonios Pitarokoilis is with Ericsson AB, Stockholm, Sweden (e-mail:[email protected]).This work was funded in part by the Swedish Foundation for StrategicResearch (SSF) under grant agreement RIT15-0091. for URLLC applications with very high reliability constraints,i.e. maximum error probability of − [2]. Furthermore, MTCrelated URLLC scenarios are among the most challengingsince in real time applications without human intervention,e.g., industrial control, extremely stringent requirements onlatency, in the order of a fraction of ms, and reliability, errorprobability less that − , must be met. [3]–[5].Although the capacity of a channel is often taken as a metricfor reliable communication, it is not a suitable metric forlatency-intolerant applications, since it gives the ultimate error-free transmission rate when the blocklength of the transmittedcodeword goes to infinity [6]. Therefore, the outage capacity,which represents the maximal transmission rate such thatthe probability of the instantaneous mutual information beinglower than the selected rate is not higher than some desiredthreshold, is sometimes studied to evaluate the performance ofa latency constrained communication [7]. However, the outagecapacity is more accurate for arbitrarily long blocklengths,which makes it suitable for latency-tolerant applications. Nev-ertheless, there has been a significant amount of progressmade in the area of non-asyptotic bounds in finite blocklengthregime. Non-asymptotical achievability and converse boundsare derived in [8], where a close approximation on the maximalachievable rate is also introduced. This approximation revealsthat a desired error probability in finite blocklength can beachieved by introducing some amount of rate penalty from theasymptotic bound, where the amount of the penalty is relatedto the channel dispersion and blocklength. A. Motivation
The bounds in finite blocklength regime show the theoreticallimits. However, achieving these limits is still a challenge.One significant determinant of this problem is the selectionof a proper channel encoder and decoder pair which canachieve performance levels close to the theoretical limits.Therefore, significant amount of work has been publishedin the recent years to investigate the performances of var-ious coding schemes in finite blocklength regime to findthe optimum selection for URLLC applications. Mainly, alist of some strong candidates for URLLC follows [9], [10]( i ) Bose–Chaudhuri–Hocquenghem (BCH) codes with order-statistics (OS) decoder [11], ( ii ) binary or non-binary low-density parity-check (LDPC) codes with OS decoder or belief-propagation algorithm [12], ( iii ) tail-biting convolutional codes(TBCC) with list or wrap-around Viterbi algorithms [13],and ( iv ) polar codes with cyclic-redundancy-check aided successive-list decoding [14]. Empirical performance resultsfor these coding schemes have been investigated and comparedin [5], [10], [15]. It is shown that linear block codes withOS decoder and TBCC with Viterbi algorithm outperform andtheir performance approach to the non-asymptotic limits. Dueto their performance in finite blocklength regime, these twocoding schemes have gained interest of the research commu-nity. Bounds on the performance of OS decoders are derivedin [16] and a novel low-complexity algorithm is proposed. Per-formance of TBCC is studied in [13] for URLLC applicationsand lists of generator polynomials leading to large minimumdistance convolutional codes are presented. On the other hand,structural delays, which is a significant factor for low-latencyapplications, for linear block codes and convolutional codesare studied in [17]. It is shown that convolutional codes haveconsiderable advantages in terms of lower structural delays.Possible implementation opportunities of polar codes in 5Gapplications are discussed in [18]. Furthermore, research ondeep-learning based coding schemes for short blocklengthsis also attracting significant interest as it is shown in [19],[20] that it is possible to create deep-learning based codingschemes for URLLC applications. Additionally, learning baseddecoder complexity reductions for convolutional codes and OSdecoders are discussed in [21] and [22], respectively.In general, performance of a decoder comes with an un-avoidable cost: its computational complexity. The trade-offbetween the performance and complexity yields many studiesto mainly focus on decreasing the computational cost of adecoder. Such trade-off, for instance, can be observed in OSdecoders and TBCCs by changing their order and memorysizes, respectively. It is shown that such decoders with higherorder/memory perform better. Besides, their complexity, interms of number of binary operations per-information-bit,exponentially increases. This trade-off has been investigated in[23] extensively and it is shown that decoding complexity hasa considerable effect in URLLC applications when processingcapabilities are taken into account. In this work, we fur-ther extent [23] and formulate a multi-objective optimizationframework for URLLC applications with the goal of max-imizing the throughput, in terms of transmitted informationrate, and energy efficiency of the system together. Such agoal is crucial for the design of URLLC IoT use cases wherereliable information transmission is required under latencyconstraint with an energy efficient communication protocols,since very long battery lifetimes are required [24]. Thus, weidentify the optimal selections of communication parameterswhere constraints on both latency and reliability are met underdecoding complexity constraint. Some of the similar worksin the literature, where the optimal parameters for URLLCare studied, can be listed as follows. Optimum rate andpower allocation for URLLC are investigated in [25]. Thiswork has been extended to non-orthogonal multiple accessnetworks to achieve a distributed rate control system designwith reliability constraints [26]. The trade-off between energyefficiency and reliability is studied in [27] for low-power short-rage communication, where a multi-objective optimizationproblem (MOOP) has been formulated and achievability ofthe optimal parameters are studied. Similarly, optimum power allocations under reliability constraints for hybrid automaticrepeat request schemes are presented in [28]. However, noconstraint on decoding complexity has been considered inthese papers. B. Contributions
This work differs considerably from the studies listed aboveas we take into account the decoding complexity as a systemdesign parameter for URLLC applications. Such an additionalparameter is relevant for most of the real-world applicationsand it is mostly the case for the URLLC IoT use cases [29]–[31]. The main contributions of this work can be mainlysummarized as • We extend our analysis in [23] where the trade-offbetween the performance of several decoders, that arepossible candidates of URLLC, and their computationalcomplexity is modeled. With the help of this model,in this paper, we show that under decoding complexityconstraint on OS decoders, a back-off from the finitemaximal achievable rate is required in order to meet thelatency and reliability constraints. We quantify this back-off both in power and rate domains. Then, based on thisback-off, the optimum transmission parameter choices arediscussed thoroughly via a MOOP framework. • Next, a MOOP for the optimum rate and power selec-tion under latency, reliability, and decoding complexityconstraints is formulated. The MOOP is then analyticallysolved and the attainable objective set and the set of opti-mal selections, namely the Pareto boundary, are derived. • Two methods for scalarization of the MOOP, namely( i ) linear weighted-sum and ( ii ) weighted Chebyshevobjective functions, are introduced and their performancesare compared. It is subsequently shown that based on theshape of the attainable objective set, some optimal pointsare not accessible, depending on the selected scalarizationmethod. This phenomena is discussed thoroughly and weanalytically derive the regions where the optimum pointsare not accessible. • Finally, the importance of the MOOP is exhibited witha case study on battery-powered communication. It isshown that the MOOP increases the energy efficiencyof the communication system while maintaining theconstraints, compared to the classical fixed parametertransmissions. We also show that weighted Chebyshevobjective function performs better than linear weighted-sum function in terms of energy efficiency.Next, we discuss the system model and formulate theproblem in Section II. The trade-off between excess powerand decoding complexity is investigated for OS decoders inSection III. Then, in Section IV, we formulate a MOOP andsolve it analytically. Two main scalarization techniques forthe MOOP are introduced in Section V. Finally, the benefitsof the MOOP is shown in a case study, where performance ofa battery-powered communication is presented.
II. P
ROBLEM D EFINITION
A. System Model
We consider communication over a discrete-time, binary-input AWGN (BI-AWGN) channel. 𝑘 number of informationbits are encoded with a linear block encoder and modulatedto form the 𝑛 symbol codeword 𝒙 = [ 𝑥 , 𝑥 , . . . , 𝑥 𝑛 ] , 𝑥 𝑖 ∈ {− , + } , (1)The codeword 𝒙 is then transmitted over BI-AWGN channeland the ratio 𝑟 = 𝑘 / 𝑛 is the code rate of the selected codebook.The observed sequence at the receiver is 𝒚 = √ 𝜌 𝒙 + 𝒛 , (2)where 𝜌 denotes the signal-to-noise ratio (SNR) and 𝒛 ∼N ( , 𝑰 𝑛 ) .We consider a communication scenario where data trans-mission is performed under latency, reliability, and decodingcomplexity constraints. The transmitter and receiver select atransmission pair, denoted as { 𝑟, 𝜌 } , according to the con-straints. The receiver captures 𝒚 and selects a decoder thatfulfills the constraints. The goal of this study is to investigatethe optimum choice of transmission parameters that guaranteesto fulfill the constraints and maximize the efficiency of thecommunication system in terms of the objective function,which will be discussed in the following Sections.Information theoretic analysis tells that there exists a max-imal limit on transmission rate where reliable communicationis possible, such that the codeword error probability (CEP)vanishes as 𝑛 → ∞ . This limit is termed as the channel capac-ity and denoted as 𝐶 . However, if the communication setup isrestricted to have arbitrarily large values of 𝑛 , such restrictionsare common for URLLC applications with stringent latencyrequirements, 𝐶 overestimates the rate of reliable informationand is not achievable. In this case the maximal transmissionrate with codewords of length 𝑛 with non-zero CEP 𝜀 can beclosely approximated by 𝑅 ( 𝑛, 𝜌, 𝜀 ) = 𝐶 − r 𝑉𝑛 𝑄 − ( 𝜀 ) log 𝑒 + 𝑂 (cid:18) log 𝑛𝑛 (cid:19) . (3)where the quantity 𝑉 is called the channel dispersion and 𝑄 − (·) is the inverse of the Gaussian 𝑄 − function. Theexpression in (3) is termed as the normal approximation tothe maximal transmission rate in finite blocklength regime.For BI-AWGN channel, 𝐶 and 𝑉 are 𝐶 = √ 𝜋 ∫ 𝑒 − 𝑧 (cid:16) − log (cid:16) + 𝑒 − 𝜌 + 𝑧 √ 𝜌 (cid:17) (cid:17) d 𝑧, (4) 𝑉 = √ 𝜋 ∫ 𝑒 − 𝑧 (cid:16) − log (cid:16) + 𝑒 − 𝜌 + 𝑧 √ 𝜌 (cid:17) − 𝐶 (cid:17) d 𝑧. (5)The second term in (3) introduces a back-off from 𝐶 toensure the transmission achieves 𝜀 CEP with 𝑛 blocklength.Hence, in order to meet the constraint on reliability, one candeduce that 𝑟 ≤ 𝑅 ( 𝑛, 𝜌, 𝜀 ) . However, this constraint is notenough to ensure that the latency constraint is fulfilled alreadyunder decoding complexity constraint, since, as will be shownthroughout the paper, the computational complexity of thedecoder exponentially increases as 𝑟 approaches to 𝑅 ( 𝑛, 𝜌, 𝜀 ) . All logarithms in this paper are with base 2.
B. Aggregate Latency and Decoding Complexity
For notational consistency, given a fixed codebook con-taining 𝑘 codewords of length 𝑛 , we denote a decoder as 𝐷 ( 𝑛, 𝑟, 𝜌 ) , where it is meant that the decoder operates onthe given codebook at a received SNR, 𝜌 , and decodes 𝑘 number of information bits. The performance of the decoderis measured by its CEP 𝜀 , denoted as 𝐸 ( 𝐷 ) = 𝜀 .It is often encountered in the literature of latency-constrained communication that the aggregate latency equalsto the time required for the transmission of a message overthe communication channel. This implies that other operationsnecessary for the successful delivery of the information, e.g.,the time required for the decoding, are assumed to happeninstantaneously. However, this assumption is not true for re-ceivers with computational complexity constraints. Therefore,we consider the aggregate latency 𝐿 𝑎 as 𝐿 𝑎 = 𝑛𝑇 𝑠 + 𝐿 𝑑 + 𝐿 𝑞 , (6)with 𝐿 𝑑 denoting the decoding latency and 𝐿 𝑞 representingthe signal propagation and other auxiliary processes such assignal processing and consequent decision making tasks. Since 𝐿 𝑞 does not depend on encoding and decoding processes, weneglect it from further analysis.For simplicity and generality, a linear relation between 𝐿 𝑑 and the total number of binary operations of the decodingprocess is assumed [23], [32]. Denoting 𝑇 𝑏 as the latencycaused by a single binary operation at the receiver, the totaltransmission and decoding latency for the transmission of acodeword of blocklength 𝑛 can be written as 𝐿 𝑎 = 𝑛𝑇 𝑠 + 𝑘𝑇 𝑏 𝐾 ( 𝐷 ) , (7)where 𝐾 ( 𝐷 ) represents the number of binary operationsper-information-bit that is required to decode 𝑘 number ofinformation bits from an 𝑛 -length noisy codeword. Notice that 𝑘𝐾 ( 𝐷 ) gives the total number of binary operations requiredfor decoding. Next, in order to address the relation of 𝐾 ( 𝐷 ) with othercommunication parameters, we define some easily verifiableproperties of the considered codes by comparing the relativeperformance of two decoders from the same decoder familyoperating on the same codebook or on the sub-codebooks. Property 1.
Let two decoders, 𝐷 ( 𝑛, 𝑟, 𝜌 ) and 𝐷 ( 𝑛, 𝑟, 𝜌 ) ,operating on the same codebook with 𝐾 ( 𝐷 ) ≤ 𝐾 ( 𝐷 ) .It follows immediately from the decoder complexities that 𝐸 ( 𝐷 ) ≥ 𝐸 ( 𝐷 ) . Intuitively, more complex decoder leads tolower CEP. Property 2.
Let two decoders, 𝐷 ( 𝑛, 𝑟, 𝜌 ) and 𝐷 ( 𝑛, 𝑟, 𝜌 ) ,operating on the same codebook with same complexity, 𝐾 ( 𝐷 ) = 𝐾 ( 𝐷 ) , but different SNR levels such that 𝜌 ≤ 𝜌 .Then, it must be true that 𝐸 ( 𝐷 ) ≥ 𝐸 ( 𝐷 ) , since higher SNRleads to lower CEP. We assume that decoding starts right after the codework transmission. A more accurate estimation on 𝐿 𝑑 can be done by investigating softwareoptimization capabilities, memory timings, parallel computation, etc. But sincethese are not in the scope of this paper, we confine (7) for further analysis.Interested readers may refer to [33] Property 3.
Take the following two decoders, 𝐷 ( 𝑛, 𝑟 , 𝜌 ) and 𝐷 ( 𝑛, 𝑟 , 𝜌 ) . Assume that 𝐷 ( 𝑛, 𝑟 , 𝜌 ) is operating over a sub-codebook of 𝐷 ( 𝑛, 𝑟 , 𝜌 ) , where 𝑟 ≤ 𝑟 , at the same SNRand complexity, i.e. 𝐾 ( 𝐷 ) = 𝐾 ( 𝐷 ) . It is true that due to thesize of the sub-codebook, 𝐸 ( 𝐷 ) ≤ 𝐸 ( 𝐷 ) . Intuitively, moreinformation leads to higher CEP. Property 4.
For the two decoders stated in Property 3, 𝐷 ( 𝑛, 𝑟 , 𝜌 ) and 𝐷 ( 𝑛, 𝑟 , 𝜌 ) , where 𝑟 ≤ 𝑟 , 𝐸 ( 𝐷 ) = 𝐸 ( 𝐷 ) can be achieved when 𝐾 ( 𝐷 ) ≤ 𝐾 ( 𝐷 ) . Thus, more informa-tion leads to higher complexity.C. Power Gap and Rate Gap Suppose that, given 𝑛 and 𝜀 , based on the SNR 𝜌 , thetransmitter and receiver agree on a transmission rate that meetsthe CEP constraint. We denote this rate-power transmissionpair as { 𝑟, 𝜌 } . The transmission efficiency, in terms of increas-ing information transmission per channel use, is maximizedby selecting the codebook that can achieve 𝑟 = 𝑅 ( 𝑛, 𝜌, 𝜀 ) .However, as shown in [10], [15], [23], this selection can leadto a very complex decoder which is not practical for receivershaving complexity constraints since it may violate the latencyconstraint or it will make the total latency too large. Forinstance, the optimum maximum likelihood decoder, whichrequires an exhaustive search over the codebook, impliesexponential complexity in 𝑘 which is not practical even forshort blocklengths.Suppose that the total latency 𝐿 𝑎 is bounded by a maximumallowed latency 𝐿 𝑚 as 𝐿 𝑎 ≤ 𝐿 𝑚 , (8)where 𝐿 𝑚 represents the maximum allowed latency. Thisconstraint imposes an upper bound on the per-information-bitdecoder complexity such that 𝐾 ( 𝐷 ) ≤ 𝜅, (9)where 𝜅 = ( 𝑘𝑇 𝑏 ) − [ 𝐿 𝑚 − 𝑛𝑇 𝑠 ] + (10)and [ 𝑧 ] + = max { , 𝑧 } . However, an upper bound on 𝐾 ( 𝐷 ) for the complexity of the decoder for fixed 𝑛 , 𝑟 and 𝜌 wouldinevitably lead to reduced reliability, i.e., increased CEP. Oneway to satisfy a desired target reliability is to spend someamount of excess power, named as the power penalty or tointroduce some amount of rate back-off. Hence, an interesting,yet complex, relation between power, rate, aggregate latency,and decoding complexity arises.Next, we introduce two definitions that will be used infurther analysis. Definition (Power penalty, Δ 𝜌 ) . Fix a codebook of 𝑘 code-words of blocklength 𝑛 . For a reference SNR, 𝜌 𝑠 , consider theML decoder that achieves a CEP of 𝜀 and the suboptimaldecoder 𝐷 ( 𝑛, 𝑟, 𝜌 ) that achieves 𝐸 ( 𝐷 ) = 𝜀 at SNR 𝜌 . Thedifference between 𝜌 and 𝜌 𝑠 is the power penalty required,such that the suboptimal decoder can achieve the same CEPas the ML decoder. Definition (Rate gap, Δ 𝑟 ) . Consider the ML decoder, oper-ating on a codebook with 𝑘 codewords of blocklenth 𝑛 ata coderate 𝑟 , achieves 𝜀 error rate at the reference SNR 𝜌 𝑠 . It is true that one can further decrease the error rate 𝜀 to 𝜀 ′ , where 𝜀 ′ < 𝜀 , by omitting sufficient amount ofcodewords from the codebook and let the ML decoder tooperate over a sub-codebook. Although this operation reducesthe coderate to 𝑟 ′ , where 𝑟 ′ < 𝑟 , it also gives the flexibilityof selecting a suboptimal decoder, 𝐷 ( 𝑛, 𝑟 ′ , 𝜌 𝑠 ) with 𝐸 ( 𝐷 ) = 𝜀 but substantially lower decoding complexity that can decode 𝑘 ′ = 𝑛𝑟 ′ number of information bits at 𝜌 𝑠 SNR with 𝜀 errorrate. Thus, the difference between 𝑟 and 𝑟 ′ is the rate gap,such that the suboptimal decoder can achieve the same CEPwith the same SNR as the ML decoder by sacrificing someamount of coderate.D. Problem Formulation Several optimization problems for URLLC with decodingconstraints have been introduced and solved in [23]. Themain logic in all those optimization problems is minimizingor maximizing a single cost function of interest subject toa set of constraints. However, in real life implementationsof URLLC, various parameters are supposed to be optimizedtogether. Here, we take our analysis in [23] one step furtherand set the optimal design of URLLC systems in a MOOPframework.A general structure of a MOOP followsminimize 𝑥 𝑓 ( 𝑥 ) , 𝑓 ( 𝑥 ) , · · · 𝑓 𝑚 ( 𝑥 ) (11a)s.t. 𝑥 ∈ 𝑋. (11b)Similar to (11), possible MOOPs for URLLC can be for-mulated by a valid combination of the following constraints,such as ( i ) minimization of 𝐿 𝑎 , ( ii ) minimization of 𝜀 , ( iii )minimization of 𝜌 , ( iv ) minimization of 𝐾 ( 𝐷 ) , and ( v ) max-imization of 𝑟 , where the optimization performs subject toconstraints on the remaining parameters. In this paper, wemainly focus on minimization of 𝜌 and maximization of 𝑟 subject to constraints on latency, reliability, and decodingcomplexity.Suppose an 𝑛 -blocklength codeword that belongs to acodebook of size 𝑛𝑟 𝑠 , where 𝑟 𝑠 stands for the referencetransmission rate, is intended to be transmitted at a CEP 𝜀 𝑚 .If no constraints on latency and decoding complexity presentor if 𝑇 𝑏 = s, which stands for infinite computation power,the reference transmission rate-power is { 𝑟 𝑠 , 𝜌 𝑠 } , where 𝜌 𝑠 = 𝑅 − ( 𝑛, 𝑟 𝑠 , 𝜀 𝑚 ) (12)represents the reference SNR. However, constraints may pre-vent to achieve { 𝑟 𝑠 , 𝜌 𝑠 } . Suppose, for instance, a selectedtransmission rate-power pair that meets the constraints is { 𝑟 𝑠 − Δ 𝑟, 𝜌 𝑠 + Δ 𝜌 } . (13)This selection arises a very significant optimization problem,being “ Given a blocklength 𝑛 and reference transmission rate-power pair, { 𝑟 𝑠 , 𝜌 𝑠 } , under latency, reliability, and decodingcomplexity constraints, what is the optimum selection of power penalty and rate gap that allows the transmission to satisfy theconstraints under some optimality criterion? ”. This MOOPcan be formulated as the followingminimize 𝑘,𝜀 Δ 𝑟 and Δ 𝜌 (14a)s.t. 𝐸 ( 𝐷 ) ≤ 𝜀 𝑚 , (14b) 𝐾 ( 𝐷 ) ≤ 𝜅, (14c) ≤ Δ 𝜌, (14d) ≤ Δ 𝑟 ≤ 𝑟 𝑠 , (14e) 𝑘 ≤ 𝑛, (14f)where (14b) and (14c) represent the reliability and latencyconstraints, respectively. (14d) and (14e) are the numericalconstraints on the variables. Since the hardware platform isassumed to be fixed in (14), 𝑇 𝑏 and 𝑇 𝑠 are fixed. Noticethat although selecting Δ 𝜌 = and Δ 𝑟 = is theoreticallyachievable, it requires very complex decoder to achieve thedesired CEP. On the other hand, selecting Δ 𝜌 > and Δ 𝑟 > yields a reduction in decoder complexity, but it will also causeperformance degradation in the transmission efficiency sincethe rate-power pair is receding from the bound.III. P OWER P ENALTY VS D ECODING C OMPLEXITY
In order to address the MOOP in (14), the relation betweenthe power gap versus the decoding complexity must be studiedthoroughly. From now on, we focus our analysis on linearblock codes with OS decoders, due to the following reasons; i ) it has been shown that these codes come very close tothe information-theoretic bounds for finite 𝑛 [9], [15], ii ) OSdecoders are easy to describe with a few parameters., iii )their decoding performance can be easily parameterized bya single parameter, i.e., the order of the decoder, 𝑠 ∈ Q , andfinally, iv ) operations that are executed during decoding canbe accurately tracked and the decoding complexity can beintuitively described. A. OS Decoders
OS decoders are universal soft-decision decoder for linearblock decoders and they are efficient in reducing the complex-ity of the decoding process by making soft decisions on a setof test codewords. Implementation of the OS decoders can besummarized as: • Find a permutation function 𝜆 ( 𝒚 ) that sorts the receivedvector, 𝒚 , with respect to the absolute amplitude values. • Using the permutation function 𝜆 (·) , reorder the columnsof the generator matrix, 𝑮 , and apply the Gauss-Jordanelimination to form the new systematic generator matrix 𝑮 𝜆 . • Set the list of test error patterns (TEPs), denoted by 𝑇 , such that it includes all possible length − 𝑘 binarysequences with Hamming distance less than or equal to 𝑠 and search over the list to find the error sequence thatmaximizes the likelihood of the codeword to the harddecoded 𝜆 ( 𝒚 ) . Overall, the decoding can be formulated as 𝒓 𝑠 = arg max { 𝒓 : 𝒓 = ( ℎ ( 𝜆 ( 𝒚 )) ⊕ 𝒕 ) ⊗ G 𝜆 , 𝒕 ∈ 𝑇 } P ( 𝒓 | ℎ ( 𝜆 ( 𝒚 )) , (15) = arg min { 𝒓 : 𝒓 = ( ℎ ( 𝜆 ( 𝒚 )) ⊕ 𝒕 ) ⊗ G 𝜆 , 𝒕 ∈ 𝑇 } k 𝒓 − ℎ ( 𝜆 ( 𝒚 )) k , (16)where ℎ (·) represents the hard-decoding function. The outputof the order − 𝑠 decoder is ˆ 𝒓 𝑠 = 𝜆 − ( 𝒓 𝑠 ) . B. Computational Complexity
The search space for the most probable codeword is con-trolled by limiting the size of the list 𝑇 , which is related withthe choice of the order 𝑠 . The total number of possible binaryerror vectors in TEP for an order − 𝑠 OS decoder, when 𝑠 isinteger, is | 𝑇 | = 𝑠 Õ 𝑖 = (cid:18) 𝑘𝑖 (cid:19) . (17)Therefore, the cardinality of 𝑇 grows exponentially in 𝑠 . Noticethat all possible codeword comparisons will be performedwhen 𝑠 = 𝑘 and the performance and complexity of the OSdecoder will be identical to the ML decoder. The number ofbinary operations per-information-bit of an OS decoder can becalculated by [23] 𝐾 ( 𝐷 ) = log ( 𝑛 ) 𝑟 + 𝑛𝑘 + | 𝑇 | (cid:16) 𝑛 − 𝑞 + 𝑞𝑛𝑘 (cid:17) , (18)where 𝑞 represents the number of quantization bits used torepresent the symbols of 𝒚 a real number. The terms in(18) represent the binary complexity values of sorting 𝒚 ,Gauss-Jordan elimination of the permuted 𝑮 , and order − 𝑠 reprocessing, respectively. Notice that the limiting complexityorder of OS decoder is O( 𝑛𝑘 𝑠 ) .Next, we validate the properties listed in Sec. II-B byanalyzing the empirical performance of OS decoders withvarious orders and transmission rates. These properties canbe exemplified in Fig. 1 where performance results of OSdecoders with orders 𝑠 = { , , } where 𝑛 = and 𝑘 = { , } are depicted. The extended BCH code [34] isused for the encoding and the error bound is derived from(3). Property 1 and 2 can be seen by fixing 𝑘 and lettingthe order − 𝑠 to vary. It is seen that as the order increases,the performance of the decoder improves and approaches theoptimal decoder, albeit at the expense of higher decodingcomplexity. Similar results are also achieved by fixing 𝑠 andlet SNR vary. Property 3 can be illustrated by fixing an orderand letting 𝑘 to vary. It can be seen that for fixed SNR, thedecoder with 𝑘 = has worse performance than the decoderwith 𝑘 = and, from (18), it also has higher per-information-bit complexity. C. Modeling the Power Penalty
One can obtain Δ 𝜌 by simply calculating the differencebetween 𝜌 ′ and 𝜌 𝑠 , where 𝜌 ′ is the the minimum SNR thatis required for a particular OS decoder to achieve the targetCEP, 𝜀 𝑚 . Thus, 𝜌 ′ can be computed by solving the followingoptimization problem 𝜌 ′ = min { 𝜌 ∈ R + , 𝐸 ( 𝐷 ) ≤ 𝜀 𝑚 } 𝜌. (19) -6 -5 -4 -3 -2 -1 Fig. 1: CEP performace of OS decoders with 𝑛 = and 𝑘 = (black) and 𝑘 = (blue) number of information bitsat different orders compared to the the normal approximationCEP bound for BI-AWGN channel.Although tight upper bounds on 𝜌 ′ have been derived in [11]and [35], computation of Δ 𝜌 is not trivial for OS decoderssince no closed-form expression of 𝜌 ′ is available.On the other hand, it is shown in [10], [15], [30], [36],[37] that for several types of coding schemes, such as linearblock codes, polar codes, convolutional codes, etc., the relationbetween computational complexity and power penalty for fixed 𝑛 subject to reliability constraint in the short block-lengthregime follows an exponential relation, where the decodingcomplexity exponentially increases as the code approaches themaximal achievable bound in (3). This behavior of computa-tional complexity of the OS decoder and its power gap hasbeen modeled in [30] and [23] by a law of the type 𝐹 ( Δ 𝜌 ) = (cid:16) 𝑎 p Δ 𝜌 + 𝑏 (cid:17) − , (20)with appropriate choices of the constants 𝑎 > and 𝑏 > ,where 𝐹 ( Δ 𝜌 ) gives a close approximation of the logarithm ofper-information-bit complexity of OS decoder that achieves thedesired CEP. The main advantage of (20) is that it describesa tractable way of the trade-off between decoding complexityand power penalty for practical OS decoders with linear blockcodes. With the help of this model, given Δ 𝜌 , it is possible toget a close approximation of the minimum per-information-bitcomplexity of the OS decoder that meets the desired CEP. Arealization of 𝐹 ( Δ 𝜌 ) is depicted in Fig. 2, where logarithmof per-information-bit complexities of the two OS decoderswith orders 𝑠 = { , , , , , } are shown with respect to theirpower gap values. It can be seen that the proposed model with 𝑎 = . and 𝑏 = . can closely describe the trade-off.Next, we show the maximal achievable information ratefor a complexity constrained receiver with OS decoder underlatency and reliability constraints. Lemma 1.
For a complexity constrained receiver with OSdecoder and aggregate latency, expressed in (7) , the maximal
Fig. 2: Power penalty values of OS decoders at different ordersversus their complexities for 𝑘 = (black markers) and 𝑘 = (blue markers), where 𝑛 = and 𝜀 = − . achievable information rate subject to latency, 𝐿 𝑎 < 𝐿 𝑚 , andreliability constraints can be expressed as 𝑀 ( 𝑛, 𝜌, 𝜀 ) = 𝑅 ( 𝑛, 𝜌 − Δ 𝜌 min , 𝜀 ) , (21) where 𝑀 ( 𝑛, 𝜌, 𝜀 ) represents the maximal information rateunder latency and complexity constraints and Δ 𝜌 min is theminimum amount of excess power that is required to fulfill theconstraints and can be computed as Δ 𝜌 min = (cid:16) ( 𝑎 log 𝜅 ) − [ − 𝑏 log 𝜅 ] + (cid:17) . (22) Proof.
Lemma 1 can be proven in accordance with Lemma 1and 2 in [23]. For the purpose of completeness of this paper,we repeat the proofs. For fixed rate and blocklength 𝑛 themaximum allowable decoding time can be calculated using(7). This in turn yields an upper bound on 𝐾 ( 𝐷 ) as given in(9). Bounding the complexity restricts the order − 𝑠 as follows 𝑠 ≤ 𝑠 𝑚 = arg max { 𝑠 | 𝑠 ∈ Q + , 𝐿 𝑎 ≤ 𝐿 𝑚 } 𝐾 ( 𝐷 ) , (23)where 𝑠 𝑚 represents the maximum allowed decoder order − 𝑠 that satisfies the decoding complexity. Although this restrictioncan be used to control the latency of decoding duration,the expense would be the reduced reliability, which can besatisfied by paying some amount of excess power. Thus, using(20) and (9) the minimum amount of required power penaltycan be computed as given in (22). Finally, 𝑀 ( 𝑛, 𝜌, 𝜀 ) can bedetermined by calculating Δ 𝜌 min for 𝑟 ∈ [ , ] and shifting 𝑅 ( 𝑛, 𝜌, 𝜀 ) by Δ 𝜌 min to the right. (cid:3) Remark 1.
Notice that Δ 𝜌 min is proportional to the pro-cessor capabilities, the latency requirements, blocklength 𝑛 ,and coderate 𝑟 . For fixed 𝑛 and 𝑇 𝑠 , as 𝑇 𝑏 decreases, i.e.more powerful processor is implemented at the receiver, Δ 𝜌 min decreases and the gap to the normal approximation shrinksand disappears when 𝑇 𝑏 ≤ 𝑘 / 𝑏 [ 𝐿 𝑚 − 𝑛𝑇 𝑠 ] + . (24) On the other hand, for fixed 𝑛 , if the transmission rate 𝑟 increases, Δ 𝜌 min also increases and the gap to the normalapproximation widens. Lemma 2.
Lemma 1 can also be introduced in terms of rategap such as 𝑀 ( 𝑛, 𝜌, 𝜀 ) = 𝑅 ( 𝑛, 𝜌, 𝜀 ) − Δ 𝑟 min , (25) where Δ 𝑟 min represents the minimum amount of rate penaltythat is required to be paid to guarantee the desired CEP forfixed 𝑛 and 𝜌 under the latency, complexity, and reliabilityconstraints. Remark 2. (20) is in accordance with the Properties listedin Sec. II-B. For instance, using the statements in Properties1 and 2, it is true that in order to achieve a desired CEP, it ispossible to use a decoder with lower computational complexityas Δ 𝜌 increases. Similar results for Δ 𝑟 can also be deducedusing Properties 3 and 4. IV. O
PTIMAL R ATE -P OWER S ELECTION
The MOOP can now be reformulated for the OS decoderwith the following additional constraintsminimize 𝑘,𝜀,𝑠 Δ 𝑟 and Δ 𝜌 (26a)s.t. (14b) , (14c) , (14f) , (26b) ≤ 𝑠 ≤ 𝑘, (26c) 𝑟 𝑚 ≤ 𝑟 𝑠 − Δ 𝑟 ≤ 𝑟 𝑠 , (26d) 𝜌 𝑠 ≤ 𝜌 𝑠 + Δ 𝜌 ≤ 𝜌 𝑚 . (26e)where (26c) is the constraint on the order − 𝑠 . Additionally,we also introduce two constraints on rate and power in (26d)and (26e), which are due to the minimum accepted trans-mission rate and the maximum short-term power budget ofthe communication system, respectively. (26) is a challengingoptimization problem due to the nonlinear relations betweenlatency, reliability and power requirements. An exhaustivesearch over OS decoders with various orders at different ratesis on the other hand a very complicated and inefficient solutionto the problem. In this section, we investigate (26) and showthe optimum solution. We first start with the following Lemma. Lemma 3.
The optimum is achieved with equality in (14b) .Proof.
We prove the lemma by using a similar analogy from[23, Lemma 6]. Without loss of generality let us first assumethat Δ 𝑟 is selected to be the optimum, Δ 𝑟 = Δ 𝑟 ∗ , and wefocus on the minimization of Δ 𝜌 . For fixed 𝜌 𝑠 , given that 𝑅 ( 𝑛, 𝜌 𝑠 , 𝜀 ) ≤ 𝑅 ( 𝑛, 𝜌 𝑠 , 𝜀 𝑚 ) , the feasible set for 𝑛 becomes thelargest for 𝜀 = 𝜀 𝑚 . Also assume that the optimal decoderis 𝐷 ∗ ( 𝑛, 𝑟 𝑠 − Δ 𝑟 ∗ , 𝜌 𝑠 + Δ 𝜌 ∗ ) with 𝐸 ( 𝐷 ∗ ) = 𝜀 ∗ < 𝜀 𝑚 .However, for some Δ 𝜌 ∗ ≥ 𝜎 > small enough, one canfind a decoder 𝐷 ′ ( 𝑛, 𝑟 𝑠 − Δ 𝑟 ∗ , 𝜌 𝑠 + Δ 𝜌 ∗ − 𝜎 ) that can achieve 𝐸 ( 𝐷 ′ ) = 𝜀 𝑚 which requires lower SNR than the optimal onewithout violating the CEP constraint, which contradicts withthe assumption.Similarly, now, we assume that Δ 𝜌 is optimum, Δ 𝜌 = Δ 𝜌 ∗ ,and the search is over Δ 𝑟 . Suppose that the optimal decoderis 𝐷 ∗ ( 𝑛, 𝑟 𝑠 − Δ 𝑟 ∗ , 𝜌 𝑠 + Δ 𝜌 ∗ ) with 𝐸 ( 𝐷 ∗ ) = 𝜀 ∗ < 𝜀 𝑚 . However, for some 𝛼 > small enough, one can find a decoder 𝐷 ′ ( 𝑛, 𝑟 𝑠 − Δ 𝑟 ∗ + 𝛼, 𝜌 𝑠 + Δ 𝜌 ∗ ) with 𝐸 ( 𝐷 ′ ) = 𝜀 𝑚 that has lowerrate gap, i.e. higher number of information bits transmitted,with higher error rate but still does not violate the CEPconstraint, which contradicts with the assumption. Hence, theoptimum is achieved with equality in (14b). (cid:3) Next, we focus on the latency constraint, 𝐿 𝑎 ≤ 𝐿 𝑚 . Recallthat this constraint is written in the form of a complexity boundon 𝐾 ( 𝐷 ) in (9). Using the model in (20), this bound can beconverted to a power penalty constraint using the the modelproposed in (20). Thus, in the light of Lemma 1 and 2, theoptimization problem now reduces tominimize 𝑘,𝑠 Δ 𝑟 and Δ 𝜌 (27a)s.t. 𝑟 𝑚 ≤ 𝑟 𝑠 − Δ 𝑟 ≤ min (cid:8) 𝑀 ( 𝑛, 𝜌 𝑠 + Δ 𝜌, 𝜀 𝑚 ) , 𝑟 𝑠 (cid:9) , (27b) ≤ Δ 𝜌 ≤ 𝜌 𝑚 − 𝜌 𝑠 , (27c) ≤ 𝑠 ≤ 𝑠 𝑚 , and (14f) , (27d)where the minimum in (27b) is due to the fact that, bydefinition, Δ 𝑟 ≥ . Notice that constraints on 𝑘 and 𝑠 canbe omitted for further analysis, since their effect have alreadybeen represented in (27b). Thus, we haveminimize Δ 𝑟 and Δ 𝜌 (28a)s.t. (27b) and (27c) . (28b)From (27), the attainable objective set follows 𝑆 = n { Δ 𝑟, Δ 𝜌 } (cid:12)(cid:12) (cid:2) 𝑟 𝑠 − 𝑀 ( 𝑛, 𝜌 𝑠 + Δ 𝜌, 𝜀 𝑚 ) (cid:3) + ≤ Δ 𝑟 ≤ 𝑟 𝑠 − 𝑟 𝑚 , for all ≤ Δ 𝜌 ≤ 𝜌 𝑚 − 𝜌 𝑠 o . (29)An illustration on the attainable objective set 𝑆 is depictedin Fig. 3, where the reference transmission rate is 𝑟 𝑠 = . and constraints on rate, power, and latency are 𝑟 𝑚 = . and 𝜌 𝑚 = dB, 𝐿 𝑚 = ms, respectively. It can be seen thatthe attainable objective set 𝑆 relies in between the constraintsdefined in (27b), (14d), and (14e). Remark 3.
For a given Δ 𝜌 , that is ≤ Δ 𝜌 ≤ 𝜌 𝑚 − 𝜌 𝑠 , if 𝑀 ( 𝑛, 𝜌 𝑠 + Δ 𝜌, 𝜀 𝑚 ) < 𝑟 𝑚 , no feasible pair can be found. Lemma 4.
The set of optimum solutions of (14) always leadsto transmission pairs that lie on 𝑀 ( 𝑛, 𝜌, 𝜀 𝑚 ) .Proof. Recall that 𝑀 ( 𝑛, 𝜌, 𝜀 𝑚 ) represents the maximal achiev-able limit under latency, realibility, and decoding complexityconstraints. Any transmission rate-power pair that violates 𝑀 ( 𝑛, 𝜌, 𝜀 𝑚 ) , also violates the constraints in (14). Thus, nobetter solution can be achieved above 𝑀 ( 𝑛, 𝜌, 𝜀 𝑚 ) and there-fore the optimum selections of Δ 𝑟 and Δ 𝜌 yield the set oftransmission pairs that lies on it. (cid:3) Lemma 4 shows that (27) is equivalent to the followingminimize Δ 𝑟 and Δ 𝜌 (30a)s.t. 𝑟 𝑠 − Δ 𝑟 = 𝑀 ( 𝑛, 𝜌 𝑠 + Δ 𝜌, 𝜀 𝑚 ) , (30b) min (cid:8) 𝜌 𝑠 − 𝜌 𝑚 , Δ 𝜌 𝑠 min (cid:9) ≥ Δ 𝜌 ≥ , (30c) min (cid:8) 𝑟 𝑠 − 𝑟 𝑚 , Δ 𝑟 𝑠 min (cid:9) ≥ Δ 𝑟 ≥ , (30d) Fig. 3: A numerical realization that shows the attainableobjective set of the MOOP, denoted as 𝑆 , where the referencetransmission rate is 𝑟 𝑠 = . and 𝑟 𝑚 = . , 𝜌 𝑚 = dB. Theultimate transmission point { 𝑟 𝑠 , 𝜌 𝑠 } is shown with the star. Asa comparison, the capacity and the maximum achievable rate,defined in (3), are also depicted with the maximum achiev-able rate subject to latency, reliability, decoding complexityconstraints, where 𝑛 = , 𝜀 𝑚 = − , 𝐿 𝑚 = ms, 𝑇 𝑠 = 𝜇 s,and 𝑇 𝑏 = ns.where Δ 𝜌 𝑠 min and Δ 𝑟 𝑠 min represent the minimum amountof power penalty and rate gap that is required to meet theconstraints at rate 𝑟 𝑠 and SNR 𝜌 𝑠 , respectively. The minimumselections in (30c) and (30d) is due to the maximum limits of Δ 𝑟 and Δ 𝜌 .As an example, let the optimum Δ 𝑟 is selected to be , whichleads to the rate 𝑟 = 𝑟 𝑠 . Assuming that 𝜌 𝑠 − 𝜌 𝑚 ≥ Δ 𝜌 𝑠 min ,setting Δ 𝑟 = only allows a horizontal movement in Fig. 3 andtherefore a certain amount of power penalty needs to be addedin order to meet the constraints. The optimum rate-power pairwould be { 𝑟 𝑠 , 𝜌 𝑠 + Δ 𝜌 𝑠 min } , (31)since 𝑟 𝑠 = 𝑀 ( 𝑛, 𝜌 𝑠 + Δ 𝜌 𝑠 min , 𝜀 𝑚 ) . Similar analysis can be ap-plied if the optimum Δ 𝜌 is . Assuming that 𝑟 𝑠 − 𝑟 𝑚 ≥ Δ 𝑟 𝑠 min ,only a vertical movement is permitted. Since 𝑟 𝑠 − Δ 𝑟 𝑠 min = 𝑀 ( 𝑛, 𝜌 𝑠 , 𝜀 𝑚 ) , the optimum rate-power pair is { 𝑟 𝑠 − Δ 𝑟 𝑠 min , 𝜌 𝑠 } . (32)Next, the Pareto boundary in a MOOP is introduced and thePareto boundary of the investigated problem is characterized. Definition (Pareto boundary) . The set of the optimal objectivesof a MOOP is called the Pareto boundary. Any objectivepair the belongs to the Pareto boundary cannot be objectivelydismissed since none of the objectives can be improved withoutdegrading the others.
Corollary 1.
For the MOOP in (26) , 𝑀 ( 𝑛, 𝜌, 𝜀 𝑚 ) , where 𝜌 𝑠 ≤ 𝜌 ≤ min (cid:8) 𝜌 𝑠 − 𝜌 𝑚 , Δ 𝜌 𝑠 min (cid:9) , represents the Pareto boundary. Fig. 4: Numerical realizations of the Pareto boundaries with 𝑟 𝑠 = . for various 𝑇 𝑏 , where 𝑛 = , 𝜀 𝑚 = − , 𝐿 𝑚 = ms,and 𝑇 𝑠 = 𝜇 s.Hence, any objective pair that belongs to the attainable objec-tive region, 𝑆 , but not on the Pareto boundary is suboptimal,since there exist other operating points that are better or atleast as good for every objective [38]. Numerical realizationsof Pareto boundaries that are achieved with different processorspeeds, i.e. different 𝑇 𝑏 values, are depicted in Fig. 4. Asone can tell the ultimate pair { , } can be achieved when 𝑇 𝑏 ≈ − s. However, as 𝑇 𝑏 increases, i.e. the processorspeed decreases, the boundary moves away from the ultimatepair and a wider Pareto boundary is experienced, which leadsto a broader set of optimal objectives.V. O BJECTIVE S CALARIZATION
Solution to the MOOP leads to a set of Pareto-optimaltransmission pairs. When selecting a single optimum ratherthan a set of optimal points is desired, scalarization techniquesare applied. The purpose of the scalarization is to aggregatethe objectives into a single objective function and reduce theproblem to a constrained single-objective optimization prob-lem [39]. Several scalarization techniques have been presentedin the literature, such as penalty-based intersection method,normal boundary intersection method, and weighted- 𝑙 𝜃 normscalarization method [40]–[42]. In this paper, the focus willbe on weighted- 𝑙 𝜃 norm scalarization technique due to itssimple structure and broad applicability. Mathematically, thistechnique can be applied to (14a) as [40]minimize (cid:2) 𝐴𝛼 ( Δ 𝑟 ) 𝜃 + 𝐵 ( − 𝛼 ) ( log Δ 𝜌 ) 𝜃 (cid:3) 𝜃 , (33)for 𝜃 ≥ and ≤ 𝛼 ≤ , where 𝛼 and ( − 𝛼 ) represent thelinear positive weight of the individual objectives and indicate the priority, 𝜃 is the value of the norm, and finally 𝐴 and 𝐵 are some constant weights. For the simplicity of the analysis,we set 𝐴 = and 𝐵 = . Observe that Δ 𝑟 and Δ 𝜌 do not sharethe same units. In the celebrated formula of Shannon for thereal AWGN channel the rate is related to the signal powervia the logarithmic function. For this reason, we also select toconsider the logarithm of Δ 𝜌 in the objective function insteadof Δ 𝜌 itself.Setting 𝛼 = , the objective function shifts to the minimiza-tion of Δ 𝑟 . In this case Δ 𝑟 = and the optimum transmissionpair yields (31). On the other hand, if 𝛼 = , the objectivefunction considers only Δ 𝜌 . Therefore, the optimum pair isthe one shown in (32). Pareto-optimal transmission pairs thatlie in between (31) and (32) can be accessed by selectingdifferent values of 𝛼 , depending on the convexity-concavityof the Pareto boundary [42].The selection of the norm, 𝜃 , is a significant determinant foraccessibility of the Pareto-optimal pairs, due to the shape of theattainable objective set 𝑆 , which depends on the objective func-tions and constraints, some optimal pairs cannot be accessedwith the selected scalarization function. This will be discussedfurther in the following Section. We continue our analysis bysetting 𝜃 = and 𝜃 = ∞ , which are the two most frequentlyused weighted- 𝑙 𝜃 norm scalarization techniques, namely linearweighted-sum and weighted Chebyshev, respectively. A. Linear Weighted-Sum Objective Function: 𝜃 = By setting 𝜃 = , the objective function reduces to theweighted sum of the objectives. The objective function canbe written as minimize 𝛼 Δ 𝑟 + ( − 𝛼 ) log Δ 𝜌. (34)The channel capacity 𝐶 for BI-AWGN channel is boundedbetween [ , ] . Due to its monotonicity in 𝜌 , it exhibits asigmoidal shape. Similar to 𝐶 , (3) also follows a sigmoidalshape, which leads to having convex and concave portionsthat are separated with an inflection point. The inflection pointrepresents the point where the shape of the maximal rate curvechanges from convex to concave. It is denoted as { 𝑟 𝑖 , 𝜌 𝑖 } andis the solution to the equation 𝜕 𝑅 ( 𝑛, 𝜌, 𝜀 ) 𝜕 𝜌 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 𝜌 = 𝜌 𝑖 = . (35)Since 𝑀 ( 𝑛, 𝜌, 𝜀 ) is the Δ 𝜌 min amount shifted version of 𝑅 ( 𝑛, 𝜌, 𝜀 ) and Δ 𝜌 min is monotonically increasing in 𝜌 , 𝑀 ( 𝑛, 𝜌, 𝜀 ) also has a sigmoidal structure, where the inflectionoccurs at 𝜌 𝑖 + Δ 𝜌 𝑖 min . Remark 4.
Due to the geometry of 𝑀 ( 𝑛, 𝜌, 𝜀 ) in 𝜌 , the Paretoboundary is concave when 𝜌 𝑠 ≤ 𝜌 𝑖 and convex when 𝜌 𝑠 >𝜌 𝑖 + Δ 𝜌 𝑖 min . However, when 𝜌 𝑖 + Δ 𝜌 𝑖 min ≥ 𝜌 𝑠 > 𝜌 𝑖 , the Paretoboundary consists of both concave and convex regions. Thus, depending on 𝜌 𝑖 and 𝜌 𝑠 , the accessibility of thePareto-optimal pairs with linear weighting objective functioncan be separated into three parts:
1) Low-SNR regime ( 𝜌 𝑠 ≤ 𝜌 𝑖 ): This is the regime where thePareto boundary is concave. Only two Pareto-optimal points,which are the end points of the Pareto boundary, described in(31) and (32), are accessible.
2) Medium-SNR regime ( 𝜌 𝑖 + Δ 𝜌 𝑖 min ≥ 𝜌 𝑠 > 𝜌 𝑖 ): Thisis the regime where the Pareto boundary is convex overa certain region and concave over a certain other region,where these regions are specified in Remark 4. Within thisregime, due to convexity, the optimal points located between 𝜌 ∈ [ 𝜌 𝑖 + Δ 𝜌 𝑖 min , 𝜌 𝑠 + Δ 𝜌 𝑠 min ] are accessible. However, dueto the concavity, optimal points between 𝜌 ∈ ( 𝜌 𝑠 , 𝜌 𝑖 + Δ 𝜌 𝑖 min ) are not accessible. Only the optimal point where 𝜌 = 𝜌 𝑠 isaccessible since it is the end point of the Pareto boundary.
3) High-SNR regime ( 𝜌 𝑠 > 𝜌 𝑖 + Δ 𝜌 𝑖 min ): This is the regimewhere the Pareto boundary is convex and Pareto-optimal pointsin between (31) and (32) are accessible.This phenomenon can be seen in Fig. 5.a, 5.b, and 5.c,where Pareto boundaries and accessible Pareto-optimal Δ 𝑟 and Δ 𝜌 values are depicted for 𝑟 𝑠 = { . , . , . } , respectively.When 𝑟 𝑠 = . , the Pareto boundary is concave. Thus, only twoPareto-optimal points are accessible with the linear weighted-sum objective function. However, when 𝑟 𝑠 = . , the Paretoboundary is convex over a part of the region and concaveover the rest of the region and therefore the Pareto-optimalpoints on the convex part are accessible. When 𝑟 𝑠 = . , thefeasible region yields a convex Pareto boundary and enablesthe objective function to attain all Pareto-optimal points byselecting different values of 𝛼 . B. Weighted Chebyshev Objective Function: 𝜃 = ∞ Another way of scalarization can be achieved by setting 𝜃 = ∞ where now the objective function transforms to a weightedmin-max formulation. This scalarization technique is knownas the weighted Chebyshev objective function, which can beformulated as the followingminimize max { 𝛼 Δ 𝑟, ( − 𝛼 ) log Δ 𝜌 } . (36)The weighted Chebyshev objective function guarantees access-ing all Pareto optimal points regardless of having concave orconvex structure. Remark 5.
The Pareto boundary and its shape are not relatedto which scalarization method is selected. Its shape is drawnaccording to the set 𝑆 which depends on the objective functionsand constraints. The scalarization method only determinesthe accessible set of the Pareto optimal points. At the end,depending on 𝐴 , 𝐵 , and 𝛼 , the scalarization function will find afinal optimal point. Therefore, the final selected Pareto-optimalpoint will be different with different scalarization functions. VI. C
ASE S TUDY : B
ATTERY -P OWERED T RANSMISSION
In this section, we exemplify the importance of the MOOPby a case study. Suppose a battery-powered transmitter iscommunicating with a complexity constrained receiver, underlatency and reliability constraints. The transmitter transmitscodewords with 𝑛 = blocklength under a latency con-straint, defined in (6). The objective is to maximize both the (a) 𝑟 𝑠 = . (b) 𝑟 𝑠 = . (c) 𝑟 𝑠 = . Fig. 5: Comparisons between the Pareto boundaries and the accessible Pareto-optimal points with the linear weighted-sumobjective function for 𝑟 𝑠 = { . , . , . } when 𝑛 = , 𝜖 = − , 𝑇 𝑠 = 𝜇 s, 𝑇 𝑏 = ns, and 𝐿 𝑚 = ms. Here, we assume that 𝑟 𝑚 and 𝜌 𝑚 are sufficiently low and high, respectively, such that they have no observable effect on the Pareto boundary.total number of information bits transmitted to the receiveruntil the battery dies and increase the energy efficiency atthe same time. The transmitter and receiver set an ultimatetransmission pair { 𝑟 𝑠 , 𝜌 𝑠 } and limits on rate and power,denoted as 𝑟 𝑚 and 𝜌 𝑚 , respectively. The rate limit representsthe minimum amount of information bits that must be sentwith every codeword transmission and power limit is due tothe power budget of the system. Then, transceivers search fora transmission pair that can meet the constraints given thebattery level.One can envision the communication environment as thefollowing, when the battery is full, the transmitter can sacrificepower to maintain the constraints instead of introducing rategap. Besides, as the remaining battery power level is decreas-ing, power is becoming more precious and the transmittermay need to introduce some amount of rate reduction insteadof sacrificing more power. This can be achieved by definingthe weight 𝛼 being related to the remaining battery powerpercentage, denoted as 𝑡 . Thus, a sigmoid relation between 𝛼 and 𝑡 , is defined as 𝛼 = − (cid:18) + (cid:16) 𝑡 + 𝑡 (cid:17) (cid:19) − . (37)From (37) , 𝛼 gets values close to when the battery level ishigh and hence the goal of the optimization problem yields theminimization of Δ 𝑟 . However, as the battery level decreases, 𝛼 increases and approaches . Therefore, the focus of theoptimization problem shifts to the minimization of Δ 𝜌 . Asignificant observation here is that with the introduction of(37), the transmission parameters are becoming adaptive withrespect to 𝑡 and constraints. An efficient algorithm to find theoptimum transmission pair is shown in Algorithm 1. A. Numerical Results
Suppose that a battery with 𝐵 = Wh capacity is usedfor data transmission at the transmitter. For each codewordtransmission, the transmitter computes 𝛼 with respect to 𝑡 and The relation between 𝛼 and 𝑡 given in (37) can be accepted as a generalexample. Different relations according to the relevance of 𝑡 in the case studycan be proposed. Algorithm 1
Multi-objective optimization
1: Given 𝑛 and 𝜀 𝑚 : compute : 𝑅 ( 𝑛, 𝜌, 𝜀 𝑚 ) using (3)2: Given ≤ 𝑟 𝑠 ≤ : compute : 𝜌 𝑠 from (12)3: compute : Δ 𝜌 min using (22), ∀ 𝑟 ∈ ( , ] compute : 𝑀 ( 𝑛, 𝜌, 𝜀 𝑚 ) using (21)5: compute : Δ 𝑟 min using (25)6: if 𝑟 𝑠 − Δ 𝑟 𝑠 min ≥ 𝑟 𝑚 then Δ 𝜌 start = else Δ 𝜌 start = 𝑀 − ( 𝑛, 𝑟 𝑚 , 𝜀 𝑚 ) end if if 𝜌 𝑠 + Δ 𝜌 𝑠 min ≤ 𝜌 𝑚 then Δ 𝜌 end = Δ 𝜌 𝑠 min else Δ 𝜌 end = 𝜌 𝑚 − 𝜌 𝑠 end if for Δ 𝜌 = Δ 𝜌 start : Δ 𝜌 end do compute : Δ 𝑟 = 𝑟 𝑠 − 𝑀 ( 𝑛, 𝜌 𝑠 + Δ 𝜌, 𝜀 𝑚 )
18: Given 𝑡 : compute : 𝛼 from (37)19: Given 𝐴 , 𝐵 , and 𝜃 : compute : 𝑧 ( 𝑖 ) = (cid:2) 𝐴𝛼 ( Δ 𝑟 ) 𝜃 + 𝐵 ( − 𝛼 ) ( log Δ 𝜌 ) 𝜃 (cid:3) 𝜃 𝑖 = 𝑖 + end for
22: Select { 𝑟 𝑠 − Δ 𝑟 , 𝜌 𝑠 + Δ 𝜌 } minimizes 𝑧 selects a transmission rate-power pair based on a target rate 𝑟 𝑠 and minimum rate 𝑟 𝑚 and the latency, reliability and decodingcomplexity constraints. Here, it is assumed that a sequence ofdata transmission is happening until the fully charged batterydies. Notice that as transmission continues, the transmittercalculates the remaining power from [43]. As the batterypower decreases, 𝛼 changes accordingly. Therefore, a newPareto-optimal transmission pair is selected at each codewordtransmission. This setup is an appropriate assumption forbattery-powered URLLC IoT use cases where event-driventransmission is occurring [44].For the empirical analysis, we assume dB average signalpower attenuation at a reference distance of m and − dBmnoise power at the receiver [45]. The distance between thetransmitter and receiver is set to m. For similarity purposeswith the previous illustrations, we set 𝐿 𝑚 = ms, 𝜀 𝑚 = − , 𝑇 𝑠 = − s, and 𝑇 𝑏 = − s.We first set 𝑟 𝑠 = . and, for simplicity, 𝑟 𝑚 = . Numericalresults on the selected Pareto-optimal transmission rate at a (a) Comparison of the rate selections with respect to the remainingbattery level, where 𝑟 𝑠 = . . (b) Ratio of the total number of codeword transmissions with differentobjective functions, where 𝑟 𝑠 = . .(c) Comparison of the objective functions in terms of energy efficiency with respect to various target rates, 𝑟 𝑠 . Fig. 6: Comparison of the objective functions: i ) minimization of Δ 𝑟 : 𝛼 = , ii ) minimization of Δ 𝜌 : 𝛼 = , iii ) linearweighted-sum function, and iv ) weighted Chebyshev function.codeword transmission with respect to 𝑡 for linear weighted-sum and weighted Chebyshev functions are depicted in Fig.6.a. Additionally, for comparison purposes, we introduce twonew objective functions: i ) minimize Δ 𝑟 , i.e. 𝛼 = , ii )minimize Δ 𝜌 , i.e. 𝛼 = , which are shown with red dashedand dash-dotted lines, respectively. These objective functionsdo not depend on 𝑡 and can be assumed as examples for fixedparameter transmission.It can be seen that setting 𝛼 = and 𝛼 = yield two distinctchoices, i.e. 𝑟 𝑠 and 𝑟 𝑠 − Δ 𝑟 𝑠 min for all 𝛼 . More interestingpictures arise, as we allow 𝛼 to vary with 𝑡 and selectother objectives. We first start with the linear weighted-sumobjective function. As expected, due to the concave structureof the attainable objective set, only two Pareto-optimal pointsare accessible 𝑟 lin = ( 𝑟 𝑠 when 𝑡 is high 𝑟 𝑠 − Δ 𝑟 𝑠 min when 𝑡 is low . (38) Notice from Fig. 6.a that a transition from 𝑟 𝑠 to 𝑟 𝑠 − Δ 𝑟 𝑠 min happens approximately when the remaining battery level fallsbelow 50 % . Besides, unlike 𝑟 lin , the optimum rate selection ofthe weighted Chebyshev function follows a smooth transitionfrom 𝑟 𝑠 to 𝑟 𝑠 − Δ 𝑟 𝑠 min and access all Pareto-optimal pointsin-between.Energy efficiency is crucial for IoT setups since verylong battery life is required [43]. Numerical results in Fig.6.a show that the selected Pareto-optimal transmission ratedecreases as 𝑡 reduces. Although this decreases the throughputof the communication system, it also allows to select lowertransmit power, which, in the long run, allow more codewordtransmissions and increase the energy efficiency. To see thiseffect, total number of codeword transmissions with a fullycharged battery are depicted in Fig. 6.b, where the vertical axisrepresents the ratio of total number of codeword transmissionsto the case where 𝛼 = . Results show that by implementing MOOP, the transmitter can transmit approximately 30% morecodewords when linear weighted-sum or weighted Chebyshevobjective functions are selected.Next, to see the long-term outcomes of the MOOP, theenergy efficiency of the proposed MOOP, in terms of totalnumber of transmitted information bits per Joule, for various 𝑟 𝑠 , are depicted in Fig. 6.c. Energy efficiency is computedaccording to the following formulaEnergy efficiency = 𝑛 Í ( 𝑟 𝑠 − Δ 𝑟 ) 𝐵 [ bits / Joule ] , (39)where the sum is over all codeword transmissions until thebattery dies. Results are depicted in Fig. 6.c. Our results arein agreement with the existing work in [46], considering thatwe also model the finite blocklength, latency, reliability anddecoding complexity constraints.Based on the results in Fig. 6.c, one can divide the figureinto two regions: low and high rate. In low rate region, theobjective function, where 𝛼 = , achieved higher energyefficiency compared to the other objectives. On the contrary,in high rate region, the maximum energy efficiency valuesare achieved when 𝛼 = . However, notice that for bothregions the linear weighted-sum and weighted Chebyshevfunctions achieve a performance that is in the middle. Alsonotice that although the weighted Chebyshev function yieldsless number of codeword transmissions compared to linearweighted-sum function, as shown in Fig. 6.b, it leads to higherenergy efficiency for all 𝑟 𝑠 . Finally, at the intersection of thelow and high rate regions, the weighted Chebyshev functionoutperforms all other objective functions and achieves thehighest energy efficient communication.VII. C ONCLUSIONS
We study the optimum transmission parameters for linearblock codes with OS decoders in finite blocklength underlatency, reliability, and decoding complexity constraints. It isshown that constraints introduce a back-off from the finitemaximal achievable rate. Based on this analysis, a MOOP isformulated and solved with the help of an empirical modelthat can accurately track the trade-off between power gapversus per-information-bit computational complexity of thedecoder. The attainable objective set and the Pareto boundaryare characterized for two different multi-objective scalariza-tion functions, namely the linear weighted-sum and weightedChebyshev functions. Then, the accessability of the Pareto-optimal points are analyzed. It is also shown that dependingon the convexity/concavity of the Pareto boundary, the Pareto-optimal points cannot be accessed by the linear weighted-sumobjective function. Finally, benefits of the MOOP are shownwith a case study on battery-powered transmission, wherethe weights for the scalarization of the MOOP is configuredaccording to the remaining battery level and thus the optimumselection of the MOOP varies with the remaining battery level.This, therefore, introduces an adaptive transmission parameterselection. Next, it is further shown that, MOOP frameworkincreases both the throughput and energy efficiency of thesystem, compared to the classical fixed parameter transmis-sion, while the constraints on latency, reliability, and decodingcomplexity are still met. R
IEEE Access , vol. 5, pp. 5917–5935, 2017.[4] S. R. Pokhrel, J. Ding, J. Park, O. S. Park, and J. Choi, “Towardsenabling critical mmtc: A review of urllc within mmtc,”
IEEE Access ,vol. 8, 2020.[5] H. B. Celebi, A. Pitarokoilis, and M. Skoglund, “Wireless communi-cation for the industrial IoT,” in
Industrial IoT: Challenges, DesignPrinciples, Applications, and Security , I. Butun, Ed. Springer Verlag,2020.[6] C. E. Shannon, “A mathematical theory of communication,”
The BellSystem Technical Journal , vol. 27, no. 3, pp. 379–423, 1948.[7] P. Popovski, C. Stefanovic, J. J. Nielsen, E. de Carvalho, M. Angjelichi-noski, K. F. Trillingsgaard, and A. Bana, “Wireless access in ultra-reliable low-latency communication (URLLC),”
IEEE Transactions onCommunications , vol. 67, no. 8, pp. 5783–5801, Aug 2019.[8] Y. Polyanskiy, H. V. Poor, and S. Verdu, “Channel coding rate in thefinite blocklength regime,”
IEEE Transactions on Information Theory ,vol. 56, no. 5, pp. 2307–2359, May 2010.[9] A. Zaidi, F. Athley, J. Medbo, U. Gustavsson, G. Durisi, and X. Chen,
5G Physical Layer: Principles, Models and Technology Components , 052018.[10] M. Shirvanimoghaddam, M. S. Mohammadi, R. Abbas, A. Minja,C. Yue, B. Matuz, G. Han, Z. Lin, W. Liu, Y. Li, S. Johnson, andB. Vucetic, “Short block-length codes for ultra-reliable low latencycommunications,”
IEEE Communications Magazine , Feb. 2019.[11] M. P. C. Fossorier and S. Lin, “Soft-decision decoding of linear blockcodes based on ordered statistics,”
IEEE Transactions on InformationTheory , vol. 41, no. 5, pp. 1379–1396, Sep. 1995.[12] H. Saeedi and A. H. Banihashemi, “Performance of belief propagationfor decoding ldpc codes in the presence of channel estimation error,”
IEEE Transactions on Communications , vol. 55, no. 1, 2007.[13] L. Gaudio, T. Ninacs, T. Jerkovits, and G. Liva, “On the performance ofshort tail-biting convolutional codes for ultra-reliable communications,”in
SCC; 11th International ITG Conference on Systems, Communica-tions and Coding , 2017, pp. 1–6.[14] K. Niu and K. Chen, “CRC-aided decoding of polar codes,”
IEEECommunications Letters , vol. 16, no. 10, pp. 1668–1671, 2012.[15] G. Liva, L. Gaudio, and T. Ninacs, “Code design for short blocks: Asurvey,” in
Proc. EuCNC , Athens, Greece, Jun 2016.[16] C. Yue, M. Shirvanimoghaddam, B. Vucetic, and Y. Li, “A revisit toordered statistic decoding: Distance distribution and decoding rules,”2020, arXiv:2004.04913.[17] T. Hehn and J. B. Huber, “LDPC codes and convolutional codes withequal structural delay: a comparison,”
IEEE Transactions on Communi-cations , vol. 57, no. 6, 2009.[18] V. Bioglio, C. Condo, and I. Land, “Design of polar codes in 5g newradio,”
IEEE Communications Surveys Tutorials , 2020.[19] X. Jiang, M. Luvisotto, Z. Pang, and C. Fischione, “Latency perfor-mance of 5G new radio for critical industrial control systems,” in , Sep. 2019, pp. 1135–1142.[20] E. Nachmani, E. Marciano, L. Lugosch, W. J. Gross, D. Burshtein, andY. Be’ery, “Deep learning methods for improved decoding of linearcodes,”
IEEE Journal of Selected Topics in Signal Processing , vol. 12,no. 1, 2018.[21] X. Ma, W. Lin, S. Cai, and B. Wei, “Statistical learning aided decodingof bmst tail-biting convolutional code,” in
IEEE International Sympo-sium on Information Theory (ISIT) , 2019.[22] B. Cavarec, H. B. Celebi, M. Bengtsson, and M. Skoglund, “Alearning-based approach to address complexity-reliability tradeoff inOS decoders,” in , 2020.[23] H. B. Celebi, A. Pitarokoilis, and M. Skoglund, “Latency and reliabilitytrade-off with computational complexity constraints: OS decoders andgeneralizations,”
IEEE Transactions on Communications , 2021.[24] A. Azari, “Serving IoT communications over cellular networks : Chal-lenges and solutions in radio resource management for massive andcritical IoT communications,” Ph.D. dissertation, KTH Royal Institute of Technology, Electrical Engineering and Computer Science Department,2018.[25] O. L. Alcaraz L´opez, H. Alves, and M. Latva-aho, “Joint powercontrol and rate allocation enabling ultra-reliability and energy efficiencyin SIMO wireless networks,”
IEEE Transactions on Communications ,vol. 67, no. 8, 2019.[26] O. L. A. L´opez, H. Alves, and M. Latva-Aho, “Distributed rate con-trol in downlink NOMA networks with reliability constraints,”
IEEETransactions on Wireless Communications , vol. 18, no. 11, 2019.[27] N. Zogovic, “Energy efficiency versus reliability tradeoff improvementin low-power wireless communications,”
IEEE Systems Journal , vol. 14,no. 3, 2020.[28] E. Dosti, M. Shehab, H. Alves, and M. Latva-Aho, “Ultra reliablecommunication via optimum power allocation for HARQ retransmissionschemes,”
IEEE Access , Aug 2019.[31] O. L. Alcaraz L´opez, N. H. Mahmood, and H. Alves, “EnablingURLLC for low-cost IoT devices via diversity combining schemes,”in
IEEE International Conference on Communications Workshops (ICCWorkshops) , 2020.[32] K. Hwang and N. Jotwani,
Advanced Computer Architecture: Paral-lelism, Scalability, Programmability , 3rd ed. McGraw-Hill HigherEducation, 2016.[33] R. Wilhelm, J. Engblom, A. Ermedahl, N. Holsti, S. Thesing, D. Whal-ley, G. Bernat, C. Ferdinand, R. Heckmann, T. Mitra, F. Mueller,I. Puaut, P. Puschner, J. Staschulat, and P. Stenstr¨om, “The worst-caseexecution-time problem: Overview of methods and survey of tools,”
ACM Trans. Embed. Comput. Syst. , vol. 7, no. 3, pp. 36:1–36:53, May2008.[34] S. R. Reed and X. Chen,
Error-Control Coding for Data Networks .Springer Science, 1999.[35] P. Dhakal, R. Garello, S. K. Sharma, S. Chatzinotas, and B. Ottersten, “On the error performance bound of ordered statistics decoding of linearblock codes,” in , May 2016.[36] B. Lian, “Performance and decoding complexity analysis of short binarycodes,” Master’s thesis, University of Toronto, Toronto, 2019.[37] S. V. Maiya, D. J. Costello, T. E. Fuja, and W. Fong, “Coding witha latency constraint: The benefits of sequential decoding,” in , 2010, pp. 201–207.[38] P. Cao, E. A. Jorswieck, and S. Shi, “Pareto boundary of the rateregion for single-stream mimo interference channels: Linear transceiverdesign,”
IEEE Transactions on Signal Processing , vol. 61, no. 20, pp.4907–4922, 2013.[39] E. Bj¨ornson, E. A. Jorswieck, M. Debbah, and B. Ottersten, “Multiob-jective signal processing optimization: The way to balance conflictingmetrics in 5G systems,”
IEEE Signal Processing Magazine , vol. 31,no. 6, pp. 14–23, 2014.[40] K. Taha, “Methods that optimize multi-objective problems: A surveyand experimental evaluation,”
IEEE Access , vol. 8, 2020.[41] K. H. Chang, “Chapter 19 - multiobjective optimization and advancedtopics,” in e-Design , K. H. Chang, Ed. Boston: Academic Press, 2015,pp. 1105–1173.[42] M. T. Emmerich and A. H. Deutz, “A tutorial on multiobjective opti-mization: Fundamentals and evolutionary methods,”
Natural Computing:An International Journal , vol. 17, no. 3, p. 585–609, 2018.[43] M. Lauridsen, R. Krigslund, M. Rohr, and G. Madueno, “An empiricalNB-IoT power consumption model for battery lifetime estimation,” in , 2018.[44] L. Feng, Y. Zi, W. Li, F. Zhou, P. Yu, and M. Kadoch, “Dynamicresource allocation with RAN slicing and scheduling for uRLLC andeMBB hybrid services,”
IEEE Access , vol. 8, 2020.[45] O. L. A. L´opez, H. Alves, R. D. Souza, and E. M. G. Fern´andez, “Ul-trareliable short-packet communications with wireless energy transfer,”
IEEE Signal Processing Letters , vol. 24, no. 4, 2017.[46] E. Bj¨ornson and E. G. Larsson, “How energy-efficient can a wirelesscommunication system become?” in2018 52nd Asilomar Conference onSignals, Systems, and Computers