# Analysis and Design of Systematic Rateless Codes in FH/BFSK System with Interference

Authors: Jingke Dai; Peng Zhao; Xiao Chen; Fenggan Zhang
Journal: Mobile Information Systems
 (2020)

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## Abstract

The asymptotic analysis of systematic rateless codes in frequency hopping (FH) systems with interference is first provided using discretized density evolution (DDE) and compared with the traditional fixed-rate scheme. A simplified analysis with Gaussian assumption of initial message is proposed in the worst case of interference, which has much lower complexity and provides a very close result to DDE. Based on this simplified analysis, the linear programming is employed to design rateless codes and the simulation results on partial-band interference channels show that the optimized codes have more powerful antijamming performance than the codes originally designed for conventional systems.


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## Body

## 1. Introduction

Frequency hopping (FH) systems have been extensively employed to combat interference in commercial and military communications [1]. To further enhance the antijamming capability, error-correction codes have also been widely used in FH systems, like convolutional codes [2], Reed–Solomon codes [3], Turbo codes [4, 5], and low-density parity-check (LDPC) codes [6, 7] and all of them are rate-fixed. Since the interference on channels may change randomly and dramatically from time to time, the required code rate or signal power must be adjusted accordingly for uninterrupted communications. Hence, rateless codes, which can automatically adapt their practical rates according to channel conditions, are considered as an alternative scheme for FH systems with interference.The first practical realization of rateless codes are Luby transform (LT) codes [8], the extension of which are Raptor codes with lower complexity [9]. They are originally designed for binary erasure channels (BEC) but also implemented on noisy channels [10]. The systematic versions of rateless codes have received extensive attention because of their efficient encoding. Systematic LT (SLT) codes and systematic Raptor (SR) codes on BEC are proposed in [9, 11], respectively. Normally, on noisy channels, SLT codes are generated in a direct way [12], i.e., the original information symbols are first transmitted followed by LT-coded symbols. If an inner SLT code is serially concatenated with an outer high-rate LDPC code, the resulting code can also be referred as the SR code [13]. Kite codes are a special class of prefix rateless LDPC codes [14], and they are combined with the multilevel coding scheme to improve transmission reliability [15]. Online fountain codes are proposed in [16], where an optimal encoding strategy is found efficiently based on the instantaneous decoding state. A modified online fountain encoding scheme is introduced in [17] to provide unequal error protection, and a theoretical framework is proposed in [18] to analyze online fountain codes more accurately.The rateless codes have been applied to FH systems in existing works. The performance of Raptor coded FH systems is investigated under partial-band interference (PBI) and full-band interference (FBI, the case of PBI with interference bandwidth fractionρ=1) in [19, 20]. The LT-coded differential frequency hopping system with PBI is discussed in [21]. All the results in above works are obtained by computer simulations, i.e., the asymptotic performance is not given. Also, those systems just use the classic codes with degree distributions originally designed for BEC, i.e., they have not been optimized for FH systems with interference. The authors of [22] use a probability-transfer method to decrease the Raptor codes’ average degree weight, but it needs many attempts and the resulting code may not have good performance.The asymptotic performance of rateless codes is usually studied by discretized density evolution (DDE) [13, 23] and Gaussian approximation (GA) methods [24–26]. As well known, the former provides accurate results but needs much higher computational complexity, whereas the latter uses a one-dimensional (1-D) analysis for faster calculation without much sacrifice in accuracy. However, most existing works focus on the coherent binary phase shift keying (BPSK) system on additive white Gaussian noise (AWGN) channels where the initial message of the log-likelihood ratio (LLR) is symmetrically Gaussian distributed [25], indicating that the GA methods may work very well. In the FH system, binary frequency shift keying (BFSK) and noncoherent (NC) demodulation are more employed, which means that the initial LLR is no longer Gaussian distributed, especially for consideration of interference on channels. Hence, DDE is a better choice to obtain the asymptotic performance of the rateless code in the FH/BFSK system with interference. Furthermore, DDE has been successfully combined with the differential evolution (DE) algorithm to design LDPC codes [7] and SR codes [13], but it is a nonlinear optimization method which costs much computation and spends a long search time.In this paper, we investigate the asymptotic performance and distribution optimization of systematic rateless codes in FH/BFSK systems with PBI. The main contributions are summarized as follows. Firstly, the asymptotic bit-error-rate (BER) performance and decoding thresholds of SLT codes and SR codes are achieved using DDE and the advantages of rateless codes over fixed-rate codes in the FH system are analyzed extensively. Secondly, since the DDE-DE method is so complicated, we propose a properly simplified method, which assumes the initial LLR of the FH/BFSK system as Gaussian distributed in the worst case of interference. With this GA-LLR, the 1-D analysis based on the extrinsic information transfer (EXIT) chart is employed to analyze the asymptotic performance which is very close to the results provided by DDE. Thirdly, the linear programming (LP) algorithm [25], which is a linear optimization method and then much faster than DE, is combined with the former 1-D analysis to design rateless codes on FBI, i.e., PBI with ρ=1. Both the asymptotic analysis and simulation results show that the optimized codes provide more powerful antijamming capability than the existing codes in the range of 0<ρ≤1.


## 2. System Model

The FH communication system model is depicted in Figure1. The original information sequence s=s1,s2,...,sK is precoded using a high-rate LDPC code to produce the intermediate sequence e=e1,e2,...,eK′, where the outer code rate is Ro=K/K′. The intermediate sequence is further SLT encoded to obtain the symbols of Raptor codes, denoted as c=c1,c2,...,cN, where the inner code rate is Ri=K′/N. Omitting the subscript, the coded symbol c∈0,1 is modulated to the BFSK symbol with energy Es=Eb·R, where Eb is the bit energy and R=RiRo is the overall rate of SR codes. The modulated symbol is transmitted over a frequency slot determined by the hopping pattern, and we only consider the case of one symbol per hop in this paper.Figure 1
Rateless coded FH/BFSK system model.The transmitted signal experiences full-band AWGN as well as the partial-band interference, which is modelled as an on-off jammer and evenly distributes its power over a fractionρ∈0,1 of the frequency range. We set the single-side power spectrum density (PSD) of noise and interference is N0 and Nj/ρ, respectively. As a result, there are essentially two states on this channel: jammed and unjammed, which are denoted as J=1 and J=0, respectively. At the receiver, the signal is assumed to be perfectly dehopped and the two branches’ outputs of the noncoherent (square-law) demodulator is denoted as y1,y0.Without loss of generality, we assume the transmitted symbolc=1 in the current hop and then the conditional PDFs of y1,y0 are given by [7]:(1)py1c=1=1Nlexp−y1+EsNlI02y1EsNl,y1>0,py0c=1=1Nlexp−y0Nl,y0>0,where Nl=N0+JNj/ρ is determined by the channel state and I0x is the modified zero-order Bessel function of the first kind. Assuming each hop is independent of each other and the channel side information is known, we have the optimal log-likelihood ratio calculated as in [6]:(2)z=logPc=1y1,y0Pc=0y1,y0=logI02y1Es/NlI02y0Es/Nl.The decoding of inner SLT codes is followed by decoding of outer LDPC codes. The SPA is implemented in the inner decoder according to the related bipartite graph, which is constructed using intermediate symbols and output symbols as variable nodes (VNs) and check nodes (CNs). The message update rules between VNs and CNs are given by [25].(3)um,nl=2tanh−1tanhzK′+n/2∏k∈Sn\mtanhvn,kl/2,(4)vn,ml+1=zm+∑k∈Sm\num,kl,where um,nl and vn,ml denote the message passed from the n-th check node to the m-th variable node and the message passed in the opposite direction at the l-th decoding iteration and Sn\m represents the set of all neighbours of the n-th node except for the m-th node. After the maximum iteration lmax, the decision message vmlmax=zm+∑k∈Smum,klmax−1 is provided as the initial LLR to decode outer LDPC codes.


## 3. Asymptotic Analysis

### 3.1. Initial LLR on PBI Channels

From (1) and (2), we know that it is hard to get close-form density expression of the initial LLR message due to Bessel functions, unlike the case of the coherent BPSK system. Thus, we use the probability mass function (PMF) instead [13]. Firstly, the continuous variables y1,y0 is quantized to y^1,y^0 with a quantization interval Δ1. Secondly, calculate the quantized initial message by a two-input function expressed as(5)z^=Fy^1,y^0=ℑlogI02y^1Es/Nl/I02y^0Es/Nl,where ℑ is the “quantization operator” ensuring that z^ takes values in a discrete set zmin,zmin+Δ2,zmin+2Δ2,...,zmin+kΔ2,...,zmax with a quantization interval Δ2. Finally, calculate the probability of z^=zmin+kΔ2 as given by(6)pz^k=∑i,j:kΔ=FiΔ,jΔpy^1ipy^0j,where the probability masses py^1i and py^0j are the quantized results from equation (1). Finally, the assemble of pz^k gives the PMF of z^. Note that Nl in (5) has two possible values, thus pz^k also has two possible values which gives the PMF of z^ denoted as pz,J and pz,NJ corresponding to jammed and unjammed states, respectively. The unconditional PMF of z^ is written as(7)pz=ρpz,J+1−ρpz,NJ.SettingΔ1=0.01, Δ2=0.05, zmin=−20, and zmax=40, the PMF of initial LLR under the channel with symbol signal-to-noise ratio (SNR) Es/N0=10dB, symbol signal-to-jamming ratio (SJR) Es/Nj=5dB, and ρ=0.7 is depicted in Figure 2. It is seen that the PMF has two peaks caused by different channel states and is no longer Gaussian distributed, which implies that the Gaussian approximation-based method may not be employed to investigate the asymptotic performance of rateless codes on PBI channels (for ρ<1, at least). Hence, we consider the DDE which does not require Gaussian initial LLR.Figure 2
PMF of initial LLR in the FH/BFSK system withEs/N0=10dB, Es/Nj=5dB, and ρ=0.7.


### 3.2. Asymptotic Performance of SLT and SR Codes

We define the generation matrix of inner SLT codes asGSLT=IK′GLT, where IK′ is an identity matrix having a size of K′×K′ and GLT of size K′×N−K′ is the generation matrix of the conventional LT code [25]. The overhead of the inner SLT code is defined by ε=N−K′/K′ which is related with the code rates as ε=1/Ri−1=Ro/R−1.The degree distributions related with the bipartite graph corresponding toGLT is defined as follows. Using polynomial notations, the check node degree distributions is written compactly as Ωx=∑j=1dcΩjxj, where dc is the maximum degree. The corresponding edge degree distribution is calculated by ωx=∑j=1dcωjxj−1=Ω′x/Ω′1 and the average degree of CNs is given by β=Ω′1. The average degree of VNs is denoted by α, and then the variable nodes and edge degrees are both assumed to be Poisson distributed as Λx=eαx−1 and λx=eαx−1, respectively [25]. However, the power series can also be truncated to obtain polynomials that is very close to the exponential, i.e., Λx=∑i=1dvΛiαxi and λx=∑i=1dvλiαxi−1, where dv is large enough to ensure Λ1≈1.We denote the PMF of the message passed from CNs and VNs bypu and pv, respectively. The updating rules of CNs and VNs are summarized as follows:(8)pu=∑j=1dcωjℜpz,ℜpv,ℜpv,...,ℜ=∑j=1dcωjℜpz,ℜj−1pv,(9)pv=∑i=1dvλipz⊗⊗i−1pu,where ℜ is a two-input operator defined in [23] to calculate the PMF of quantized message emanating from check nodes and ⊗ denotes convolution calculation. The DDE of outer LDPC codes is similar to the above rules, only that the channel message PMF pz is removed at the CNs. The detail of DDE on the SR code is referred to [13], and here we just need to replace the initial LLR of the BPSK system by equation (7).The asymptotic BER performance of SLT codes and SR codes obtained by DDE under different PBI bandwidth fractionρ is shown in Figure 3, where SNR is fixed to 10 dB and the outer code is fixed to a 0.98-rate regular (4, 200)-LDPC code. The degree distribution of the inner code is fixed to the optimized rate 1/2 results in [13], which is rewritten as(10)Ω1x=0.00477x+0.26101x2+0.09240x3+0.06913x8+0.51223x9+0.06046x60.Figure 3
Asymptotic BER of SLT codes and SR codes withEs/N0=10dB. The dotted lines are BERs obtained after inner decoding and the solid lines are those after outer decoding. (a) R=1/2. (b) Es/Nj=5dB.
(a)
(b)In Figure3(a), the overall code rate is fixed to 1/2 and SJR is variable, whereas in Figure 3(b), SJR is fixed to 5 dB and the overhead of the inner code ε (then the code rate R) is variable. In the range of small SJR or overhead, the performance of SLT codes and SR codes is very close to each other, indicating that the outer code cannot correct residual errors from the inner code. The BER of the SLT code decreases as either SJR or overhead increases but high error floors are observed.Once the BER of the inner SLT code achieves a critical BER, which is firstly introduced in [23], the BER of the SR code decreases very sharply. It is easy to calculate the critical BER using DDE [23], which is equal to 0.0039 for the 0.98-rate regular LDPC code. In addition, the case of ρ=1 has the worst performance, where PBI has actually become the full-band interference.


### 3.3. Comparison of Rateless Codes and Fixed-Rate Codes

To compare the rateless codes and conventional fixed-rate codes in the FH system, their decoding thresholds are investigated under different PBI bandwidth fractionsρ. From Figure 3(b), we know the BERs of SR codes tend to zero as the inner BER achieves critical BER and the corresponding overhead can be referred as the decoding threshold. According to the relation between overhead and code rate presented in Section 2, we plot the rate thresholds of SR codes for different ρ with various Es/Nj in Figure 4. The classic 1/2-rate regular (3, 6) LDPC is taken as the fixed-rate code for comparison, and the SJR-decoding thresholds associated with the right axis are also depicted in Figure 4.Figure 4
Rate thresholds of Raptor codes and SJR-thresholds of LDPC codes for differentρ.It is seen that the SJR-threshold of the LDPC code grows asρ increases and the worst case appears at ρ=1 corresponding to the threshold of 5.71 dB. As ρ may change randomly over time, the SJR of the fixed-rate coded FH system must be set to be larger than the threshold of the worst case, which in essence is a waste of power or rate for the cases of ρ<1. However, Raptor codes can “automatically” choose the proper rates for different ρ, e.g., under the SJR of 5.71 dB, the rate thresholds of Raptor codes are 0.74 and 0.51 for ρ=0.1 and ρ=1, respectively. Meanwhile, the Raptor code is more adaptable to environment with dramatic SJR changes. When SJR decreases, e.g., to 4 dB, the Raptor can still work as long as we continue to send coded symbols (decreasing actual rate) but the specific fixed-rate code cannot. When SJR increases, e.g., to 7 dB, the Raptor code can work with a higher rate than the 1/2-rate LDPC code. In general, rateless codes are more flexible and more robust than the fixed-rate codes on PBI channels.


## 3.1. Initial LLR on PBI Channels

From (1) and (2), we know that it is hard to get close-form density expression of the initial LLR message due to Bessel functions, unlike the case of the coherent BPSK system. Thus, we use the probability mass function (PMF) instead [13]. Firstly, the continuous variables y1,y0 is quantized to y^1,y^0 with a quantization interval Δ1. Secondly, calculate the quantized initial message by a two-input function expressed as(5)z^=Fy^1,y^0=ℑlogI02y^1Es/Nl/I02y^0Es/Nl,where ℑ is the “quantization operator” ensuring that z^ takes values in a discrete set zmin,zmin+Δ2,zmin+2Δ2,...,zmin+kΔ2,...,zmax with a quantization interval Δ2. Finally, calculate the probability of z^=zmin+kΔ2 as given by(6)pz^k=∑i,j:kΔ=FiΔ,jΔpy^1ipy^0j,where the probability masses py^1i and py^0j are the quantized results from equation (1). Finally, the assemble of pz^k gives the PMF of z^. Note that Nl in (5) has two possible values, thus pz^k also has two possible values which gives the PMF of z^ denoted as pz,J and pz,NJ corresponding to jammed and unjammed states, respectively. The unconditional PMF of z^ is written as(7)pz=ρpz,J+1−ρpz,NJ.SettingΔ1=0.01, Δ2=0.05, zmin=−20, and zmax=40, the PMF of initial LLR under the channel with symbol signal-to-noise ratio (SNR) Es/N0=10dB, symbol signal-to-jamming ratio (SJR) Es/Nj=5dB, and ρ=0.7 is depicted in Figure 2. It is seen that the PMF has two peaks caused by different channel states and is no longer Gaussian distributed, which implies that the Gaussian approximation-based method may not be employed to investigate the asymptotic performance of rateless codes on PBI channels (for ρ<1, at least). Hence, we consider the DDE which does not require Gaussian initial LLR.Figure 2
PMF of initial LLR in the FH/BFSK system withEs/N0=10dB, Es/Nj=5dB, and ρ=0.7.


## 3.2. Asymptotic Performance of SLT and SR Codes

We define the generation matrix of inner SLT codes asGSLT=IK′GLT, where IK′ is an identity matrix having a size of K′×K′ and GLT of size K′×N−K′ is the generation matrix of the conventional LT code [25]. The overhead of the inner SLT code is defined by ε=N−K′/K′ which is related with the code rates as ε=1/Ri−1=Ro/R−1.The degree distributions related with the bipartite graph corresponding toGLT is defined as follows. Using polynomial notations, the check node degree distributions is written compactly as Ωx=∑j=1dcΩjxj, where dc is the maximum degree. The corresponding edge degree distribution is calculated by ωx=∑j=1dcωjxj−1=Ω′x/Ω′1 and the average degree of CNs is given by β=Ω′1. The average degree of VNs is denoted by α, and then the variable nodes and edge degrees are both assumed to be Poisson distributed as Λx=eαx−1 and λx=eαx−1, respectively [25]. However, the power series can also be truncated to obtain polynomials that is very close to the exponential, i.e., Λx=∑i=1dvΛiαxi and λx=∑i=1dvλiαxi−1, where dv is large enough to ensure Λ1≈1.We denote the PMF of the message passed from CNs and VNs bypu and pv, respectively. The updating rules of CNs and VNs are summarized as follows:(8)pu=∑j=1dcωjℜpz,ℜpv,ℜpv,...,ℜ=∑j=1dcωjℜpz,ℜj−1pv,(9)pv=∑i=1dvλipz⊗⊗i−1pu,where ℜ is a two-input operator defined in [23] to calculate the PMF of quantized message emanating from check nodes and ⊗ denotes convolution calculation. The DDE of outer LDPC codes is similar to the above rules, only that the channel message PMF pz is removed at the CNs. The detail of DDE on the SR code is referred to [13], and here we just need to replace the initial LLR of the BPSK system by equation (7).The asymptotic BER performance of SLT codes and SR codes obtained by DDE under different PBI bandwidth fractionρ is shown in Figure 3, where SNR is fixed to 10 dB and the outer code is fixed to a 0.98-rate regular (4, 200)-LDPC code. The degree distribution of the inner code is fixed to the optimized rate 1/2 results in [13], which is rewritten as(10)Ω1x=0.00477x+0.26101x2+0.09240x3+0.06913x8+0.51223x9+0.06046x60.Figure 3
Asymptotic BER of SLT codes and SR codes withEs/N0=10dB. The dotted lines are BERs obtained after inner decoding and the solid lines are those after outer decoding. (a) R=1/2. (b) Es/Nj=5dB.
(a)
(b)In Figure3(a), the overall code rate is fixed to 1/2 and SJR is variable, whereas in Figure 3(b), SJR is fixed to 5 dB and the overhead of the inner code ε (then the code rate R) is variable. In the range of small SJR or overhead, the performance of SLT codes and SR codes is very close to each other, indicating that the outer code cannot correct residual errors from the inner code. The BER of the SLT code decreases as either SJR or overhead increases but high error floors are observed.Once the BER of the inner SLT code achieves a critical BER, which is firstly introduced in [23], the BER of the SR code decreases very sharply. It is easy to calculate the critical BER using DDE [23], which is equal to 0.0039 for the 0.98-rate regular LDPC code. In addition, the case of ρ=1 has the worst performance, where PBI has actually become the full-band interference.


## 3.3. Comparison of Rateless Codes and Fixed-Rate Codes

To compare the rateless codes and conventional fixed-rate codes in the FH system, their decoding thresholds are investigated under different PBI bandwidth fractionsρ. From Figure 3(b), we know the BERs of SR codes tend to zero as the inner BER achieves critical BER and the corresponding overhead can be referred as the decoding threshold. According to the relation between overhead and code rate presented in Section 2, we plot the rate thresholds of SR codes for different ρ with various Es/Nj in Figure 4. The classic 1/2-rate regular (3, 6) LDPC is taken as the fixed-rate code for comparison, and the SJR-decoding thresholds associated with the right axis are also depicted in Figure 4.Figure 4
Rate thresholds of Raptor codes and SJR-thresholds of LDPC codes for differentρ.It is seen that the SJR-threshold of the LDPC code grows asρ increases and the worst case appears at ρ=1 corresponding to the threshold of 5.71 dB. As ρ may change randomly over time, the SJR of the fixed-rate coded FH system must be set to be larger than the threshold of the worst case, which in essence is a waste of power or rate for the cases of ρ<1. However, Raptor codes can “automatically” choose the proper rates for different ρ, e.g., under the SJR of 5.71 dB, the rate thresholds of Raptor codes are 0.74 and 0.51 for ρ=0.1 and ρ=1, respectively. Meanwhile, the Raptor code is more adaptable to environment with dramatic SJR changes. When SJR decreases, e.g., to 4 dB, the Raptor can still work as long as we continue to send coded symbols (decreasing actual rate) but the specific fixed-rate code cannot. When SJR increases, e.g., to 7 dB, the Raptor code can work with a higher rate than the 1/2-rate LDPC code. In general, rateless codes are more flexible and more robust than the fixed-rate codes on PBI channels.


## 4. Simplified Analysis under FBI

### 4.1. Gaussian Assumption of Initial LLR

Although the asymptotic BER and decoding thresholds have been obtained by DDE, the employed degree distribution is originally designed for the coherent BPSK system, which means that stronger antijamming capability may be achieved if we can optimize the degree distribution for the noncoherent FH system. The conventional method combined with DDE is differential evolution, but it has a pretty high complexity and long search time. The linear programming method can also be used to design Raptor/LT codes, and it is much simpler and faster than DE. But the LP method is usually combined with the one-dimensional analysis method, e.g., EXIT chart, which requires the initial LLR being Gaussian distributed.From Figure2, we know that LLR on the PBI channel is not Gaussian but we find that the PMF of initial LLR in the FH/BFSK system can be approximated by a symmetric Gaussian density when ρ=1, which happened to be the worst case of interference. Using (7), we have pz=pz,J at this time and the mean of z, denoted by mz, can be easily calculated based on pz. We approximate pz by a Gaussian density with mean mz and variance 2mz and plot the actual PMF as well as the approximated density shown in Figure 5. The curves in one pair are very close to each other, which indicate that the one-dimensional analysis could be employed to investigate asymptotic performance and be further combined with LP to design codes.Figure 5
Actual PMF of initial LLR and their Gaussian approximated PDF underρ=1.


### 4.2. One-Dimensional Analysis

Since the initial LLR is approximately Gaussian distributed whenρ=1, the EXIT chart can be implemented to analyze the asymptotic performance of the inner SLT codes. Assuming the all-zero codeword is transmitted, the mutual information of z is computed by [24] as follows:(11)Jσ=1−12πσ∫−∞∞exp−ξ−σ2/222σ2log21+e−ξdξ,where Jσ is a monotonically increasing function and has an inverse function σ=J−1I. Both J⋅ and J−1⋅ do not have close-form expressions, but they can be closely approximated as suggested in [27].From (4) we can calculate the mutual information of VNs at the l-th iteration as(12)IE,VlIE,Cl−1=∑i=1dvλiαJσch12+i−1σV2,where σch12=2mz is the variance of initial LLR and σV2=J−1IE,Cl−12 is the message variance of CNs in the previous iteration. The mutual information outputted by CNs at the l-th iteration is given by [24] as follows:(13)IE,ClIE,Vl=∑j=1dcωj1−Jσch22+j−1σC2,where σch2=J−11−J2mz and σC=J−11−IE,Vl. Combining (12) and (13), we can get the iteration expression of IE,Cl as the function of IE,Cl−1, σ, and α as given by(14)IE,Cl=TIE,Cl−1,σ,α=∑j=1dcωj1−Jσch22+j−1J−11−IE,VlIE,Cl−12.At the end of iterationlmax, the inner decoder outputs the extrinsic information provided to the outer decoder as given by(15)Iext=∑i=1dvΛiJ4/σ2+iJ−1IE,Clmax2,and the corresponding error probability is calculated as(16)PeSLT=∑i=1dvΛiαQ1/σ2+iJ−1IE,Clmax2/4,where the function Qx=1/2π∫x+∞e−t2/2dt.Using the above method and settingEs/N0=10dB and Es/Nj=4dB, we depict the asymptotic BERs of the inner SLT code of Ω1x in Figure 6, which is represented by “EXIT”. Also shown are those achieved by DDE and the simulated BERs of SLT codes with finite length K′ equal to 2000, 4000, and 10000, respectively. We observe that the EXIT curve is very close to the DDE curve and both of them can be taken as the asymptotic performance of simulation results, indicating that the proposed Gaussian assumption of initial LLR under ρ=1 is very effective. In Figure 6, the critical BER of the outer codes is also presented, and its intersection points with asymptotic curves correspond to the decoding thresholds of Raptor codes. It is seen that the threshold of EXIT is lower than DDE which is opposite in the ordinary BPSK system [13]. This is because the initial LLR of the FH/BFSK system is assumed to be Gaussian, which is different from the real situation (see Figure 5), but the initial LLR of the ordinary BPSK system is really Gaussian and the gap between EXIT and DDE is caused by Gaussian assumption of the message from CNs. Though the asymptotic performance by EXIT is slightly different from that by DDE, it can still be employed to design codes with good performance.Figure 6
The comparison of asymptotic performance and simulation results of SLT codes.


## 4.1. Gaussian Assumption of Initial LLR

Although the asymptotic BER and decoding thresholds have been obtained by DDE, the employed degree distribution is originally designed for the coherent BPSK system, which means that stronger antijamming capability may be achieved if we can optimize the degree distribution for the noncoherent FH system. The conventional method combined with DDE is differential evolution, but it has a pretty high complexity and long search time. The linear programming method can also be used to design Raptor/LT codes, and it is much simpler and faster than DE. But the LP method is usually combined with the one-dimensional analysis method, e.g., EXIT chart, which requires the initial LLR being Gaussian distributed.From Figure2, we know that LLR on the PBI channel is not Gaussian but we find that the PMF of initial LLR in the FH/BFSK system can be approximated by a symmetric Gaussian density when ρ=1, which happened to be the worst case of interference. Using (7), we have pz=pz,J at this time and the mean of z, denoted by mz, can be easily calculated based on pz. We approximate pz by a Gaussian density with mean mz and variance 2mz and plot the actual PMF as well as the approximated density shown in Figure 5. The curves in one pair are very close to each other, which indicate that the one-dimensional analysis could be employed to investigate asymptotic performance and be further combined with LP to design codes.Figure 5
Actual PMF of initial LLR and their Gaussian approximated PDF underρ=1.


## 4.2. One-Dimensional Analysis

Since the initial LLR is approximately Gaussian distributed whenρ=1, the EXIT chart can be implemented to analyze the asymptotic performance of the inner SLT codes. Assuming the all-zero codeword is transmitted, the mutual information of z is computed by [24] as follows:(11)Jσ=1−12πσ∫−∞∞exp−ξ−σ2/222σ2log21+e−ξdξ,where Jσ is a monotonically increasing function and has an inverse function σ=J−1I. Both J⋅ and J−1⋅ do not have close-form expressions, but they can be closely approximated as suggested in [27].From (4) we can calculate the mutual information of VNs at the l-th iteration as(12)IE,VlIE,Cl−1=∑i=1dvλiαJσch12+i−1σV2,where σch12=2mz is the variance of initial LLR and σV2=J−1IE,Cl−12 is the message variance of CNs in the previous iteration. The mutual information outputted by CNs at the l-th iteration is given by [24] as follows:(13)IE,ClIE,Vl=∑j=1dcωj1−Jσch22+j−1σC2,where σch2=J−11−J2mz and σC=J−11−IE,Vl. Combining (12) and (13), we can get the iteration expression of IE,Cl as the function of IE,Cl−1, σ, and α as given by(14)IE,Cl=TIE,Cl−1,σ,α=∑j=1dcωj1−Jσch22+j−1J−11−IE,VlIE,Cl−12.At the end of iterationlmax, the inner decoder outputs the extrinsic information provided to the outer decoder as given by(15)Iext=∑i=1dvΛiJ4/σ2+iJ−1IE,Clmax2,and the corresponding error probability is calculated as(16)PeSLT=∑i=1dvΛiαQ1/σ2+iJ−1IE,Clmax2/4,where the function Qx=1/2π∫x+∞e−t2/2dt.Using the above method and settingEs/N0=10dB and Es/Nj=4dB, we depict the asymptotic BERs of the inner SLT code of Ω1x in Figure 6, which is represented by “EXIT”. Also shown are those achieved by DDE and the simulated BERs of SLT codes with finite length K′ equal to 2000, 4000, and 10000, respectively. We observe that the EXIT curve is very close to the DDE curve and both of them can be taken as the asymptotic performance of simulation results, indicating that the proposed Gaussian assumption of initial LLR under ρ=1 is very effective. In Figure 6, the critical BER of the outer codes is also presented, and its intersection points with asymptotic curves correspond to the decoding thresholds of Raptor codes. It is seen that the threshold of EXIT is lower than DDE which is opposite in the ordinary BPSK system [13]. This is because the initial LLR of the FH/BFSK system is assumed to be Gaussian, which is different from the real situation (see Figure 5), but the initial LLR of the ordinary BPSK system is really Gaussian and the gap between EXIT and DDE is caused by Gaussian assumption of the message from CNs. Though the asymptotic performance by EXIT is slightly different from that by DDE, it can still be employed to design codes with good performance.Figure 6
The comparison of asymptotic performance and simulation results of SLT codes.


## 5. Code Design and Simulation Results

Technically, the optimal distributions of inner SLT codes are different for differentρ, but it is unrealistic for practical communications to switch distributions in real time according to the current channel conditions. Hence, the optimized distributions for the worst case of interference, naturally, are a good choice for all situations. Besides, we are lucky that the worst case of interference occurs at ρ=1 so that the PMF of initial LLR can be approximated by a Gaussian density and the EXIT analysis can be combined with the standard LP method, which is described by the following model:(17)min:∑i=1dcωjj,s.t.TIE,Cl-1,σ,α>IE,Cl-1,IE,Cl-1∈0,IE,Cmax⁡,l=1,2,…,L,∑i=1dcωj=1,ωj≥0,j=1,…,dc.In the optimization model, the design objective is to minimize the overheadε=α/β=α∑j=1dcωj/j. The second and third constraints ensure the distributions validity. According to (14), the first constraint ensures that the tracking variable converges to a fixed value IE,Cmax in iterations.We fix the SNR as 10 dB and set the critical BER as 0.0039 and search the good distributions of SLT codes using LP under different SJR. The resulting degree distributions and corresponding overhead thresholds are listed in Table1, in which εE denotes the required overhead of inner codes achieving critical BER by the EXIT method and εD denotes the required overhead by the DDE method. We can see that the overhead thresholds obtained by two methods are very close to each other, confirming that the optimized codes indeed have good performance. Specifically, εE=εD for SJR = 6 dB and SJR = 7 dB is not a surprising result since the approximated initial LLR is almost the same with the real one (see Figure 5).Table 1
The optimized degree distributions of inner SLT codes with SNR = 10 dB.
SJR
4 dB
5 dB
6 dB
7 dB
Ω
opt
1
x
α
=
13.5
Ω
opt
2
x
α
=
11
Ω
opt
3
x
α
=
9
Ω
opt
4
x
α
=
8
Ω
5
0.59654
Ω
6
0.53805
Ω
7
0.29357
Ω
9
0.33414
Ω
6
0.28390
Ω
7
0.36372
Ω
8
0.61311
Ω
10
0.58986
Ω
30
0.09375
Ω
44
0.07748
Ω
59
0.06836
Ω
83
0.01170
Ω
87
0.01826
Ω
88
0.00901
Ω
60
0.02496
Ω
84
0.06430
Ω
88
0.00755
Ω
89
0.01174
ε
E
1.39
1.00
0.73
0.53
ε
D
1.43
1.03
0.73
0.53It is noted that all the optimized distributions are obtained with conditionρ=1. To investigate the performance of optimized distributions under ρ<1, we use DDE to get the overhead thresholds of optimized codes and code with Ω1x under SJR = 4 dB and SJR = 7 dB which are compared in Figure 7. When SJR = 7 dB, the optimized code has smaller thresholds than Ω1x for all ρ. At the point of ρ=1, the overhead thresholds of two codes are 0.53 and 0.71, respectively. If the outer code rate is fixed to 0.98, the corresponding overall rate of the optimized code increases about 11.7% compared to the code with Ω1x. When SJR = 4 dB, the optimized code also has better performance than the code with Ω1x for large ρ but they are very close when ρ is small. Considering that the PBI mainly works on the range of large ρ, we say that our optimized codes have more powerful antijamming capacity than the traditional code.Figure 7
Overhead thresholds of inner SLT codes for differentρ.We apply the optimized degree distribution to the codes with information lengthK=4000 under serious interference (SJR = 4 dB). The outer code is set as the 0.98-rate regular LDPC code, and SNR is fixed to 10 dB. The simulated BERs of Raptor codes in the FH/BFSK system with ρ=0.7 and ρ=1 are plotted in Figure 8. As shown in this figure, the optimized code provides better performance than the existing code for both PBI and FBI conditions, verifying the effectiveness of our design method.Figure 8
Simulated BER of SR codes versus overhead under SJR = 4 dB and SNR = 10 dB.


## 6. Conclusions

The asymptotic performance of systematic rateless codes in the FH/BFSK system with PBI has been obtained and compared with the fixed-rate LDPC codes. The rateless codes are more flexible and more robust than fixed-rate codes on PBI channels. The worst case of interference for the rateless-coded FH/BFSK system usually appears at the point ofρ=1 where PBI become to FBI. Under FBI, the initial LLR is assumed to be Gaussian distributed and then the 1-D analysis based on the EXIT chart is employed to investigate the asymptotic performance which is very close to the result by DDE. The code degree distributions are designed using linear programming which is combined with the 1-D analysis. On PBI channels, the optimized codes in the FH/BFSK system provide outstanding antijamming performance compared with the code originally designed for the conventional BPSK system. In addition, the coding scheme of the FH system in this paper concentrates on the SR codes and some novel rateless codes, such as Kite codes and online fountain codes, will be considered in our future works.



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*Source: 9049284-2020-02-10.xml*