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

. 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.


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 lowdensity 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. e first practical realization of rateless codes are Luby transform (LT) codes [8], the extension of which are Raptor codes with lower complexity [9]. ey are originally designed for binary erasure channels (BEC) but also implemented on noisy channels [10]. e 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. e rateless codes have been applied to FH systems in existing works. e 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 fraction ρ � 1) in [19,20]. e LTcoded 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. e 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. e asymptotic performance of rateless codes is usually studied by discretized density evolution (DDE) [13,23] and Gaussian approximation (GA) methods [24][25][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. e main contributions are summarized as follows. Firstly, the asymptotic biterror-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.
irdly, 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.

System
Model e FH communication system model is depicted in Figure 1. e original information sequence s � s 1 , s 2 , ..., s K is precoded using a high-rate LDPC code to produce the intermediate sequence e � e 1 , e 2 , ..., e K′ , where the outer code rate e intermediate sequence is further SLT encoded to obtain the symbols of Raptor codes, denoted as c � c 1 , c 2 , ..., c N , where the inner code rate is Omitting the subscript, the coded symbol c ∈ 0, 1 { } is modulated to the BFSK symbol with energy E s � E b · R, where E b is the bit energy and R � R i R o is the overall rate of SR codes. e 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. e 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 fraction ρ ∈ (0, 1] of the frequency range. We set the single-side power spectrum density (PSD) of noise and interference is N 0 and N j /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 y 1 , y 0 .
Without loss of generality, we assume the transmitted symbol c � 1 in the current hop and then the conditional PDFs of y 1 , y 0 are given by [7]: where N l � N 0 + J(N j /ρ) is determined by the channel state and I 0 (x) 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]: e decoding of inner SLT codes is followed by decoding of outer LDPC codes. e 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). e message update rules between VNs and CNs are given by [25].
where u (l) m,n and v (l) n,m 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 S n \m represents the set of all neighbours of the n-th node except for the m-th node. After the maximum iteration l max , the decision message v is provided as the initial LLR to decode outer LDPC codes. (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. us, we use the probability mass function (PMF) instead [13]. Firstly, the continuous variables y 1 , y 0 is quantized to y 1 , y 0 with a quantization interval Δ 1 . Secondly, calculate the quantized initial message by a two-input function expressed as

Initial LLR on PBI Channels. From
where I is the "quantization operator" ensuring that z takes values in a discrete set z min , z min + Δ 2 , z min + 2Δ 2 , ... , z min + kΔ 2 , ... , z max } with a quantization interval Δ 2 . Finally, calculate the probability of z � z min + kΔ 2 as given by where the probability masses p y 1 [i] and p y 0 [j] are the quantized results from equation (1). Finally, the assemble of p z k gives the PMF of z. Note that N l in (5) has two possible values, thus p z k also has two possible values which gives the PMF of z denoted as p z,J and p z,NJ corresponding to jammed and unjammed states, respectively. e unconditional PMF of z is written as Setting Δ 1 � 0.01, Δ 2 � 0.05, z min � − 20, and z max � 40, the PMF of initial LLR under the channel with symbol signal-to-noise ratio (SNR) E s /N 0 � 10 dB, symbol signalto-jamming ratio (SJR) E s /N j � 5 dB, 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.

Asymptotic Performance of SLT and SR Codes.
We define the generation matrix of inner SLT codes as is the generation matrix of the conventional LT code [25]. e overhead of the inner SLT code is defined by ε � ( e degree distributions related with the bipartite graph corresponding to G LT is defined as follows. Using polynomial notations, the check node degree distributions is written compactly as and the average degree of CNs is given by β � Ω ′ (1). e average degree of VNs is denoted by α, and then the variable nodes and edge degrees are both assumed to be Poisson distributed as Λ(x) � e α(x− 1) and λ(x) � e α(x− 1) , respectively [25]. However, the power series can also be truncated to obtain polynomials that is very close to the exponential, i.e., We denote the PMF of the message passed from CNs and VNs by p u and p v , respectively. e updating rules of CNs and VNs are summarized as follows: where R is a two-input operator defined in [23]   the SR code is referred to [13], and here we just need to replace the initial LLR of the BPSK system by equation (7). e asymptotic BER performance of SLT codes and SR codes obtained by DDE under different PBI bandwidth fraction ρ 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. e degree distribution of the inner code is fixed to the optimized rate 1/2 results in [13], which is rewritten as In Figure 3(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. e 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 fullband interference.

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 fractions ρ. 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 E s /N j in Figure 4. e 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.
It is seen that the SJR-threshold of the LDPC code grows as ρ 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.

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. e conventional method combined with DDE is differential evolution, but it has a pretty high complexity and long search time. e 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 Figure 2, 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 p z � p z,J at this time and the mean of z, denoted by m z , can be easily calculated based on p z . We approximate p z by a Gaussian density with mean m z and variance 2m z and plot the actual PMF as well as the approximated density shown in Figure 5. e 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.

One-Dimensional Analysis.
Since the initial LLR is approximately Gaussian distributed when ρ � 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: where J(σ) is a monotonically increasing function and has an inverse function σ � J − 1 (I). 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 where σ 2 ch1 � 2m z is the variance of initial LLR and σ 2 is the message variance of CNs in the previous iteration. e mutual information outputted by CNs at the lth iteration is given by [24] as follows: (12) and (13), we can get the iteration expression of I (l) E,C as the function of I (l− 1) E,C , σ, and α as given by At the end of iteration l max , the inner decoder outputs the extrinsic information provided to the outer decoder as given by and the corresponding error probability is calculated as where the function Q(x) � 1/ �� � 2π √ +∞ x e − t 2 /2 dt. Using the above method and setting E s /N 0 � 10 dB and E s /N j � 4 dB, we depict the asymptotic BERs of the inner  SLTcode of Ω 1 (x) 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]. is 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. ough the asymptotic performance by EXIT is slightly different from that by DDE, it can still be employed to design codes with good performance.

Code Design and Simulation Results
Technically, the optimal distributions of inner SLT codes are different for different ρ, 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: In the optimization model, the design objective is to minimize the overhead ε � α/β � α d c j�1 ω j /j. e second and third constraints ensure the distributions validity. According to (14), the first constraint ensures that the tracking variable converges to a fixed value I max E,C 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. e resulting degree distributions and corresponding overhead thresholds are listed in Table 1, in which ε E denotes the required overhead of inner codes achieving critical BER by the EXIT method and ε D 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, ε E � ε D 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).
It is noted that all the optimized distributions are obtained with condition ρ � 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 Ω 1 (x) 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 Ω 1 (x) 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 Ω 1 (x). When SJR � 4 dB, the optimized code also has better performance than the code with Ω 1 (x) 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.
We apply the optimized degree distribution to the codes with information length K � 4000 under serious interference (SJR � 4 dB).
e outer code is set as the 0.98-rate regular LDPC code, and SNR is fixed to 10 dB. e 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.

Conclusions
e asymptotic performance of systematic rateless codes in the FH/BFSK system with PBI has been obtained and compared with the fixed-rate LDPC codes. e rateless codes are more flexible and more robust than fixed-rate codes on PBI channels. e worst case of interference for the rateless-coded FH/BFSK system usually appears at the point of ρ � 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. e 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.
Data Availability e numerical and simulation data used to support the findings of this study are included within the supplementary information file.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.   Mobile Information Systems 7