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.
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 fraction ) 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 . 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 .
2. System Model
The FH communication system model is depicted in Figure 1. The original information sequence is precoded using a high-rate LDPC code to produce the intermediate sequence , where the outer code rate is . The intermediate sequence is further SLT encoded to obtain the symbols of Raptor codes, denoted as , where the inner code rate is . Omitting the subscript, the coded symbol is modulated to the BFSK symbol with energy , where is the bit energy and 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.
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 fraction of the frequency range. We set the single-side power spectrum density (PSD) of noise and interference is and , respectively. As a result, there are essentially two states on this channel: jammed and unjammed, which are denoted as and , 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 .
Without loss of generality, we assume the transmitted symbol in the current hop and then the conditional PDFs of are given by [7]:where is determined by the channel state and 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]:
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].where and 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 represents the set of all neighbours of the n-th node except for the m-th node. After the maximum iteration , the decision message 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 is quantized to with a quantization interval . Secondly, calculate the quantized initial message by a two-input function expressed aswhere is the “quantization operator” ensuring that takes values in a discrete set with a quantization interval . Finally, calculate the probability of as given bywhere the probability masses and are the quantized results from equation (1). Finally, the assemble of gives the PMF of . Note that in (5) has two possible values, thus also has two possible values which gives the PMF of denoted as and corresponding to jammed and unjammed states, respectively. The unconditional PMF of is written as
Setting , , , and , the PMF of initial LLR under the channel with symbol signal-to-noise ratio (SNR) , symbol signal-to-jamming ratio (SJR) , and 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 , at least). Hence, we consider the DDE which does not require Gaussian initial LLR.
3.2. Asymptotic Performance of SLT and SR Codes
We define the generation matrix of inner SLT codes as , where is an identity matrix having a size of and of size is the generation matrix of the conventional LT code [25]. The overhead of the inner SLT code is defined by which is related with the code rates as .
The degree distributions related with the bipartite graph corresponding to is defined as follows. Using polynomial notations, the check node degree distributions is written compactly as , where is the maximum degree. The corresponding edge degree distribution is calculated by and the average degree of CNs is given by . The average degree of VNs is denoted by , and then the variable nodes and edge degrees are both assumed to be Poisson distributed as and , respectively [25]. However, the power series can also be truncated to obtain polynomials that is very close to the exponential, i.e., and , where is large enough to ensure .
We denote the PMF of the message passed from CNs and VNs by and , respectively. The updating rules of CNs and VNs are summarized as follows:where 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 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 fraction 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
(a)
(b)
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 (then the code rate ) 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 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 fractions . 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 with various 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.
It is seen that the SJR-threshold of the LDPC code grows as increases and the worst case appears at corresponding to the threshold of 5.71 dB. As 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 . However, Raptor codes can “automatically” choose the proper rates for different , e.g., under the SJR of 5.71 dB, the rate thresholds of Raptor codes are 0.74 and 0.51 for and , 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 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 , which happened to be the worst case of interference. Using (7), we have at this time and the mean of , denoted by , can be easily calculated based on . We approximate by a Gaussian density with mean and variance 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.
4.2. One-Dimensional Analysis
Since the initial LLR is approximately Gaussian distributed when , 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 is computed by [24] as follows:where is a monotonically increasing function and has an inverse function . Both and 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 aswhere is the variance of initial LLR and 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:where and . Combining (12) and (13), we can get the iteration expression of as the function of , , and as given by
At the end of iteration , the inner decoder outputs the extrinsic information provided to the outer decoder as given byand the corresponding error probability is calculated aswhere the function .
Using the above method and setting and , we depict the asymptotic BERs of the inner SLT code of 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 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 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.
5. Code Design and Simulation Results
Technically, the optimal distributions of inner SLT codes are different for different , 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 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 . 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 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 Table 1, in which denotes the required overhead of inner codes achieving critical BER by the EXIT method and 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, 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 . To investigate the performance of optimized distributions under , we use DDE to get the overhead thresholds of optimized codes and code with 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 for all . At the point of , 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 . When SJR = 4 dB, the optimized code also has better performance than the code with for large but they are very close when is small. Considering that the PBI mainly works on the range of large , 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 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 and 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.
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 of 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.
Data Availability
The numerical and simulation data used to support the findings of this study are included within the supplementary information file.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 61501469) and the China Postdoctoral Science Foundation (Grant no. 2015M572782).