Analysis and Design of Analog Fountain Codes for Short Packet Communications in IoT
Wen Jun Lim, Mahyar Shirvanimoghaddam, Rana Abbas, Yonghui Li, Branka Vucetic
aa r X i v : . [ c s . I T ] F e b Analysis and Design of Analog Fountain Codes forShort Packet Communications in IoT
Wen Jun Lim,
Student Member, IEEE , Mahyar Shirvanimoghaddam,
Senior Member, IEEE , Rana Abbas,
Member, IEEE , Yonghui Li,
Fellow, IEEE , Branka Vucetic
Fellow, IEEE
Abstract —In this paper, we focus on the design and analysis ofthe Analog Fountain Code (AFC) for short packet communica-tions. We first propose a density evolution (DE) based framework,which tracks the evolution of the probability density function ofthe messages exchanged between variable and check nodes ofAFC in the belief propagation decoder. Using the proposed DEframework, we formulate an optimisation problem to find theoptimal AFC code parameters, including the weight-set, whichminimizes the bit error rate at a given signal-to-noise ratio (SNR).Our results show the superiority of our code design in comparisonto existing code designs and thus the validity of the proposed DEframework in the asymptotic block length regime. We then focuson the selection of the precoder to improve the performance ofAFC at short block lengths. Simulation results show that lowerprecoder rates obtain better realised rates over a wide SNR rangefor short information block length.
Index Terms —Analog fountain code (AFC), density evolution,differential evolution optimization, rateless codes.
I. I
NTRODUCTION T HE third generation partnership project (3GPP) has de-fined 3 main services for the fifth generation (5G) ofmobile communications [1] . These include enhanced mobilebroadband (eMBB), ultra-reliable low-latency communica-tions (URLLC), and massive machine type communications(mMTC). 5G aims at providing higher data rate (e.g., upto 10 Gbps for eMBB), shorter end-to-end latency (e.g., ≤ This paper was presented in part at the IEEE 90th Vehicular TechnologyConference, Honolulu, HI, USA, Sep. 2019.The authors are with the Centre for IoT and Telecommunications, Schoolof Electrical and Information Engineering, The University of Sydney,NSW 2006, Australia (e-mail: { wenjun.lim; mahyar.shirvanimoghaddam;rana.abbas; yonghui.li; branka.vucetic } @sydney.edu.au). large consensus that traditional channel codes, adopted in cur-rent cellular networks, are strictly sub-optimal for short packetcommunications [7]. Modern channel coding techniques havediverged away from the traditional theories of Shannon, dueto this new shift alongside other challenges imposed by the5G services, e.g., higher reliability requirements (lower than − ) and higher energy efficiency. We refer the readers tothe survey paper in [7] for a comprehensive review of modernchannel coding techniques. Amongst the proposed techniques,the 3GPP Release-15 standard [1] has indicated that eMBBwill be using low density parity check (LDPC) codes in thedata channels and Polar codes in the control channels [8].Currently, the standardisation of the channel coding techniquesfor URLLC and mMTC is still underway [9].In this paper, we focus on modern rateless codes as apotential candidate for 5G. When employing fixed-rate channelcodes, rate adaptation to the varying wireless channel requiresthe receiver to frequently feedback channel state informationto the transmitter to select the best coding and modulationscheme. For short block lengths, the overhead incurred by thisfeedback can be costly both in terms of spectrum efficiency formMTC and in terms of latency for URLLC [10]. This is whereself-adaptive codes, also known as rateless codes, come in asan attractive candidate. In rateless codes, the rate is determinedon-the-fly without the need for the transmitter to be aware ofthe channel conditions. This is particularly favourable in fastvarying channels as well as non-reciprocal channels. A. Related Works
The original research on rateless codes, such as LT codes[11] and Raptor codes [12], have shown that rateless code canachieve the channel capacity over binary erasure channels. Amodified version of Raptor codes, called RaptorQ, has beenstandardised for mobile wireless communication broadcastand multicast, as well as DVB-H standards for IP datacast[13]. However, all design extensions of rateless codes tonoisy channels, e.g., additive white Gaussian noise (AWGN)channel, have shown to be channel dependent. To the bestof our knowledge, only a few rateless codes to date arenear capacity-achieving over a wide range of SNRs for anasymptotically long block length, such as analog fountaincodes [14] and Spinal codes [15]. This gap to the capacityis significantly degraded for short block lengths [5]. Theapplication of short rateless codes in 5G NR services has beendemonstrated in several works, including protograph-basedraptor-like LDPC codes in [16], which showed satisfactory performance in the short block length regime. Meanwhile, theresearch carried out in [17] identified the challenges for thedesign of channel codes to address the specification of mMTC,and the work in [18] proposed bi-interleave coded multipleaccess (BICMA) as a multiple access scheme in the physicallayer of mMTC, where it can support high demanding loadwith high energy efficiency and low complexity. Althoughtheses works demonstrated good performance of short ratelesscodes for mMTC application, the underlying rateless codesare mainly binary and heavily dependent on SNR. On theother hand, the research in [19] evaluates the techniques inphysical layer and MAC layer to increase reliability and reducelatency, and provide numerical evaluation of URLLC whichis enabled by coexistence of LTE in unlicensed spectrum.The work carried out in [20], [21] demonstrated guideline forRate Compatible Modulation (RCM) in selection of weightsaccording to channel condition for reduced complexity withoutloss of performance and also parallel belief propagation (BP),respectively. However, the proposed designs therein weremostly heuristic and did not provide meaningful insights intothe code design for short block lengths.Recent results on short AFC [22] have shown impressiveresults for 5G URLLC in terms of comparable latency to thePolyanskiy-Poor and Verdu (PPV) normal approximation [23]as well as reliability down to − . However, the proposedencoding scheme therein as well as the weight set design arebased on heuristics rather than a solid analytical framework.In this paper, we aim to bridge this gap in the literature byproviding a solid analytical framework as well as an optimi-sation framework for AFC. To the best of our knowledge,the only work that has attempted to do so can be found in[24]. Authors in [24] proposed a modified weight selectionscheme for short block length weight-adaption AFC (SWA-AFC), which achieves significant coding gains, close to thePPV bound. In particular, the extrinsic information transfer(EXIT) chart analysis [25] was modified in [24], to addressthe issue of performance degradation of EXIT analysis in thefinite block length regime. The design in [24], however, relieson approximate densities and the assumption of symmetricGaussian distribution for the messages exchanged in each it-eration of the message passing decoder. These assumptions arenot accurate [26], particularly at low rates, which negativelyimpact the search for the optimal code ensembles.In this work, we employ the density evolution (DE) analysisto track the evolution of the probability distribution function( pdf ) of messages exchanged between the nodes in eachiteration of the belief propagation (BP) decoding algorithm.DE was previously used in [26] to analyse Raptor codes asa multi-edge type LDPC code. The degree distribution ofthe Raptor code was then optimized using this framework,which significantly outperformed existing Raptor codes mainlydesigned using linear programming based on approximate DEalgorithms [27] or Gaussian approximations [28], [29]. In whatfollows, we explain our main contributions in this work. B. Main Contributions1) A Density Evolution-based Framework to Analyse AFC:
A DE framework is proposed in this work to analyse the performance of the iterative message-passing decoder of AFC.The algorithm is designed in the way to track the evolutionof pdf of log-likelihood ratios (LLR) which are exchangedbetween variable and check nodes during the belief propa-gation decoding. Due to the complexity of the variable andcheck node updating rules, we proposed a Monte-Carlo basedanalysis to derived the pdf s in each iteration of DE. Ourresults show that the proposed DE framework can preciselyapproximate the pdf of LLRs in each iteration of BP.
2) A Differential Evolution Algorithm to Optimize AFCparameters:
We propose an optimization problem based on theDE analysis to find the optimal weight set of AFC. We usedifferential evolution to solve the optimization problem andfind the optimal weight set for AFC with different degrees.Simulation results showed that by applying the optimizedweight set, the performance of the AFC is improved regardlessin asymptotic long block length and finite block length regime,thus proving the validity of the DE framework proposed.
3) AFC optimization in the short block length regime:
To further enhance the performance of AFC at short blocklengths, we evaluate different precoders with different coderates. In particular, we consider BCH codes with an orderedstatistics decoder (OSD) [30] and LDPC codes with BP de-coder. The combined code is simulated under different channelconditions where we show that a low rate precoder, offersbetter reliability at a wide range of SNRs compared to high-rate precoders. We also show that the precoded AFC performsclose to the normal approximation benchmark [23] in the finiteblock length regime over a wide range of SNRs.
C. Paper Organization
The rest of the paper is organized as follows. In SectionII, we explain the encoder and decoder of the concatenatedAFC code. We then propose the density evolution frameworkfor the analysis and design of AFC in Section III. Section IVpresents the differential evolution optimization framework forthe AFC weight-set optimization. The design of AFC codesin the short block length will be studies in Section V. Finally,conclusions are drawn in Section VI.II. A
NALOG F OUNTAIN C ODES
AFC was originally proposed in [14], which is mainlycharacterized by a weight set, degree distribution function, andthe message length. AFC has linear-complexity encoding anddecoding processes in terms of the block length. The codeis rateless in nature and can generate a potentially limitlessnumber of coded symbols; thus, achieving any desired rate on-the-fly. In what follows, we explain the encoding and decodingprocesses of AFC.
A. The Encoder
The precoded AFC is a concatenation of a fixed-rate precodeand the AFC code. An information block of length k bits,denoted by b , is first encoded by using a fixed-rate ( n, k )code of rate R pre = k/n , to generate a codeword of length n bits, denoted by u , referred to as intermediate symbols . The generator matrix of the fixed-rate code is denoted by G pre ;therefore, we have: u = G pre b . (1)The precode serves as the outer code for the precoded AFCcode. Intermediate symbols are then modulated by using aBPSK modulation, to generate n modulated symbols, v i , thatis v i = ( − u i , for i = 1 , · · · , n. (2)Next, by using an AFC code a potentially limitless numberof AFC coded symbols, also referred to as output symbols ,are generated. AFC is mainly characterized by a weight set W and a degree distribution function Ω( x ) = P ni =1 Ω i x i .In order to generate an AFC coded symbol, a degree d isdrawn based on the degree distribution function Ω( x ) . Then d modulated intermediate symbols are randomly selected andlinearly combined in the real domain with a set of d real weightcoefficients selected from the weight set W . For simplicity, weconsider that the degree d is fixed, i.e., Ω( x ) = x d , and theweight set is predefined and given by W = { w , w , · · · , w d } .The i th AFC coded symbol, denoted by c i , is then given by: c i = d X j =1 w j v i,j , (3)where v i,j ∈ V i and V i is the set of modulated intermediatesymbols that have been chosen to generate the i th outputsymbol. We further assume that P di =1 w i = 1 , thereforewe have E [ | c | ] = 1 , where E [ . ] is the expectation operand.Fig. 1 shows the bipartite graph representation of the AFCcode truncated at length m . We refer to AFC coded symbolsand intermediate symbols in the bipartite graph of AFC bycheck and variable nodes, respectively. The degree of a check(variable) node is defined as the number of variable (check)nodes connected to it in the bipartite graph. A regular AFChas constant variable node degree d v and check node degree d c .AFC coded symbols are sent over the channel. Once thereceiver received m symbols, it performs the decoding. If thedecoding failed, it collects δ additional AFC coded symbolsand run the decoder again. This process will continue until thedecoding succeeds. The realize rate of the precoded AFC isgiven by R = k E [ m s ] = R pre × n E [ m s ] , (4)where m s is the number of AFC coded symbols collected toperform a successful decoding. B. The Decoder
We consider the additive Gaussian noise (AWGN) channel,where the channel output y i is given by y i = c i + n i , for i = 1 , , · · · , (5)where n i is AWGN with zero mean and variance σ . Thesignal-to-noise ratio (SNR), denoted by γ , is then given by γ = 1 /σ . b b b b n c c c c m w , w , w , w , w ,n w m, w m, Fig. 1. Bipartite graph representation of and AFC code.
The decoding is performed in two stages. First, the BPdecoding algorithm is applied to AFC to find the log-likelihoodratios (LLRs) of intermediate symbols. We use the BP al-gorithm originally proposed in [31] and further modified in[32] to decode AFC. Second, the LLRs are passed to thedecoder of the precode to find the original k informationsymbols. If the decoding failed, the BP decoding is repeatedwith a longer block of AFC symbols which includes newlyarrived symbols. The LLRs are passed again to the decoderof the precode. This continues until the decoding succeededor the maximum number of AFC symbols are sent. In thispaper, we consider both Bose, Chaudhuri, and Hocquenghem(BCH) codes and LDPC codes as the precoder. BCH codeis a powerful cyclic error correcting code with a variety ofblock lengths and corresponding code rates [7]. BCH codeshave strong error correcting capability which can correct allrandom patterns of t errors, where t is the design parameter.BCH codes are effective on preventing error floor due tothe large minimum Hamming distance [7]. However there isa shortcoming of BCH codes, which is not flexible enoughdue the fact that block length and information length cannotbe selected arbitrarily. For BCH precoded AFC, we use theordered statistics decoder (OSD), which is computationallycomplex. LDPC on the other hand offers a lower complexitydecoder, as BP can be effectively used to decode them.LDPC however cannot offer the same level of error correctioncapability as BCH codes, particularly at short block lengths.III. T HE D ENSITY E VOLUTION A NALYSIS OF
AFCDensity Evolution (DE) [33] is a powerful tool used toanalyse the belief propagation algorithm and has been exten-sively applied to graph-based codes with static uniformity. Ina nutshell, this technique involves tracking the distribution ofthe messages exchanged along the edges of the bipartite graphin each iteration of the message passing algorithm (MPA) [34].In this section, we first rephrase the message passing decoderfor AFC and provide a modification to AFC in order to meetthe requirement for DE. We then present the updating rulesat the variable and check nodes of the bipartite graph of AFCand explain how we implement it in a practical manner.
A. The Message Passing Decoding of AFC
We consider a regular AFC with constant variable and checknode degree, d v and d c , respectively. In each iteration of MPA(i.e., the BP decoder), messages are exchanged between thecheck and variable nodes and vice versa. We use the log-likelihood ratio (LLR) as the message which is exchangesbetween nodes in each iteration of MPA. Let m ( ℓ ) c → v ( w v ) denote the message sent from check node c to variable node v along the edge with weight w v in the ℓ thiteration of MPA. It can be calculated as follows: m ( ℓ ) c → v ( w v ) =ln P b v ′ ∈{− , } v ′ ∈M c \ v e − y − wv − P v ′∈M c \ v wv ′ bv ′ σ Q v ′ ∈M c \ v p ( ℓ − v ′ → c ( b v ′ ) P b v ′ ∈{− , } v ′ ∈M c \ v e − y + wv − P v ′∈M c \ v wv ′ bv ′ σ Q v ′ ∈M c \ v p ( ℓ − v ′ → c ( b v ′ ) , (6)where y = c + n is the received signal corresponds to checknode c , M c \ v denote the set of variable node connected tocheck node c except variable node v and p ( ℓ ) v ′ → c ( b v ′ ) = (cid:18) e − m ( ℓ ) v ′→ c (cid:19) − , if b v ′ = 1 , (cid:18) e m ( ℓ ) v ′→ c (cid:19) − , if b v ′ = − , (7)and m ( ℓ ) v → c denote be the message sent from a variable node v to check node c in the ℓ th iteration of MPA, which is givenby m ( ℓ ) v → c = X c ′ ∈N v \ c m ( ℓ − c ′ → v ( w v ′ ) (8)where N v \ c represents the set of check nodes that are con-nected to variable node v except check node c .The messages are exchanged in an iterative manner betweenvariable and check nodes for a predefined number of iterationsor until the decoding achieves convergence. The final LLRvalue of each variable node after L iterations of MPA iscalculated as follows: m ( L ) v = X c ∈N v m ( L ) c → v ( w v ) . (9)These LLRs are then passed to the decoder of the precode tofind the original k information symbols. B. The Density Evolution Analysis
Without loss of generality, we assume that all incomingmessages in a node are independent and identically distributed(i.i.d). This assumption was first considered in [33] and furtherjustified in [35]. Based on this assumption, the bipartitegraph can be viewed as a set of independent sub-trees withindependently distributed messages, making the analysis morefeasible. With probability arbitrarily approaching to 1 when n goes to infinity, a cycle-free bipartite graph emerged [35].The other assumption which has been widely consideredfor the DE analysis, is the all-zero codeword transmission.For this condition to be met, the output channel LLRs shouldbe symmetric that is the bit error rate is independent of thetransmitted codeword [36]. However, AFC does not meet thiscondition as the average power of the output symbols dependson the information sequence. For example, when the all-zero information sequence is being encoded using AFC, each codedsymbol will be equal to P d c i =1 w i which is equivalent tothe signal with the highest power assuming that w i ’s are allpositive. For an information sequences with some non-zerosymbols, some of the coded symbols will have lower power.This results in an unequal protection of information sequences.To solve this problem and meet the requirements for the DEanalysis, we adopt the idea of the independent and identicallydistributed (i.i.d) channel adapter [37].In particular, we slightly modify the encoding process ofAFC in (3) by multiplying a binary random number, i.e., +1 and − , with the weight associated to each edge in thebipartite graph. The i th AFC coded symbols is then generatedas follows: c i = d X j =1 ( − t j w j v i,j , (10)where t j ’s are uniformly and randomly drawn from set { , } .In the rest of the paper, we use this modified encoder unlessotherwise specified.
1) DE Check Node Updating Rule:
It is important to notethat unlike binary graph-based codes which are mainly char-acterized by the degree distribution function in the asymptoticblock length regime, the analysis of AFC should also take intoaccount the weights associated with each edge in the graph.To do so, we assume that the messages passed from checkto variable nodes are weight dependent. Let m ( ℓ ) c → v ( w ) denotethe message passed from check node c to variable node v along the edge with weight w in the ℓ th iteration of MPA. Let f ( ℓ ) cv ( w, m ) denote the pdf of m ( ℓ ) c → v ( w ) . In DE, we track theevolution of f ( ℓ ) cv ( w, m ) .To calculate f ( ℓ ) cv ( w, m ) , we need to find the pdf of m ( ℓ ) c → v ( w ) which is derived in (6). Due to the complexityof this equation, it is not straightforward to find f ( ℓ ) cv ( w, m ) .To address this, we propose to collect the exchanged mes-sages by random sampling, i.e., by performing Monte-Carlosimulations. In particular, we randomly generate samples of y assuming that an all zero-codeword is being sent. It isimportant to note that for DE we consider channel adaptersand therefore the modified encoder in (10). We also randomlygenerate samples of p ( ℓ − v → c ( b v ) from the pdf of m ( ℓ − v → c ,denoted by f ( ℓ − vc ( m ) .More specifically, to generate samples of p ( ℓ − v → c ( b v ) , wefirst draw a random number m ( ℓ − v → c from f ( ℓ − vc ( m ) . Then byusing (7), p ( ℓ − v → c ( b v ) is calculated for b v = 1 and b v = − .These will be inserted into (6) to calculate one sample of m ( ℓ ) c → v ( w ) . Once a large number of samples are generated, f ( ℓ ) cv ( w, m ) can be approximated by finding the histogram ofsamples of m ( ℓ ) c → v ( w ) .
2) DE Variable Node Updating Rule:
Let f ( ℓ ) vc ( m ) denotethe pdf of the m ( ℓ ) v → c , which is given in (8). As we assumethat the messages passed along the edges are independent inan asymptotic long block length regime, we can find f ( ℓ ) vc ( m ) -20 -10 0 10 20 30 40 50 60 70 8000.020.040.060.080.1 Fig. 2. The variable node densities calculated by DE at different decodingiterations for an AFC code with information length n = 8000 , weight set W = { . , . , . , . } , check node degree d c = 4 , andrate R = 0 . , when SNR = 10 dB. Solid and dashed curves respectively showDE analytical results and simulation results. as follows: f ( ℓ ) vc ( m ) = 12 d c d v − O i =1 O w ∈W ± f ( ℓ ) cv ( w, m ) , (11)where W ± = { w, − w : w ∈ W} is the set of all positiveand negative weight coefficients due to channel adapter. Thisequation follows from the fact that each variable node has d v connected check nodes and the edges have weights which arerandomly drawn from W ± . Since each edge can randomlychoose one of the d c available weight coefficients, we nor-malize the density by multiplying it with d c . Fig. 2 showsthe densities m ( ℓ ) v at different iterations of MPA for an AFCcode with weight set W = { . , . , . , . } ,degree d c = 4 , and rate R = 0 . , when SNR = 10 dB. As canbe seen in this figure, the densities are shifting towards rightwhen the iteration number increases.
3) Approximation of the bit error rate using DE:
Weassume that DE converges after a few iterations and thecheck to variable node densities converge to f ( ∞ ) cv ( w, m ) . Thevariable node density denoted by f ( ∞ ) v ( m ) is then given by: f ( ∞ ) v ( m ) = 12 d c d v O i =1 O w ∈W ± f ( ∞ ) cv ( w, m ) , (12)The bit error rate (BER) of the AFC code, denoted by ǫ ( d c , d v , W ) , can then be calculated as follows: ǫ ( d c , d v , W ) = Z −∞ f ( ∞ ) v ( x ) dx, (13)which directly follows from the assumption of all-zero infor-mation sequence and that a bit error occurs when the LLRis calculated to be negative. Fig. 3 shows the approximationof BER at different decoding iterations using the DE analysisfor an AFC code at different SNRs when n = 8000 . As canbe seen, (13) provides a tight approximation for the BER forAFC codes when the information block length n is large.IV. AFC W EIGHT S ET O PTIMISATION
In this section, we use the DE analysis and define anoptimization problem to minimize the bit error rate of the -6 -5 -4 -3 -2 -1 B it E rr o r R a t e Fig. 3. The BER of AFC at different decoding iterations with the in-formation length n = 8000 , check node degree d c = 4 , weight set W = { . , . , . , . } , and rate R = 0 . .AFC and find the optimal weight set for a given check nodedegree d c , variable node degree d v , and channel SNR. Theoptimization problem can be summarized as follows: min W ǫ ( d c , d v , W ) (14) s . t . C : w i > , for i = 1 , · · · , d c , C : d c X i =1 w i ≈ , where (14) is found via (13), condition C is to make surethat all weight coefficients are positive, and condition C itto make sure the average power of the AFC coded symbolsis 1. Traditionally, the optimization of a graph-based channelcodes involved maximizing the decoding threshold. This isequivalent to minimizing the bit error rate, which we considerin this paper for AFC. We further note that we are interested infinding the optimal weight set for a given fixed code degree d c .A more general optimization can be easily defined to jointlydetermine the optimal degree distribution function and weightset. This is however out of the scope of this paper.We use the differential evolution algorithm [38] to solve(14). The differential evolution is a simple yet powerfuloptimization tool based on the population stochastic searchtechnique. There are three main parameters that control theoptimization algorithm, which include, the scaling factor,crossover probability, and population size. The setting ofthese three parameters would directly affect the time andperformance of the optimization process. The population ofdifferential evolution may move through different region ofsearch space to find suitable candidates. Although this ap-proach is time consuming, it suffices our purpose as we donot aim for efficient optimization and instead would like tofind some good weight sets for AFC.For the case of AFC, the optimization process involved thetuning of weight-set with prefixed rate at the given SNR tominimize the BER as in (14). Table I shows the optimizedAFC weight set for different degree d c , when the rate is R AFC = 2 and SNR is 15 dB. In this optimization, weperformed 100 iterations of the density evolution and generate1000 samples of LLRs to calculated the densities in (11). As
TABLE IO
PTIMISED WEIGHT SETS OBTAINED BY (14),
FOR
AFC AT SNR = 15 D B WHEN R AFC = 2 . d c Weight Set W { }W { }W { }W { }W { }W { } TABLE IIB
ENCHMARK WEIGHT SETS IN THE LITERATURE FOR
AFC
WHEN d c = 4 .Name Weight Set ¯ V [14] { } ¯ V [14] { } ¯ V [24] { } ¯ V [32] { } for the differential evolution, we considered the population sizeof approximately 50, crossover probability of 1, and mutationfactor of 0.85.When optimizing the weight set of the AFC using (14),we need to specify the rate and SNR. In other words, theweight set obtained through (14) depend on the rate andchannel SNR. For example, when R AFC = 0 . and γ =5 dB, the optimised weight set we obtained for d c = 4 is W ∗ = { . , . , . , . } . When R AFC = 2 and γ = 15 dB, the optimised weight set we obtained for d c = 4 will be W = { . , . , . , . } .The optimized weight coefficients seem to be closer to eachother when optimizing AFC at lower SNRs and rates.Fig. 4 compares the BER performance of AFC with thesetwo weight sets when R AFC = 0 . . As can be seen, the weightset optimised at higher SNR and higher rate performs bettercompared to weight set optimised at a low SNR and low rate.This observation provide us a guideline on the input parameterselection for the optimisation process, i.e., the rate and SNRshould be as high as possible within the region of investigationin order to obtain better performance. We also compared theoptimized weight sets with with some weight sets which werepreviously used in the literature (see Table II) [14], [24], [32].Fig.4 shows the superiority of the optimized weight sets forAFC in terms of BER.V. D ESIGN OF
AFC
FOR S HORT P ACKET C OMMUNICATIONS
In this section, we evaluate the precoded AFC in the shortblock length regime. The main focus of this section wouldbe determining the precode, where we specifically focus onBCH codes for very short block lengths and LDPC codes formoderate to long block lengths.We use the normal approximation [23] developed byPolyanskiy-Poor and Verdu (PPV) as a benchmark for com-paring the performance of AFC at short block lengths. For theAWGN channel, the normal approximation for the achievablerate is given by [23]: R ≈ C − r Vn Q − ( ǫ ) + log n n , (15) SNR [dB] -4 -3 -2 -1 B it E rr o r R a t e Fig. 4. The BER of an AFC with n = 8000 , R AFC = 0 . ,and optimised weight set W (Table I), in comparison to previously de-signed weight sets in the literature (Table II). The weight set W ∗ = { . , . , . , . } was optimised for AFC at R AFC = 0 . and γ = 5 dB. where C = log (1 + γ ) is the channel capacity, γ is thechannel SNR, V = log ( e ) γ ( γ +2)2( γ +1) is the channel dispersion, ǫ is the block error rate, and Q ( x ) = √ π R ∞ x e − x dx is thestandard Q -function. A. Block Error Rate of AFC at Fixed Rate
We first investigate the block error rate (BLER) perfor-mance of precoded AFC truncated at a fixed block length.In particular, we treat AFC as a fixed-rate code in orderto investigate the achievable reliability guarantees. This isessential to understand the effect of the precode rate on theoverall performance of AFC codes.Fig. 5 shows the performance of the precoded AFC whena BCH(63,57) code is used as a precoder. As can be seenin this figure, the optimized weight set ( W in Table I)outperforms other weight sets previously designed for AFC.The improvement in terms of BLER is consistent when theAFC is operating at different overall rates. Fig. 6 shows theBLER performance of precoded AFC where a BCH(127,57)code is used as a precoder. Results are consistent with what weobserved in Fig. 5 and that the optimized weight set obtainedin this paper outperforms weight sets previously designed forAFC.Fig. 7 shows the comparison between AFC codes precodedwith different BCH codes, i.e., BCH(63,57) and BCH(127,57),where the message length is k = 57 , and at different overallrates. As can be seen, the AFC code with a low-rate precoderachieves a lower BLER. This is mainly because the lower rateprecoder is more capable of correcting residual errors that AFCis unable to recover, especially when the SNR is low. Fig. 8shows the same trend at high SNRs, where the lower rateprecoder performs better in terms of BLER at different rates.Fig. 7 also shows the performance of precoded AFC usingdifferent weight sets. It can be observed that the lower rateprecode performs better that the high rate precode when usingdifferent weight sets with different degrees. It is important tonote that when a high rate precode, i.e., BCH(63,57) is beingused, the weight set W with degree d c = 3 outperforms theother weight sets. However, when the higher rate precode, i.e.,
15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
SNR [dB] -4 -3 -2 -1 B l o c k e E rr o r R a t e Fig. 5. The BLER of a precoded AFC using weight W (Table I), incomparison to previously designed weight sets (Table II). The precoder usedhere is the a BCH(63,57) code.
15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
SNR [dB] -4 -3 -2 -1 B l o c k E rr o r R a t e Fig. 6. The BLER of a precoded AFC using weight W (Table I), incomparison to previously designed weight sets (Table II). The precoder usedhere is the BCH(127,57) code. BCH(127,57) is being used, the weight set W with degree d c = 4 outperforms the other weight sets in different overallrates. This shows that the degree of the AFC code should bechosen carefully and that depends on the precode rate, in orderto minimize the block error rate.Simulation results demonstrate that a similar trend is ob-tained when using a LDPC code as the precoder, as can beseen in Fig. 9 and Fig. 10. These results further strengthenthe statement that we made, which is AFC with a low rateprecoder outperformes AFC with the high rate precoder. B. Achievable Realised Rate
We compare the realised rates achievable by the optimisedweights versus previously designed weight sets in Table II.For all simulations, we assumed that the receiver attempts adecoding everytime it receives δ = 5 additional AFC codedsymbols. The first decoding attempts occurs when the receivercollects m = k log(1+ γ ) AFC coded symbols,where γ is thechannel SNR. The decoder sends an acknowledgment to the SNR [dB] -4 -3 -2 -1 B l o c k E rr o r R a t e Fig. 7. BLER versus SNR for precoded AFC, when BCH(63,57) andBCH(127,57) are used as precoder, at low SNRs with optimized weight W .
15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
SNR [dB] -5 -4 -3 -2 -1 B l o c k E rr o r R a t e Fig. 8. BLER versus SNR for precoded AFC, when BCH(63,57) andBCH(127,57) are used as precoder, at high SNRs with weight set W . SNR [dB] -4 -3 -2 -1 B l o c k E rr o r R a t e Fig. 9. BLER versus SNR for precoded AFC when LDPC(384,320) andLDPC(384,192) are used as precoder at low SNRs with weight W .
15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20
SNR [dB] -4 -3 -2 -1 B l o c k E rr o r R a t e Fig. 10. BLER versus SNR for precoded AFC when LDPC(384,320) andLDPC(384,192) are used as precoder at high SNRs with weight W . transmitter when the decoding succeeds and accordingly thetransmission is terminated.Results are shown in Fig. 11. It is clear that our opti-mised weight set W is superior in performance to previousAFC weight set designs, with similar degrees, i.e., similarcomplexity. In particular, for an AFC code precoded with aBCH(63,57), our optimised weight set W can achieve about4.35% and 9.68% higher realised rate than weight sets V and V , respectively, in the high SNR regime (around 20 dB),and 13.96% and 18.49% higher realized rates, respectively,in the low SNR regime (around 5 dB). When compared withbenchmark weight sets V and V , the performances of thebenchmark weight sets are quiet close with our optimisedweight set W , but our optimised weight set still demonstratedsuperiority compared to benchmark weight sets. Our optimisedweight set W can achieved about 0.84% and 3.37% higherrealised rate than weight sets V and V , respectively, in thehigh SNR regime (around 20 dB), and 1.49% and 1.66%higher realized rates, respectively, in the low SNR regime(around 5 dB).It is important to note that, in theory, the realised rateis defined for the zero-error transmission. However, due tocomputational limitations, such a realised rate cannot becalculated in practice. Thus, here we assume that the plottedrealised rates correspond to block error rates less than − ,i.e. for messages, no errors are exhibited. Furthermore,since the transmitted block length varies from one frame to thenext, we also plot the cumulative distribution function ( cdf )of the block length at different SNRs to better understandwhat latency guarantees AFC can provide. As can be seenin Fig. 12, our optimised code exhibits a smaller variance inthe block length than previously designed AFC codes. Thus,our optimised weight set can provide better latency guarantees,particularly for delay sensitive applications with little tolerancefor jitters.Rsults are shown in Fig. 13, where we consider twoprecoders a BCH(63,57) code and a BCH(127,57) code. Wechoose these two precoders such that they have the same -5 0 5 10 15 20 SNR [dB] R ea li ze d R a t e Fig. 11. The realized rates of precoded AFC using our optimised weightset W (Table I), in comparison to previously designed weight sets in theliterature (Table II). The precoder used here is the BCH(63,57) code. Block Length (n) C D F SNR= 10 dB SNR = 0 dB
Fig. 12. The cdf of the block length for a precoded AFC using our optimisedweight set W (Table I), in comparison to previously designed weight sets inthe literature (Table II). The precoder used here is the BCH(63,57) code. message length but different rates. For AFC, we use theoptimised weight set W from Table I. As can be seen, therealised rate of AFC with a lower precoder rate is higher thanan AFC with a higher rate precoder, over a wide range ofSNRs. At SNR of 20 dB, AFC with BCH(127,57) has a gap of9.57% to the PPV bound compared to AFC with BCH(63,57)which has a gap of 13.68% to the bound. At SNR of 5 dB,AFC with BCH(127,57) is closer to bound, i.e., it has a gap of7.14% to the bound compared to AFC with BCH(63,57) thathas a gap of 9.57% to the bound. Similarly, we observe in Fig.14 that the CDF of the block length has a smaller variance,i.e. steeper gradient, when the precoder rate is lower. A similarobservation was made in [26] for the case of LDPC precodedRaptor codes.The observation that lower precoder rates have the potentialto achieve higher realised rates over a wide range of SNRsis further validated in Fig. 15. In Fig. 15, we use a range ofLDPC precoders with different rates and longer block lengths.Results for these medium length blocks show the same trend -5 0 5 10 15 20 SNR [dB] R ea li ze d R a t e Fig. 13. The realized rates of an AFC precoded with a BCH code usingweight set W .Fig. 14. The cumulative distribution function of the block length of a precodedAFC with weight set W (ref. Table I) and different precoder rates. as those of short blocks in Fig. 13. Furthermore, we alsoshowed that the cdf of block length of the respective AFCwith different precoder rates at two regions of SNR in Fig.16 which demonstrated that lower precoder rate of AFC hassteeper gradient compared to higher precoder AFC, furtherstrengthen our claim that lower precoder rate have superiorperformance compared to higher precoder rate.For the sake of completion, we would now investigate theperformance of the our optimised precoded-AFC using differ-ent degrees. It is well established that the degree of the AFCcode plays a significant role in its achievable rates, i.e., thereis a trade-off between the maximum achievable rate and theallowed encoding/decoding complexity. More specifically, ina noise-free environment, the AFC code can achieve a rate of d c . However, its decoding complexity increases exponentiallywith d c as can be clearly inferred from Section III.We expect that the check node degree to be the upperbound on the maximum achievable rate when the SNR issufficiently high. When the BCH(127,57) code is used asthe precoder, the achievable rates are plotted in Fig. 17.For medium message lengths, we use the LDPC precoder SNR [dB] R ea li ze d R a t e Fig. 15. The realized rates of a precoded AFC using weight set W (ref.Table I) and different LDPC precoder rates. Block length (n) C D F SNR = 0 dBSNR =15 dB LDPC(384,192) AFC, SNR = 0 dBLDPC(384,192) AFC, SNR = 15 dBLDPC(384,320) AFC, SNR = 0 dBLDPC(384,320) AFC, SNR = 15 dB
Fig. 16. The cumulative distribution function of the block length of a precodedAFC with weight set W (ref. Table I) and different LDPC precoder rates. due to complexity constraints of the OSD decoder of BCHcode. More specifically, when the LDPC(384,192) is used asthe precoder the achievable rates are plotted in Fig. 18. Weconsider three different weight sets, W , W , and W fromTable I, with for AFC code with degree d c = 2 , d c = 4 , and d c = 6 , respectively. As can be seen in Fig. 17 and Fig. 18,the AFC code with the larger degree achieves a higher realizedrate over a wide range of SNRs. C. Threshold-based Decoder for BCH
The precoded AFC requires two decoders, i.e, the BPdecoder for AFC and the decoder for precode. The decodingprocess is carried out in a way that every time that a newset of AFC coded symbols are received, the decoder needs torun both BP and the decoder for precode. This process willstop only when the decoding succeeds. This leads to a hugecomplexity at the receiver side and accordingly the decodingtime increases dramatically. The problem becomes more chal-lenging when an OSD algorithm is being used for decoding theprecode. When a low rate BCH code is used as the precode, we SNR [dB] R ea li ze d R a t e Fig. 17. Realized rate comparison for different degree of optimised weight: W , W and W with BCH (127,57) SNR [dB] R ea li ze d R a t e Fig. 18. The realized rates of a precoded AFC using different degrees withLDPC (384,192) as a precoder. usually need a high-order OSD to be able to achieve a nearmaximum-likelihood decoding performance. The complexityhowever increases with O ( k ℓ ) , where k is the message lengthand ℓ is the order of OSD. For a code with minimum Hammingdistance d H , order ℓ = ⌈ d H / ⌉ is asymptotically optimal,i.e., it can achieve near-ML performance [30].To reduce the decoding complexity, we propose a threshold-based decoding algorithm for the precoded AFC code. Inthis modified algorithm, we pass the soft information to theprecode decoder and perform the algorithm only when theaverage reliability of the soft information is above a predefinedthreshold value. Further details of this algorithm can be foundin Algorithm 1. Details on how to find the threshold valuecan be found in [39]. It is important to note that in Step 9 ofAlgorithm 1, we check CRCs to verify whether the decodingsucceeds or not. This is because the output of the OSD isalways a valid codeword. Therefore, we need to add CRC bitsto verify wethere the output of the OSD is the transmittedcodeword or not.In order to show the efficiency of the proposed threshold-based decoder, we evaluate the number of times that we need Algorithm 1:
Threshold-based Decoding for BCH Inputs: y , G pre , δ , m , γ th initialization: m = m while (CRC check fails) do Request for additional δ AFC coded symbols m = m + δ Perform BP decoding using m AFC coded symbols Calculate the average LLR of the output of the BPdecoder, E [ | L | ] if E [ | L | ] ≥ γ th then Perform OSD using G pre and L and checkCRCs else Go to step 4 end end return ˆ b Threshold value [dB] R ea li ze d R a t e O S D Realized Rate, SNR=10dB
Fig. 19. Number of OSD instances versus the threshold SNR for the BCH-AFC code at different SNRs when BCH(63,57) is used as a precoder. to run the OSD algorithm versus the threshold SNR. Resultsare shown in Fig. 19, where the average number of OSDinstances and the realized rate are plotted versus the thresholdSNR for a precoded AFC at different channel SNRs whenBCH(63,57) was used as a precoder. As can be seen in thisfigure, the number of times that we run OSD is not verysensitive to the threshold value in high SNRs. However, thelower the threshold value, the higher the realized rate. At lowSNRs, the threshold value cannot be chosen very small asit significantly increases the number of OSD instances. Infact, one can choose a very low threshold value and get avery high realized rate, but on the other hand the complexitywould significantly increase. It is important to note that ifthe threshold SNR is chosen very low, every time that thereceiver receives a new set of AFC symbols, it will runthe OSD algorithm, no matter what the reliability of AFCdecoder outputs is. The proposed decoder can effectively limitthe number of OSD instances to almost 1, when a properthreshold value is considered with a negligible degradationin the realized rate [39]. VI. C
ONCLUSION
In this paper, we proposed a density evolution (DE) analysisframework for analog fountain codes. In the DE framework,we tracked the evolution of messages exchanged betweenthe variable and check nodes of AFC and characterized theirprobability density functions. Using the proposed framework,we defined an optimization problem to find the weight set ofAFC. Results show that for the asymptotically long messagelengths, the optimized weight sets outperform existing weightsets previously designed for AFC. We also studied the designof the precoder for AFC to optimize the performance at shortblock lengths. We particularly focused on BCH and LDPCcodes and showed via simulations that a lower rate precodercan achieve a lower block error rate under the same overallrate and also a higher realized rate over a wide range of signalto noise ratios (SNRs). We further showed that our optimizedweight sets outperform existing weight sets in the literature atboth low and high SNRs in the short block length regime. Wefurther shed lights on how to reduce the decoding complexityof precoded AFC in the rateless setting. The proposed pre-coded AFC achieves near optimal performance when the codeparameters, such as the code degree, weight set, and precoderate, are chosen properly. The proposed code can be effectivelyused for rateless transmission of short information sequences,which have applications in many mMTC scenarios. The factthat these codes do not need channel state information at thetransmitter side, makes them a strong candidate to reducingsignificant overhead associated with channel estimation andfeedback in mMTC applications.R
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