AnalysisandDesignofSystematicRatelessCodesinFH/BFSK ...downloads.hindawi.com/journals/misy/2020/9049284.pdf · Received 19 October 2019; Revised 25 December 2019; Accepted 31 December

Research ArticleAnalysis and Design of Systematic Rateless Codes in FHBFSKSystem with Interference

Jingke Dai Peng Zhao Xiao Chen and Fenggan Zhang

School of Military Operational Support Rocket Force University of Engineering Xirsquoan 710025 China

Correspondence should be addressed to Jingke Dai djk029163com

Received 19 October 2019 Revised 25 December 2019 Accepted 31 December 2019 Published 10 February 2020

Academic Editor Ramon Aguero

Copyright copy 2020 Jingke Dai et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)e asymptotic analysis of systematic rateless codes in frequency hopping (FH) systems with interference is first provided usingdiscretized density evolution (DDE) and compared with the traditional fixed-rate scheme A simplified analysis with Gaussianassumption of initial message is proposed in the worst case of interference which has much lower complexity and provides a veryclose result to DDE Based on this simplified analysis the linear programming is employed to design rateless codes and thesimulation results on partial-band interference channels show that the optimized codes have more powerful antijammingperformance than the codes originally designed for conventional systems

1 Introduction

Frequency hopping (FH) systems have been extensivelyemployed to combat interference in commercial and mili-tary communications [1] To further enhance the antijam-ming capability error-correction codes have also beenwidely used in FH systems like convolutional codes [2]ReedndashSolomon codes [3] Turbo codes [4 5] and low-density parity-check (LDPC) codes [6 7] and all of them arerate-fixed Since the interference on channels may changerandomly and dramatically from time to time the requiredcode rate or signal power must be adjusted accordingly foruninterrupted communications Hence rateless codeswhich can automatically adapt their practical rates accordingto channel conditions are considered as an alternativescheme for FH systems with interference

)e first practical realization of rateless codes areLuby transform (LT) codes [8] the extension of whichare Raptor codes with lower complexity [9] )ey areoriginally designed for binary erasure channels (BEC)but also implemented on noisy channels [10] )e sys-tematic versions of rateless codes have received extensiveattention because of their efficient encoding SystematicLT (SLT) codes and systematic Raptor (SR) codes on BECare proposed in [9 11] respectively Normally on noisy

channels SLT codes are generated in a direct way [12]ie the original information symbols are first trans-mitted followed by LT-coded symbols If an inner SLTcode is serially concatenated with an outer high-rateLDPC code the resulting code can also be referred as theSR code [13] Kite codes are a special class of prefixrateless LDPC codes [14] and they are combined with themultilevel coding scheme to improve transmission re-liability [15] Online fountain codes are proposed in [16]where an optimal encoding strategy is found efficientlybased on the instantaneous decoding state A modifiedonline fountain encoding scheme is introduced in [17] toprovide unequal error protection and a theoreticalframework is proposed in [18] to analyze online fountaincodes more accurately

)e rateless codes have been applied to FH systems inexisting works )e performance of Raptor coded FH sys-tems is investigated under partial-band interference (PBI)and full-band interference (FBI the case of PBI with in-terference bandwidth fraction ρ 1) in [19 20] )e LT-coded differential frequency hopping system with PBI isdiscussed in [21] All the results in above works are obtainedby computer simulations ie the asymptotic performance isnot given Also those systems just use the classic codes withdegree distributions originally designed for BEC ie they

HindawiMobile Information SystemsVolume 2020 Article ID 9049284 8 pageshttpsdoiorg10115520209049284

have not been optimized for FH systems with interference)e authors of [22] use a probability-transfer method todecrease the Raptor codesrsquo average degree weight but itneeds many attempts and the resulting code may not havegood performance

)e asymptotic performance of rateless codes isusually studied by discretized density evolution (DDE)[13 23] and Gaussian approximation (GA) methods[24ndash26] As well known the former provides accurateresults but needs much higher computational complexitywhereas the latter uses a one-dimensional (1-D) analysisfor faster calculation without much sacrifice in accuracyHowever most existing works focus on the coherentbinary phase shift keying (BPSK) system on additive whiteGaussian noise (AWGN) channels where the initialmessage of the log-likelihood ratio (LLR) is symmetricallyGaussian distributed [25] indicating that the GAmethodsmay work very well In the FH system binary frequencyshift keying (BFSK) and noncoherent (NC) demodulationare more employed which means that the initial LLR is nolonger Gaussian distributed especially for considerationof interference on channels Hence DDE is a better choiceto obtain the asymptotic performance of the rateless codein the FHBFSK system with interference FurthermoreDDE has been successfully combined with the differentialevolution (DE) algorithm to design LDPC codes [7] andSR codes [13] but it is a nonlinear optimization methodwhich costs much computation and spends a long searchtime

In this paper we investigate the asymptotic perfor-mance and distribution optimization of systematic ratelesscodes in FHBFSK systems with PBI )e main contribu-tions are summarized as follows Firstly the asymptotic bit-error-rate (BER) performance and decoding thresholds ofSLT codes and SR codes are achieved using DDE and theadvantages of rateless codes over fixed-rate codes in the FHsystem are analyzed extensively Secondly since the DDE-DE method is so complicated we propose a properlysimplified method which assumes the initial LLR of theFHBFSK system as Gaussian distributed in the worst caseof interference With this GA-LLR the 1-D analysis basedon the extrinsic information transfer (EXIT) chart isemployed to analyze the asymptotic performance which isvery close to the results provided by DDE )irdly thelinear programming (LP) algorithm [25] which is a linearoptimization method and then much faster than DE iscombined with the former 1-D analysis to design ratelesscodes on FBI ie PBI with ρ 1 Both the asymptoticanalysis and simulation results show that the optimizedcodes provide more powerful antijamming capability thanthe existing codes in the range of 0lt ρle 1

2 System Model

)e FH communication system model is depicted in Figure 1)e original information sequence s s1 s2 sK1113864 1113865 is pre-coded using a high-rate LDPC code to produce the interme-diate sequence e e1 e2 eKprime1113864 1113865 where the outer code rateis Ro KKprime )e intermediate sequence is further SLT

encoded to obtain the symbols of Raptor codes denoted asc c1 c2 cN1113864 1113865 where the inner code rate is Ri KprimeNOmitting the subscript the coded symbol c isin 0 1 is mod-ulated to the BFSK symbol with energyEs Eb middot R whereEb isthe bit energy and R RiRo is the overall rate of SR codes)emodulated symbol is transmitted over a frequency slot de-termined by the hopping pattern andwe only consider the caseof one symbol per hop in this paper

)e transmitted signal experiences full-band AWGNas well as the partial-band interference which is modelledas an on-off jammer and evenly distributes its power overa fraction ρ isin (0 1] of the frequency range We set thesingle-side power spectrum density (PSD) of noise andinterference is N0 and Njρ respectively As a result thereare essentially two states on this channel jammed andunjammed which are denoted as J 1 and J 0 re-spectively At the receiver the signal is assumed to beperfectly dehopped and the two branchesrsquo outputs of thenoncoherent (square-law) demodulator is denoted asy1 y01113864 1113865

Without loss of generality we assume the transmittedsymbol c 1 in the current hop and then the conditionalPDFs of y1 y01113864 1113865 are given by [7]

p y11113868111386811138681113868 c 11113872 1113873

1Nl

exp minusy1 + Es

Nl

1113888 1113889I02

y1Es

1113968

Nl

1113888 1113889 y1 gt 0

p y01113868111386811138681113868 c 11113872 1113873

1Nl

exp minusy0

Nl

1113888 1113889 y0 gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where Nl N0 + J(Njρ) is determined by the channelstate and I0(x) is the modified zero-order Bessel functionof the first kind Assuming each hop is independent ofeach other and the channel side information is knownwe have the optimal log-likelihood ratio calculated asin [6]

z logP c 1 | y1 y0( 1113857

P c 0 | y1 y0( 1113857 log

I0 2y1Es

1113968Nl1113872 1113873

I0 2y0Es

1113968Nl1113872 1113873

(2)

)e decoding of inner SLT codes is followed bydecoding of outer LDPC codes )e SPA is implementedin the inner decoder according to the related bipartitegraph which is constructed using intermediate symbolsand output symbols as variable nodes (VNs) and checknodes (CNs) )e message update rules between VNs andCNs are given by [25]

Outerencoder

Innerencoder

BFSK Hopper

PBI AWGN

DehopperNCdemod

Innerdecoder

Outerdecoder

Figure 1 Rateless coded FHBFSK system model

2 Mobile Information Systems

u(l)mn 2 tanhminus 1 tanh zKprime+n2( 1113857 1113945

kisin Sn m

tanh v(l)nk21113872 1113873

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎣ (3)

v(l+1)nm zm + 1113944

kisin Sm n

u(l)mk

(4)

where u(l)mn and v(l)

nm denote the message passed from the n-thcheck node to the m-th variable node and the messagepassed in the opposite direction at the l-th decoding iterationand Sn1113864 1113865m represents the set of all neighbours of the n-thnode except for them-th node After the maximum iterationlmax the decision message v

(lmax)m zm + 1113936kisin Sm u

(lmaxminus 1)

mk isprovided as the initial LLR to decode outer LDPC codes

3 Asymptotic Analysis

31 Initial LLR on PBI Channels From (1) and (2) we knowthat it is hard to get close-form density expression of theinitial LLR message due to Bessel functions unlike the caseof the coherent BPSK system )us we use the probabilitymass function (PMF) instead [13] Firstly the continuousvariables y1 y01113864 1113865 is quantized to 1113954y1 1113954y01113864 1113865 with a quanti-zation interval Δ1 Secondly calculate the quantized initialmessage by a two-input function expressed as

1113954z F 1113954y1 1113954y0( 1113857

I log I0 21113954y1Es

1113969

Nl1113874 1113875I0 21113954y0Es

1113969

Nl1113874 11138751113874 11138751113874 1113875(5)

whereI is the ldquoquantization operatorrdquo ensuring that 1113954z takesvalues in a discrete set zmin zmin + Δ2 zmin +1113864

2Δ2 zmin + kΔ2 zmax with a quantization intervalΔ2 Finally calculate the probability of 1113954z zmin + kΔ2 asgiven by

p1113954zk 1113944

(ij) kΔF(iΔjΔ)p1113954y1

[i]p1113954y0[j]

(6)

where the probability masses p 1113954y1[i] and p 1113954y0

[j] are thequantized results from equation (1) Finally the assembleof p1113954zk

gives the PMF of 1113954z Note that Nl in (5) has two possiblevalues thus p1113954zk

also has two possible values which gives thePMF of 1113954z denoted as pzJ and pzNJ corresponding to jammedand unjammed states respectively)e unconditional PMF of1113954z is written as

pz ρpzJ +(1 minus ρ)pzNJ (7)

Setting Δ1 001 Δ2 005 zmin minus 20 and zmax 40the PMF of initial LLR under the channel with symbolsignal-to-noise ratio (SNR) EsN0 10 dB symbol signal-to-jamming ratio (SJR) EsNj 5 dB and ρ 07 is depictedin Figure 2 It is seen that the PMF has two peaks caused bydifferent channel states and is no longer Gaussian distrib-uted which implies that the Gaussian approximation-basedmethod may not be employed to investigate the asymptoticperformance of rateless codes on PBI channels (for ρlt 1 atleast) Hence we consider the DDE which does not requireGaussian initial LLR

32 Asymptotic Performance of SLT and SR Codes Wedefine the generation matrix of inner SLT codes asGSLT [IKprimeGLT] where IKprime is an identity matrix having a sizeof Kprime times Kprime and GLT of size Kprime times (N minus Kprime) is the generationmatrix of the conventional LT code [25] )e overhead of theinner SLTcode is defined by ε (N minus Kprime)Kprime which is relatedwith the code rates as ε 1Ri minus 1 RoR minus 1

)e degree distributions related with the bipartite graphcorresponding to GLT is defined as follows Using polyno-mial notations the check node degree distributions iswritten compactly as Ω(x) 1113936

dc

j1Ωjxj where dc is the

maximum degree )e corresponding edge degree distri-bution is calculated by ω(x) 1113936

dc

j1ωjxjminus 1 Ωprime(x)Ωprime(1)

and the average degree of CNs is given by β Ωprime(1) )eaverage degree of VNs is denoted by α and then the variablenodes and edge degrees are both assumed to be Poissondistributed as Λ(x) eα(xminus 1) and λ(x) eα(xminus 1) respectively[25] However the power series can also be truncated toobtain polynomials that is very close to the exponential ieΛ(x) 1113936

dv

i1Λi(α)xi and λ(x) 1113936dv

i1λi(α)ximinus 1 where dv islarge enough to ensure Λ(1) asymp 1

We denote the PMF of the message passed fromCNs andVNs by pu and pv respectively )e updating rules of CNsand VNs are summarized as follows

pu 1113944

dc

j1ωj R pzR pvR pv R( 1113857( 1113857( 1113857

1113944

dc

j1ωjR pzR

jminus 1pv1113872 1113873

(8)

pv 1113944

dv

i1λi pz otimes otimes

iminus 1pu1113874 11138751113876 1113877 (9)

whereR is a two-input operator defined in [23] to calculate thePMF of quantized message emanating from check nodes andotimes denotes convolution calculation )e DDE of outer LDPCcodes is similar to the above rules only that the channelmessage PMF pz is removed at the CNs)e detail of DDE on

002

004

006

008

01

012

014

016

018

pz

0ndash20 ndash10 0 10 20 30 40

z

Figure 2 PMF of initial LLR in the FHBFSK system withEsN0 10 dB EsNj 5 dB and ρ 07

Mobile Information Systems 3

the SR code is referred to [13] and here we just need to replacethe initial LLR of the BPSK system by equation (7)

)e asymptotic BER performance of SLT codes and SRcodes obtained by DDE under different PBI bandwidthfraction ρ is shown in Figure 3 where SNR is fixed to 10 dBand the outer code is fixed to a 098-rate regular (4 200)-LDPC code)e degree distribution of the inner code is fixedto the optimized rate 12 results in [13] which is rewritten as

Ω1(x) 000477x + 026101x2

+ 009240x3

+ 006913x8

+ 051223x9

+ 006046x60

(10)

In Figure 3(a) the overall code rate is fixed to 12 and SJR isvariable whereas in Figure 3(b) SJR is fixed to 5dB and theoverhead of the inner code ε (then the code rate R) is variableIn the range of small SJR or overhead the performance of SLTcodes and SR codes is very close to each other indicating thatthe outer code cannot correct residual errors from the innercode )e BER of the SLT code decreases as either SJR oroverhead increases but high error floors are observed

Once the BER of the inner SLT code achieves a criticalBER which is firstly introduced in [23] the BER of the SRcode decreases very sharply It is easy to calculate the criticalBER using DDE [23] which is equal to 00039 for the 098-rateregular LDPC code In addition the case of ρ 1 has theworst performance where PBI has actually become the full-band interference

33 Comparison of Rateless Codes and Fixed-Rate CodesTo compare the rateless codes and conventional fixed-ratecodes in the FH system their decoding thresholds are in-vestigated under different PBI bandwidth fractions ρ FromFigure 3(b) we know the BERs of SR codes tend to zero asthe inner BER achieves critical BER and the correspondingoverhead can be referred as the decoding thresholdAccording to the relation between overhead and code ratepresented in Section 2 we plot the rate thresholds of SRcodes for different ρ with various EsNj in Figure 4 )eclassic 12-rate regular (3 6) LDPC is taken as the fixed-ratecode for comparison and the SJR-decoding thresholds as-sociated with the right axis are also depicted in Figure 4

It is seen that the SJR-threshold of the LDPC code growsas ρ increases and the worst case appears at ρ 1 corre-sponding to the threshold of 571 dB As ρ may changerandomly over time the SJR of the fixed-rate coded FHsystem must be set to be larger than the threshold of theworst case which in essence is a waste of power or rate forthe cases of ρlt 1 However Raptor codes can ldquoautomati-callyrdquo choose the proper rates for different ρ eg under theSJR of 571 dB the rate thresholds of Raptor codes are 074and 051 for ρ 01 and ρ 1 respectively Meanwhile theRaptor code is more adaptable to environment with dra-matic SJR changes When SJR decreases eg to 4 dB theRaptor can still work as long as we continue to send codedsymbols (decreasing actual rate) but the specific fixed-ratecode cannot When SJR increases eg to 7 dB the Raptorcode can work with a higher rate than the 12-rate LDPC

code In general rateless codes are more flexible and morerobust than the fixed-rate codes on PBI channels

4 Simplified Analysis under FBI

41 Gaussian Assumption of Initial LLR Although the as-ymptotic BER and decoding thresholds have been ob-tained by DDE the employed degree distribution isoriginally designed for the coherent BPSK system whichmeans that stronger antijamming capability may beachieved if we can optimize the degree distribution for thenoncoherent FH system )e conventional methodcombined with DDE is differential evolution but it has apretty high complexity and long search time )e linearprogramming method can also be used to design RaptorLT codes and it is much simpler and faster than DE Butthe LP method is usually combined with the one-di-mensional analysis method eg EXIT chart which re-quires the initial LLR being Gaussian distributed

From Figure 2 we know that LLR on the PBI channel isnot Gaussian but we find that the PMF of initial LLR in theFHBFSK system can be approximated by a symmetricGaussian density when ρ 1 which happened to be theworst case of interference Using (7) we have pz pzJ atthis time and the mean of z denoted by mz can be easilycalculated based on pz We approximate pz by a Gaussiandensity with mean mz and variance 2mz and plot the actualPMF as well as the approximated density shown in Figure 5)e curves in one pair are very close to each other whichindicate that the one-dimensional analysis could beemployed to investigate asymptotic performance and befurther combined with LP to design codes

42 One-Dimensional Analysis Since the initial LLR isapproximately Gaussian distributed when ρ 1 the EXITchart can be implemented to analyze the asymptotic per-formance of the inner SLT codes Assuming the all-zerocodeword is transmitted the mutual information of z iscomputed by [24] as follows

J(σ) 1 minus12π

radicσ

1113946infin

minus infinexp minus

ξ minus σ22( 11138572

2σ21113890 1113891log2 1 + e

minus ξ1113872 1113873dξ

(11)

where J(σ) is a monotonically increasing function and hasan inverse function σ Jminus 1(I) Both J(middot) and Jminus 1(middot) do nothave close-form expressions but they can be closely ap-proximated as suggested in [27]

From (4) we can calculate the mutual information ofVNs at the l-th iteration as

I(l)EV I

(lminus 1)EC1113960 1113961 1113944

dv

i1λi(α)J

σ2ch1 +(i minus 1)σ2V1113969

1113874 1113875 (12)

where σ2ch1 2mz is the variance of initial LLR and σ2V

Jminus 1[I(lminus 1)EC ]1113966 1113967

2is the message variance of CNs in the previous

iteration)emutual information outputted by CNs at the l-th iteration is given by [24] as follows


I(l)EC I

(l)EV1113872 1113873 1113944

dc

j1ωj 1 minus J

σ2ch2 +(j minus 1)σ2C1113969

1113874 11138751113876 1113877 (13)

where σch2 Jminus 1[1 minus J(2mz

1113968)] and σC Jminus 1[1 minus I

(l)EV]

Combining (12) and (13) we can get the iteration expressionof I

(l)EC as the function of I

(lminus 1)EC σ and α as given by

I(l)EC T I

(lminus 1)EC σ α1113872 1113873

1113944

dc

j1ωj 1 minus J

σ2ch2 +(j minus 1) Jminus 1 1 minus I(l)EV I

(lminus 1)EC1113872 11138731113872 11138731113872 1113873

21113970

1113888 11138891113890 1113891

(14)

At the end of iteration lmax the inner decoder outputs theextrinsic information provided to the outer decoder as givenby

Iext 1113944

dv

i1ΛiJ

4σ2 + i Jminus 1 Ilmax( )

EC1113874 11138751113874 11138752

1113971

⎛⎝ ⎞⎠ (15)

and the corresponding error probability is calculated as

PSLTe 1113944

dv

i1Λi(α)Q


EC1113874 11138751113874 111387524

1113971

⎡⎢⎢⎣ ⎤⎥⎥⎦ (16)

where the function Q(x) 12π

radic1113938

+infinx

eminus t22dtUsing the above method and setting EsN0 10 dB and

EsNj 4 dB we depict the asymptotic BERs of the inner

EsN0 = 10dB EsNj = 4dB

EsN0 = 10dB EsNj = 7dB EsN0 = 20dB EsNj = 10dB

z

0

005

01

015

02

025

ndash25 ndash20 ndash15 ndash10 ndash 5 0 5 10 15 20 25

Actual PMFApproximated PDF

p(z)

Figure 5 Actual PMF of initial LLR and their Gaussian approx-imated PDF under ρ 1

Raptor EsNj = 7dBRaptor EsNj = 571dB

Raptor EsNj = 4dBLDPC R = 12

PBI bandwidth fraction (ρ)

Cod

e rat

e (R)

035

04

045

05

055

06

065

07

075

0

1

2

3

4

5

6

7

8

01 02 03 04 05 06 07 08 09 1

E sN

j (dB

)

Figure 4 Rate thresholds of Raptor codes and SJR-thresholds ofLDPC codes for different ρ

100 1 2 3 4 5 6 7 8 9EsNj (dB)

ρ = 07ρ = 10 ρ = 05

Critical BER

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100A

sym

ptot

ic B

ER

(a)

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Asy

mpt

otic

BER

05 06 07 08 09 1 11 12 13 14 15Overhead of inner codes (ε)

ρ = 07ρ = 10 ρ = 05

Critical BER

(b)

Figure 3 Asymptotic BER of SLTcodes and SR codes with EsN0 10 dB)e dotted lines are BERs obtained after inner decoding and thesolid lines are those after outer decoding (a) R 12 (b) EsNj 5 dB


SLTcode ofΩ1(x) in Figure 6 which is represented by ldquoEXITrdquoAlso shown are those achieved byDDE and the simulated BERsof SLT codes with finite length Kprime equal to 2000 4000 and10000 respectively We observe that the EXIT curve is veryclose to the DDE curve and both of them can be taken as theasymptotic performance of simulation results indicating thatthe proposedGaussian assumption of initial LLR under ρ 1 isvery effective In Figure 6 the critical BER of the outer codes isalso presented and its intersection points with asymptoticcurves correspond to the decoding thresholds of Raptor codesIt is seen that the threshold of EXITis lower thanDDEwhich isopposite in the ordinary BPSK system [13] )is is because theinitial LLR of the FHBFSK system is assumed to be Gaussianwhich is different from the real situation (see Figure 5) but theinitial LLR of the ordinary BPSK system is really Gaussian andthe gap between EXIT and DDE is caused by Gaussian as-sumption of the message from CNs )ough the asymptoticperformance by EXIT is slightly different from that by DDE itcan still be employed to design codes with good performance

5 Code Design and Simulation Results

Technically the optimal distributions of inner SLTcodes aredifferent for different ρ but it is unrealistic for practicalcommunications to switch distributions in real timeaccording to the current channel conditions Hence theoptimized distributions for the worst case of interferencenaturally are a good choice for all situations Besides we arelucky that the worst case of interference occurs at ρ 1 sothat the PMF of initial LLR can be approximated by aGaussian density and the EXIT analysis can be combinedwith the standard LP method which is described by thefollowing model

min 1113944

dc

i1

ωj

j

st TIlminus 1EC σ αgt I

lminus 1EC I

lminus 1EC isin 0 I

maxEC1113960 1113961

l 1 2 L

1113944

dc

i1ωj 1

ωj ge 0 j 1 dc

(17)

In the optimization model the design objective is tominimize the overhead ε αβ α1113936

dc

j1ωjj )e second andthird constraints ensure the distributions validity According to(14) the first constraint ensures that the tracking variableconverges to a fixed value Imax

EC in iterationsWe fix the SNR as 10dB and set the critical BER as 00039

and search the good distributions of SLTcodes using LP underdifferent SJR )e resulting degree distributions and corre-sponding overhead thresholds are listed in Table 1 in which εE

denotes the required overhead of inner codes achieving criticalBER by the EXIT method and εD denotes the required over-head by the DDE method We can see that the overheadthresholds obtained by two methods are very close to eachother confirming that the optimized codes indeed have goodperformance Specifically εE εD for SJR 6dB andSJR 7dB is not a surprising result since the approximatedinitial LLR is almost the same with the real one (see Figure 5)

It is noted that all the optimized distributions are obtainedwith condition ρ 1 To investigate the performance of op-timized distributions under ρlt 1 we use DDE to get theoverhead thresholds of optimized codes and code with Ω1(x)

under SJR 4dB and SJR 7 dB which are compared inFigure 7 When SJR 7 dB the optimized code has smallerthresholds than Ω1(x) for all ρ At the point of ρ 1 theoverhead thresholds of two codes are 053 and 071 respec-tively If the outer code rate is fixed to 098 the correspondingoverall rate of the optimized code increases about 117compared to the code with Ω1(x) When SJR 4dB theoptimized code also has better performance than the code withΩ1(x) for large ρ but they are very close when ρ is smallConsidering that the PBI mainly works on the range of large ρwe say that our optimized codes have more powerful anti-jamming capacity than the traditional code

We apply the optimized degree distribution to the codeswith information length K 4000 under serious interfer-ence (SJR 4 dB) )e outer code is set as the 098-rateregular LDPC code and SNR is fixed to 10 dB)e simulatedBERs of Raptor codes in the FHBFSK system with ρ 07and ρ 1 are plotted in Figure 8 As shown in this figure theoptimized code provides better performance than theexisting code for both PBI and FBI conditions verifying theeffectiveness of our design method

15 16 17 18 19 242 21 22 23 25Overhead

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

BER

Kprime = 2000Kprime = 4000Kprime = 10000

EXTDDECritical BER

Figure 6 )e comparison of asymptotic performance and sim-ulation results of SLT codes


6 Conclusions

)e asymptotic performance of systematic rateless codes inthe FHBFSK system with PBI has been obtained andcompared with the fixed-rate LDPC codes )e rateless

codes are more flexible and more robust than fixed-ratecodes on PBI channels )e worst case of interference for therateless-coded FHBFSK system usually appears at the pointof ρ 1 where PBI become to FBI Under FBI the initial LLRis assumed to be Gaussian distributed and then the 1-Danalysis based on the EXIT chart is employed to investigatethe asymptotic performance which is very close to the resultby DDE )e code degree distributions are designed usinglinear programming which is combined with the 1-Danalysis On PBI channels the optimized codes in the FHBFSK system provide outstanding antijamming perfor-mance compared with the code originally designed for theconventional BPSK system In addition the coding schemeof the FH system in this paper concentrates on the SR codesand some novel rateless codes such as Kite codes and onlinefountain codes will be considered in our future works

Data Availability

)e numerical and simulation data used to support thefindings of this study are included within the supplementaryinformation file

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant no 61501469) and the ChinaPostdoctoral Science Foundation (Grant no 2015M572782)

References

[1] C Kim J S No J Park et al ldquoAnti-jamming partially regularLDPC codes for follower jamming with Rayleigh block fadingin frequency hopping spread spectrumrdquo in Proceedings of theInternational Conference on Information amp CommunicationTechnology Convergence pp 583ndash587 Jeju South KoreaOctober 2016

[2] J S Lee R H French and L E Miller ldquoError-correctingcodes and nonlinear diversity combining against worst casepartial-band noise jamming of frequency-hopping MFSKsystemsrdquo IEEE Transactions on Communications vol 36no 4 pp 471ndash478 1988

[3] M Pursley and W Stark ldquoPerformance of reed-solomoncoded frequency-hop spread-spectrum communications in

Table 1 )e optimized degree distributions of inner SLT codes with SNR 10 dB

SJR4 dB 5 dB 6 dB 7 dB

Ωopt1(x) α 135 Ωopt2(x) α 11 Ωopt3(x) α 9 Ωopt4(x) α 8

Ω5 059654 Ω6 053805 Ω7 029357 Ω9 033414Ω6 028390 Ω7 036372 Ω8 061311 Ω10 058986Ω30 009375 Ω44 007748 Ω59 006836 Ω83 001170Ω87 001826 Ω88 000901 Ω60 002496 Ω84 006430Ω88 000755 Ω89 001174

εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old

EsNj = 4dB Ω1(x)EsNj = 4dB Ωopt1(x)


1

Figure 7 Overhead thresholds of inner SLT codes for different ρ

1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)

Figure 8 Simulated BER of SR codes versus overhead underSJR 4 dB and SNR 10 dB


partial-band interferencerdquo IEEE Transactions on Communi-cations vol 33 no 8 pp 767ndash774 1985

[4] U-C Fiebig and P Robertson ldquoSoft-decision and erasuredecoding in fast frequency-hopping systems with convolu-tional turbo and Reed-Solomon codesrdquo IEEE Transactionson Communications vol 47 no 11 pp 1646ndash1654 1999

[5] D Torrieri S Cheng and M C Valenti Robust FrequencyHopping for Interference and Fading Channels in Proceedingsof the IEEE International Conference on Communicationspp 1ndash7 Glasgow UK June 2007

[6] X C Wu X Zhao You and S Li ldquoRobust diversity-combingreceivers for LDPC coded FFH-SS with partial-band inter-ferencerdquo IEEE Communications Letters vol 11 no 7pp 613ndash615 2007

[7] J Dai Y Jing and M Yao ldquoAnalysis of LDPC code in the FHsystem with partial-band interferencerdquo IET Communicationsvol 11 no 17 pp 2585ndash2595 2017

[8] M Luby ldquoLTcodesrdquo in Proceedings of the 43rd Symposium onFoundations of Computer Science ser FOCS rsquo02 pp 271ndash280Washington DC USA 2002

[9] A Shokrollahi ldquoRaptor codesrdquo IEEE Transactions on Infor-mation eory vol 52 no 6 pp 2551ndash2567 2006

[10] B Sivasubramanian and H Leib ldquoFixed-rate raptor codesover Rician fading channelsrdquo IEEE Transactions On VehicularTechnology vol 57 no 6 pp 3905ndash3911 2008

[11] X Yuan and L Ping ldquoOn systematic LT codesrdquo IEEECommunications Letters vol 12 no 9 pp 681ndash683 2008

[12] T D Nguyen L L Yang and L Hanzo ldquoSystematic Lubytransform codes and their soft decodingrdquo in Proceedings of theIEEE Workshop on Signal Processing Systems pp 67ndash72Shanghai China October 2007

[13] A Kharel and L Cao ldquoAsymptotic analysis and optimizationdesign of physical layer systematic rateless codesrdquo in Pro-ceedings of 15th IEEE Consumer Communications and Net-working Conference pp 1ndash6 Las Vegas NV USA January2018

[14] B Bai B Bai and X Ma ldquoSimple rateless error-correctingcodes for fading channelsrdquo Science China Information Sci-ences vol 55 no 10 pp 2194ndash2206 2012

[15] D Jia C Shangguan Z Fei et al ldquoMultilevel coding tech-nology based on Kite code in AWGN channelrdquo Transactionsof Beijing Institute of Technology vol 37 no 4 pp 360ndash3642017

[16] Y Cassuto and A Shokrollahi ldquoOnline fountain codes withlow overheadrdquo IEEE Transactions on Information eoryvol 61 no 6 pp 3137ndash3149 2015

[17] J Huang Z Fei C Cao M Xiao and D Jia ldquoOn-linefountain codes with unequal error protectionrdquo IEEE Com-munications Letters vol 21 no 6 pp 1225ndash1228 2017

[18] J Huang Z Fei C Cao M Xiao and D Jia ldquoPerformanceanalysis and improvement of online fountain codesrdquo IEEETransactions on Communications vol 66 no 12 pp 5916ndash5926 2018

[19] F Gao X Zeng and X Bu ldquoResearch of anti-interferenceperformance of digital fountain codes in Frequency hoppingcommunicationsrdquo Transactions of Beijing Institute of Tech-nology vol 32 no 4 pp 420ndash424 2012

[20] X Bu P Li Y Wang et al ldquoTransmission design and per-formance of a strong anti-jamming frequency hopping systembased on fountain codesrdquo Transactions of Beijing Institute ofTechnology vol 34 no 11 pp 1186ndash1190 2014

[21] W Zhu B Yi and L Gan ldquoPerformance research of fountain-DFH concatenated coding systems over AWGN with partial-

band noise jammingrdquo Systems Engineering and Electronicsvol 38 no 3 pp 665ndash671 2016

[22] F Gao X Zeng and X Bu ldquoImproved scheme for short-length Raptor codes based on frequency hopping commu-nicationrdquo Transactions of Beijing Institute of Technologyvol 33 no 4 pp 403ndash407 2013

[23] A Kharel and L Cao ldquoAnalysis and design of physical layerraptor codesrdquo IEEE Communications Letters vol 22 no 3pp 450ndash453 2018

[24] I Hussain X Xiao and L K Rasmussen ldquoError floor analysisof LT codes over the additive white Gaussian noise channelrdquoin Proceedings of IEEE Global Telecommunications Conferencepp 1ndash5 Kathmandu Nepal 2011

[25] S Xu and D Xu ldquoOptimization design and asymptoticanalysis of systematic Luby transform codes over BIAWGNchannelsrdquo IEEE Transactions on Communications vol 64no 8 pp 3160ndash3168 2016

[26] D Deng D Xu and S Xu ldquoOptimisation design of systematicLT codes over AWGN multiple access channelrdquo IET Com-munications vol 12 no 11 pp 1351ndash1358 2018

[27] S T Brink G Kramer and A Ashikhmin ldquoDesign of low-density parity-check codes for modulation and detectionrdquoIEEE Transactions on Communications vol 52 no 4pp 670ndash678 2004


have not been optimized for FH systems with interference)e authors of [22] use a probability-transfer method todecrease the Raptor codesrsquo average degree weight but itneeds many attempts and the resulting code may not havegood performance

)e asymptotic performance of rateless codes isusually studied by discretized density evolution (DDE)[13 23] and Gaussian approximation (GA) methods[24ndash26] As well known the former provides accurateresults but needs much higher computational complexitywhereas the latter uses a one-dimensional (1-D) analysisfor faster calculation without much sacrifice in accuracyHowever most existing works focus on the coherentbinary phase shift keying (BPSK) system on additive whiteGaussian noise (AWGN) channels where the initialmessage of the log-likelihood ratio (LLR) is symmetricallyGaussian distributed [25] indicating that the GAmethodsmay work very well In the FH system binary frequencyshift keying (BFSK) and noncoherent (NC) demodulationare more employed which means that the initial LLR is nolonger Gaussian distributed especially for considerationof interference on channels Hence DDE is a better choiceto obtain the asymptotic performance of the rateless codein the FHBFSK system with interference FurthermoreDDE has been successfully combined with the differentialevolution (DE) algorithm to design LDPC codes [7] andSR codes [13] but it is a nonlinear optimization methodwhich costs much computation and spends a long searchtime

In this paper we investigate the asymptotic perfor-mance and distribution optimization of systematic ratelesscodes in FHBFSK systems with PBI )e main contribu-tions are summarized as follows Firstly the asymptotic bit-error-rate (BER) performance and decoding thresholds ofSLT codes and SR codes are achieved using DDE and theadvantages of rateless codes over fixed-rate codes in the FHsystem are analyzed extensively Secondly since the DDE-DE method is so complicated we propose a properlysimplified method which assumes the initial LLR of theFHBFSK system as Gaussian distributed in the worst caseof interference With this GA-LLR the 1-D analysis basedon the extrinsic information transfer (EXIT) chart isemployed to analyze the asymptotic performance which isvery close to the results provided by DDE )irdly thelinear programming (LP) algorithm [25] which is a linearoptimization method and then much faster than DE iscombined with the former 1-D analysis to design ratelesscodes on FBI ie PBI with ρ 1 Both the asymptoticanalysis and simulation results show that the optimizedcodes provide more powerful antijamming capability thanthe existing codes in the range of 0lt ρle 1

2 System Model

)e FH communication system model is depicted in Figure 1)e original information sequence s s1 s2 sK1113864 1113865 is pre-coded using a high-rate LDPC code to produce the interme-diate sequence e e1 e2 eKprime1113864 1113865 where the outer code rateis Ro KKprime )e intermediate sequence is further SLT

encoded to obtain the symbols of Raptor codes denoted asc c1 c2 cN1113864 1113865 where the inner code rate is Ri KprimeNOmitting the subscript the coded symbol c isin 0 1 is mod-ulated to the BFSK symbol with energyEs Eb middot R whereEb isthe bit energy and R RiRo is the overall rate of SR codes)emodulated symbol is transmitted over a frequency slot de-termined by the hopping pattern andwe only consider the caseof one symbol per hop in this paper

)e transmitted signal experiences full-band AWGNas well as the partial-band interference which is modelledas an on-off jammer and evenly distributes its power overa fraction ρ isin (0 1] of the frequency range We set thesingle-side power spectrum density (PSD) of noise andinterference is N0 and Njρ respectively As a result thereare essentially two states on this channel jammed andunjammed which are denoted as J 1 and J 0 re-spectively At the receiver the signal is assumed to beperfectly dehopped and the two branchesrsquo outputs of thenoncoherent (square-law) demodulator is denoted asy1 y01113864 1113865

Without loss of generality we assume the transmittedsymbol c 1 in the current hop and then the conditionalPDFs of y1 y01113864 1113865 are given by [7]

p y11113868111386811138681113868 c 11113872 1113873

1Nl

exp minusy1 + Es

Nl

1113888 1113889I02

y1Es

1113968

Nl

1113888 1113889 y1 gt 0

p y01113868111386811138681113868 c 11113872 1113873

1Nl

exp minusy0

Nl

1113888 1113889 y0 gt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where Nl N0 + J(Njρ) is determined by the channelstate and I0(x) is the modified zero-order Bessel functionof the first kind Assuming each hop is independent ofeach other and the channel side information is knownwe have the optimal log-likelihood ratio calculated asin [6]

z logP c 1 | y1 y0( 1113857

P c 0 | y1 y0( 1113857 log

I0 2y1Es

1113968Nl1113872 1113873

I0 2y0Es

1113968Nl1113872 1113873

(2)

)e decoding of inner SLT codes is followed bydecoding of outer LDPC codes )e SPA is implementedin the inner decoder according to the related bipartitegraph which is constructed using intermediate symbolsand output symbols as variable nodes (VNs) and checknodes (CNs) )e message update rules between VNs andCNs are given by [25]

Outerencoder

Innerencoder

BFSK Hopper

PBI AWGN

DehopperNCdemod

Innerdecoder

Outerdecoder

Figure 1 Rateless coded FHBFSK system model



kisin Sn m

tanh v(l)nk21113872 1113873

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎣ (3)

v(l+1)nm zm + 1113944

kisin Sm n

u(l)mk

(4)




(lmaxminus 1)




1113954z F 1113954y1 1113954y0( 1113857

I log I0 21113954y1Es

1113969

Nl1113874 1113875I0 21113954y0Es

1113969

Nl1113874 11138751113874 11138751113874 1113875(5)



p1113954zk 1113944


[i]p1113954y0[j]

(6)









dc



dc



dv




pu 1113944

dc

j1ωj R pzR pvR pv R( 1113857( 1113857( 1113857

1113944

dc

j1ωjR pzR

jminus 1pv1113872 1113873

(8)

pv 1113944

dv


iminus 1pu1113874 11138751113876 1113877 (9)


002

004

006

008

01

012

014

016

018

pz

0ndash20 ndash10 0 10 20 30 40

z





Ω1(x) 000477x + 026101x2

+ 009240x3

+ 006913x8

+ 051223x9

+ 006046x60

(10)










J(σ) 1 minus12π

radicσ

1113946infin


ξ minus σ22( 11138572

2σ21113890 1113891log2 1 + e

minus ξ1113872 1113873dξ

(11)



I(l)EV I

(lminus 1)EC1113960 1113961 1113944

dv

i1λi(α)J

σ2ch1 +(i minus 1)σ2V1113969

1113874 1113875 (12)






I(l)EC I

(l)EV1113872 1113873 1113944

dc

j1ωj 1 minus J

σ2ch2 +(j minus 1)σ2C1113969

1113874 11138751113876 1113877 (13)



(l)EV]




I(l)EC T I

(lminus 1)EC σ α1113872 1113873

1113944

dc

j1ωj 1 minus J


(lminus 1)EC1113872 11138731113872 11138731113872 1113873

21113970

1113888 11138891113890 1113891

(14)


Iext 1113944

dv

i1ΛiJ


EC1113874 11138751113874 11138752

1113971

⎛⎝ ⎞⎠ (15)


PSLTe 1113944

dv

i1Λi(α)Q


EC1113874 11138751113874 111387524

1113971

⎡⎢⎢⎣ ⎤⎥⎥⎦ (16)


radic1113938

+infinx





z

0

005

01

015

02

025



p(z)





Cod

e rat

e (R)

035

04

045

05

055

06

065

07

075

0

1

2

3

4

5

6

7

8

01 02 03 04 05 06 07 08 09 1

E sN

j (dB

)


100 1 2 3 4 5 6 7 8 9EsNj (dB)

ρ = 07ρ = 10 ρ = 05

Critical BER

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100A

sym

ptot

ic B

ER

(a)

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Asy

mpt

otic

BER


ρ = 07ρ = 10 ρ = 05

Critical BER

(b)






min 1113944

dc

i1

ωj

j


lminus 1EC I

lminus 1EC isin 0 I

maxEC1113960 1113961

l 1 2 L

1113944

dc

i1ωj 1

ωj ge 0 j 1 dc

(17)


dc








15 16 17 18 19 242 21 22 23 25Overhead

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


EXTDDECritical BER



6 Conclusions



Data Availability




Acknowledgments


References








εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old



1


1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)































kisin Sn m

tanh v(l)nk21113872 1113873

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎣ (3)

v(l+1)nm zm + 1113944

kisin Sm n

u(l)mk

(4)




(lmaxminus 1)




1113954z F 1113954y1 1113954y0( 1113857

I log I0 21113954y1Es

1113969

Nl1113874 1113875I0 21113954y0Es

1113969

Nl1113874 11138751113874 11138751113874 1113875(5)



p1113954zk 1113944


[i]p1113954y0[j]

(6)









dc



dc



dv




pu 1113944

dc

j1ωj R pzR pvR pv R( 1113857( 1113857( 1113857

1113944

dc

j1ωjR pzR

jminus 1pv1113872 1113873

(8)

pv 1113944

dv


iminus 1pu1113874 11138751113876 1113877 (9)


002

004

006

008

01

012

014

016

018

pz

0ndash20 ndash10 0 10 20 30 40

z





Ω1(x) 000477x + 026101x2

+ 009240x3

+ 006913x8

+ 051223x9

+ 006046x60

(10)










J(σ) 1 minus12π

radicσ

1113946infin


ξ minus σ22( 11138572

2σ21113890 1113891log2 1 + e

minus ξ1113872 1113873dξ

(11)



I(l)EV I

(lminus 1)EC1113960 1113961 1113944

dv

i1λi(α)J

σ2ch1 +(i minus 1)σ2V1113969

1113874 1113875 (12)






I(l)EC I

(l)EV1113872 1113873 1113944

dc

j1ωj 1 minus J

σ2ch2 +(j minus 1)σ2C1113969

1113874 11138751113876 1113877 (13)



(l)EV]




I(l)EC T I

(lminus 1)EC σ α1113872 1113873

1113944

dc

j1ωj 1 minus J


(lminus 1)EC1113872 11138731113872 11138731113872 1113873

21113970

1113888 11138891113890 1113891

(14)


Iext 1113944

dv

i1ΛiJ


EC1113874 11138751113874 11138752

1113971

⎛⎝ ⎞⎠ (15)


PSLTe 1113944

dv

i1Λi(α)Q


EC1113874 11138751113874 111387524

1113971

⎡⎢⎢⎣ ⎤⎥⎥⎦ (16)


radic1113938

+infinx





z

0

005

01

015

02

025



p(z)





Cod

e rat

e (R)

035

04

045

05

055

06

065

07

075

0

1

2

3

4

5

6

7

8

01 02 03 04 05 06 07 08 09 1

E sN

j (dB

)


100 1 2 3 4 5 6 7 8 9EsNj (dB)

ρ = 07ρ = 10 ρ = 05

Critical BER

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100A

sym

ptot

ic B

ER

(a)

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Asy

mpt

otic

BER


ρ = 07ρ = 10 ρ = 05

Critical BER

(b)






min 1113944

dc

i1

ωj

j


lminus 1EC I

lminus 1EC isin 0 I

maxEC1113960 1113961

l 1 2 L

1113944

dc

i1ωj 1

ωj ge 0 j 1 dc

(17)


dc








15 16 17 18 19 242 21 22 23 25Overhead

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


EXTDDECritical BER



6 Conclusions



Data Availability




Acknowledgments


References








εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old



1


1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)
































Ω1(x) 000477x + 026101x2

+ 009240x3

+ 006913x8

+ 051223x9

+ 006046x60

(10)










J(σ) 1 minus12π

radicσ

1113946infin


ξ minus σ22( 11138572

2σ21113890 1113891log2 1 + e

minus ξ1113872 1113873dξ

(11)



I(l)EV I

(lminus 1)EC1113960 1113961 1113944

dv

i1λi(α)J

σ2ch1 +(i minus 1)σ2V1113969

1113874 1113875 (12)






I(l)EC I

(l)EV1113872 1113873 1113944

dc

j1ωj 1 minus J

σ2ch2 +(j minus 1)σ2C1113969

1113874 11138751113876 1113877 (13)



(l)EV]




I(l)EC T I

(lminus 1)EC σ α1113872 1113873

1113944

dc

j1ωj 1 minus J


(lminus 1)EC1113872 11138731113872 11138731113872 1113873

21113970

1113888 11138891113890 1113891

(14)


Iext 1113944

dv

i1ΛiJ


EC1113874 11138751113874 11138752

1113971

⎛⎝ ⎞⎠ (15)


PSLTe 1113944

dv

i1Λi(α)Q


EC1113874 11138751113874 111387524

1113971

⎡⎢⎢⎣ ⎤⎥⎥⎦ (16)


radic1113938

+infinx





z

0

005

01

015

02

025



p(z)





Cod

e rat

e (R)

035

04

045

05

055

06

065

07

075

0

1

2

3

4

5

6

7

8

01 02 03 04 05 06 07 08 09 1

E sN

j (dB

)


100 1 2 3 4 5 6 7 8 9EsNj (dB)

ρ = 07ρ = 10 ρ = 05

Critical BER

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100A

sym

ptot

ic B

ER

(a)

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Asy

mpt

otic

BER


ρ = 07ρ = 10 ρ = 05

Critical BER

(b)






min 1113944

dc

i1

ωj

j


lminus 1EC I

lminus 1EC isin 0 I

maxEC1113960 1113961

l 1 2 L

1113944

dc

i1ωj 1

ωj ge 0 j 1 dc

(17)


dc








15 16 17 18 19 242 21 22 23 25Overhead

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


EXTDDECritical BER



6 Conclusions



Data Availability




Acknowledgments


References








εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old



1


1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)






























I(l)EC I

(l)EV1113872 1113873 1113944

dc

j1ωj 1 minus J

σ2ch2 +(j minus 1)σ2C1113969

1113874 11138751113876 1113877 (13)



(l)EV]




I(l)EC T I

(lminus 1)EC σ α1113872 1113873

1113944

dc

j1ωj 1 minus J


(lminus 1)EC1113872 11138731113872 11138731113872 1113873

21113970

1113888 11138891113890 1113891

(14)


Iext 1113944

dv

i1ΛiJ


EC1113874 11138751113874 11138752

1113971

⎛⎝ ⎞⎠ (15)


PSLTe 1113944

dv

i1Λi(α)Q


EC1113874 11138751113874 111387524

1113971

⎡⎢⎢⎣ ⎤⎥⎥⎦ (16)


radic1113938

+infinx





z

0

005

01

015

02

025



p(z)





Cod

e rat

e (R)

035

04

045

05

055

06

065

07

075

0

1

2

3

4

5

6

7

8

01 02 03 04 05 06 07 08 09 1

E sN

j (dB

)


100 1 2 3 4 5 6 7 8 9EsNj (dB)

ρ = 07ρ = 10 ρ = 05

Critical BER

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100A

sym

ptot

ic B

ER

(a)

10ndash6

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Asy

mpt

otic

BER


ρ = 07ρ = 10 ρ = 05

Critical BER

(b)






min 1113944

dc

i1

ωj

j


lminus 1EC I

lminus 1EC isin 0 I

maxEC1113960 1113961

l 1 2 L

1113944

dc

i1ωj 1

ωj ge 0 j 1 dc

(17)


dc








15 16 17 18 19 242 21 22 23 25Overhead

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


EXTDDECritical BER



6 Conclusions



Data Availability




Acknowledgments


References








εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old



1


1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)

































min 1113944

dc

i1

ωj

j


lminus 1EC I

lminus 1EC isin 0 I

maxEC1113960 1113961

l 1 2 L

1113944

dc

i1ωj 1

ωj ge 0 j 1 dc

(17)


dc








15 16 17 18 19 242 21 22 23 25Overhead

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

BER


EXTDDECritical BER



6 Conclusions



Data Availability




Acknowledgments


References








εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old



1


1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)






























6 Conclusions



Data Availability




Acknowledgments


References








εE 139 100 073 053εD 143 103 073 053

01 02 03 04 05ρ

06 07 08 0902

04

06

08

1

12

14

16

18

Ove

rhea

d th

resh

old



1


1 12 14 16 18 2 22

10ndash4

10ndash2

10ndash3

10ndash1

BER

ε

ρ = 10 Ω1(x)ρ = 10 Ωopt1(x)

ρ = 07 Ω1(x)ρ = 07 Ωopt1(x)

























































Documents

AnalysisandDesignofSystematicRatelessCodesinFH/BFSK ...downloads.hindawi.com/journals/misy/2020/9049284.pdf · Received 19 October 2019; Revised 25 December 2019; Accepted 31 December