- Research
- Open Access
Jointly optimized multiple Reed-Muller codes for wireless half-duplex coded-cooperative network with joint decoding
- Saqib Ejaz^{1}Email author,
- Fengfan Yang^{1},
- Hongjun Xu^{2} and
- Shunwai Zhang^{3}
https://doi.org/10.1186/s13638-015-0334-1
© Ejaz et al.; licensee Springer. 2015
Received: 6 May 2014
Accepted: 17 March 2015
Published: 22 April 2015
Abstract
In this paper, we present a novel technique to use Reed-Muller (RM) codes for the wireless half-duplex coded-cooperative network. Plotkin’s construction allows RM codes to be used in a coded-cooperative scheme. To improve the cooperation provided by the relay in a coded-cooperative scheme, a design criterion and an efficient algorithm to achieve the design objective are also suggested. Moreover, union bounds for average error probability are determined, for both the cooperative and the non-cooperative schemes based on RM codes in the Rayleigh fading channel. To generalize the proposed RM coded-cooperative scheme, we examined different RM codes at the source and at the relay. At the destination, soft decision maximum likelihood decoding (SD-MLD) and majority logic decoding are used. Theoretical analysis and Monte-Carlo simulations show that the proposed RM coded-cooperative scheme clearly outperforms the RM non-cooperative scheme.
Keywords
- Average error probability
- Coded-cooperative diversity
- Joint multi-RM code design
- Joint decoding
- Majority logic decoding
- Maximum likelihood decoding
- Plotkin’s construction
- Partial encoding
- Reed-Muller codes
1 Introduction
In modern wireless communications, one of the most important aspects is to reduce the channel impairments over the signal propagation. The signal is distorted in a number of ways as it propagates through a wireless channel. Many phenomena like reflection, diffraction, and scattering are responsible for the loss of quality of service (QoS) [1]. Various diversity techniques such as time, frequency, polarization, and space are suggested in the literature to improve the quality of a wireless channel [1]. However, spatial diversity has proven to be the most effective diversity technique to combat the channel effects [2-4]. Unfortunately, many mobile communication devices are not capable to exploit this spatial diversity due to the constraints such as power, size, and hardware complexity. However, novel ideas such as the three-terminal communication [5] and the user cooperation to provide uplink diversity via single antenna sharing [6-8] provide suitable alternative and allow the mobile devices to take advantages of the spatial diversity.
The idea of coded-cooperative diversity was first introduced in [9]. In order to achieve the coded-cooperative diversity, many distributed coding schemes have been reported in the literature such as the convolutional codes [10,11], distributed space-time coding [12,13], distributed low-density parity-check codes (D-LDPC) [14,15] and distributed turbo codes (DTC) [16,17] and more recently the polar codes [18,19]. However, the BER performance of binary turbo and the LDPC codes largely depends on the information block size, the longer the better and vice versa, whereas for the short non-binary turbo and LDPC codes, reasonable performances under the iterative decoding are reported in [20,21]. In many existing and emerging applications (such as device-to-device and sensor networks), it is possible to have scenarios, which may transmit small information block size. Thus, it provides one of the many motivations for our work to develop such a coded-cooperative scheme, which may be useful particularly for applications having small information block size and require low encoding and decoding complexity. For any communication system, there is and always will be a trade-off between the complexity and performance as suggested in [22].
In this paper, we present a novel coded-cooperative scheme based on Reed-Muller (RM) codes [23,24]. The recursive algebraic structure of RM codes makes them different and in many ways superior over other linear block codes. RM codes have low encoding and decoding complexity which is a desirable feature in most of the practical applications [25]. Their algebraic structure and construction allows them to be used as suitable channel codes in a coded-cooperative communication system. There are various methods already suggested in the literature about the recursive construction of RM codes [26]. However, in order to use RM codes in a coded-cooperative diversity scheme, we use Plotkin’s construction [27]. In a coded-cooperative scheme, the source and the relay terminal jointly contribute to build good codes at the destination. The code is good at the destination as it is superior in its decoding properties as compared to the code transmitted at the source without any coded cooperation. To get the maximum coding gains from this joint construction, a joint decoding scheme is established at the destination. The relay plays a vital role in the code construction at the destination and a good code design at the relay can greatly improve the overall performance of the coded-cooperative scheme. Based on this fact, we also propose an efficient algorithm to design a good code at the relay. In [28,29], Plotkin’s construction is used in conjunction with the superposition coding for cooperative broadcasting in wireless networks. In superposition coding [30], two modulated subcode sequences, produced by the two coordinated and independent sources, are combined at the antenna of the receiver (also referred as over the air mixing [31]). The received modulated signals from the two cooperative broadcasting sources are added over the field of real numbers (or depending on the signal constellation used). That scheme also showed promising BER performance gains; however, the scheme proposed in this paper is entirely different from the former in many ways. Firstly, we use Plotkin’s construction to construct a new code at the destination, and binary addition of unmodulated codewords takes place at the relay instead of real number addition as suggested in [28,29]. Secondly, there is only one source in our scheme, and there is no over the air mixing, whereas in [28,29], two independent source nodes broadcast their information to the destination node and over the air mixing is performed.
This paper is organized as follows: Section 2 presents a generalized three-terminal-based coded-cooperative network and gives the channel description. In Section 3, preliminaries related to the RM codes and Plotkin’s construction are presented. Section 4 presents the encoding scheme for RM coded-cooperative and non-cooperative networks. In Section 5, the code design for the single-relay coded cooperation is presented. Performance analysis for RM-code-based cooperative and non-cooperative schemes over the Rayleigh fading channel is presented in Section 6. Section 7 presents various Monte-Carlo simulations and shows the significance of the proposed RM coded-cooperative scheme. Section 8 presents the conclusion of the article.
2 Three-terminal-based coded-cooperative communication model
During the second time slot, the recovered information bits at the relay node (received in the first time slot), are re-encoded using the code C _{2}. After BPSK modulation of the binary codeword, the relay transmits its signal \( {\mathbf{x}}^r=\left[{x}_1^r,{x}_2^r,\dots, {x}_L^r\right] \) to the destination node, where x ^{ r } is a vector of length L and each entry in x ^{ r } is \( {x}_l^r\in\ \left\{-1,+1\right\},\ l=1,2,\dots, L \). The transmission of x ^{ r } to the destination during the second time slot is represented with a dashed line as shown in Figure 1.
3 Fundamentals of RM codes
RM codes belong to a family of linear block codes, which have very nice mathematical properties. Before we discuss the motivation of this paper, let us present some preliminaries related to the RM codes and Plotkin’s construction. Mathematically, RM codes are best described using Boolean function as follows:
Plotkin’s construction is also referred to as |u|u + v| construction.
4 RM coded-cooperative and non-cooperative schemes
However, this time, the codes used at the source and at the relay are two distinct RM codes C _{1}(n, k _{1}, d _{1}) = ℛ(r + 1, m) and C _{2}(n, k _{2}, d _{2}) = ℛ(r, m), respectively. The information sequence m _{1} generated at the source takes two time slots to be recovered at the destination. During the first time slot, k _{1} message bits are encoded using the ℛ(r + 1, m) code, and the resultant codeword vector u of length n = 2^{ m } is then BPSK-modulated. The modulated signal x ^{s} is broadcasted simultaneously to the relay and the destination nodes. The relay decodes the received signal y _{1} to recover the information sequence transmitted at the source. The recovered information sequence \( {\tilde{\mathbf{m}}}_1=\left[{\tilde{m}}_1,{\tilde{m}}_2,\dots, {\tilde{m}}_{k_1}\right] \) may or may not be error free, depending on the S-R link.
The first part in (9) is in fact a codeword generated at the source, and the second part is a codeword generated at the relay. The second part provides additional redundancy, which is exploited by the joint decoding at the destination. The word ‘joint’ here refers to the fact that the signals received from the source y _{2} and the relay y _{3} during different time slots are jointly decoded as a single received vector y = |y _{2}|y _{3}|. The code rate of the overall distributed code is \( {R}_c^0={k}_1/2n \).
In a non-cooperative scheme, both the source and the relay units assumed in the coded-cooperative scheme are considered to be a single source unit, i.e., the source employs two encoders C _{1} = ℛ(r + 1, m) and C _{2} = ℛ(r, m); information bit selection for encoding by the C _{2} = ℛ(r, m) encoder is similar to the coded-cooperative scheme, explained in the next section. The direct sum of the two codewords u + v generated by each encoder is then concatenated with the codeword u generated by the first encoder C _{1} = ℛ(r + 1, m) by an additional block constructing the |u|u + v| codeword, which is then modulated and sent to the destination.
5 Code design for partial encoding at the relay
Therefore, to avoid this worst case also referred to as the first worst case scenario hereafter, we propose a design criterion as follows:
‘Select a subset C _{3} of \( {\tilde{C}}_3=\mathcal{R}\left(r+1,\kern0.5em m+1\right) \) code at the destination with as few as possible codewords of minimum Hamming distance d _{3} = 2^{ m − r } .’
- i.
The first worst case scenario is defined as an event when the codeword generated at the source achieves minimum Hamming weight, i.e., wt(u) = d _{1} and at the relay, the all-zero codeword, i.e., v = 0 ⇒ wt(v) = 0 is obtained. Let K _{1} represent the number of occurrences for the first worst case scenario.
- ii.
The second worst case scenario is defined as an event in which the codeword of minimum Hamming distance \( {d}_3^{1\mathrm{s}\mathrm{t}}={d}_3 \) is obtained at the destination, due to the joint encoding of the source and the relay nodes. Let K _{2} represent the number of occurrences for the second worst case scenario, where K _{2} = K _{ b } is also referred to as the error coefficient of any linear block code.
- iii.
Similarly, the third worst case scenario is defined as an event in which the codeword of Hamming distance \( {d}_3^{2\mathrm{n}\mathrm{d}} \) just greater than the \( {d}_3^{1\mathrm{s}\mathrm{t}} \) is obtained at the destination, due to the joint encoding of the source and the relay nodes, where \( {d}_3^{1\mathrm{s}\mathrm{t}}<{d}_3^{2\mathrm{n}\mathrm{d}}<\dots <{d}_3^{N\mathrm{t}\mathrm{h}} \), and \( {d}_3^{N\mathrm{t}\mathrm{h}} \) is the maximum Hamming distance of a codeword. Let K _{3} represent the number of occurrences for the third worst case scenario.
- iv.
|Ω| represents the cardinality of any set Ω.
- v.
a → b represents that the selection of quantity a results in the quantity b.
- 1)
Determine A = {m _{ κ }}, which is a set of all message blocks that result in the codewords of minimum Hamming distance d _{1} at the source, where κ = 1, 2, …, K _{ b }, and ϑ = |A|.
- 2)Determine B = {λ _{ g }}, which is a set of unique combinations \( {\boldsymbol{\uplambda}}_g=\left[{\lambda}_1,{\lambda}_2,\dots, {\lambda}_{k_2}\right] \), where g = 1, 2, …, S and each λ _{ g } is a vector of length k _{2,} S = |B| and is determined as:$$ S=\left(\begin{array}{c}\hfill {k}_1\hfill \\ {}\hfill {k}_2\hfill \end{array}\right)=\frac{k_1!}{k_2!\left({k}_1-{k}_2\right)!} $$(13)
- 3)
Determine the first worst case scenarios K _{1}, ∀ m _{ κ } ∈ A and ∀ λ _{ g } ∈ B by keeping each unique combination λ _{ g } intermediately fixed at the relay.
- 4)
Select λ _{ g } → min(K _{1}) and store it in the set C. If |C| = 1, then go to step 9 else proceed to step 5.
- 5)
Determine the second worst case scenarios K _{2}, ∀ m _{ κ } ∈ A and ∀ λ _{ g } ∈ C by keeping each unique combination λ _{ g } intermediately fixed at the relay.
- 6)
Select λ _{ g } → min(K _{2}) and store it in the set D. If |D| = 1, then go to step 9 else proceed to step 7.
- 7)
Determine the third worst case scenarios K _{3}, ∀ m _{ κ } ∈ A and ∀ λ _{ g } ∈ D by keeping each unique combination λ _{ g } intermediately fixed at the relay.
- 8)
Select λ _{ g } → min(K _{3}) and store it in the set E. If |E| = 1, then go to step 9 else arbitrarily choose any unique combination λ _{ g } ∈ E and proceed to step 9.
- 9)
The optimum combination λ ^{ o } = λ _{ g } is selected. End of the algorithm.
Finally, the optimum combination λ ^{ o } is fixed at the relay, and only the k _{2} information bits defined by λ ^{ o } are further encoded using ℛ(r, m) code to get the codeword v.
For a better understanding of the proposed algorithm, we present the following illustrative example.
5.1 Example 1
Consider a RM coded-cooperative scheme in which the source employs C _{1}(n _{1} = 16, k _{1} = 11, d _{1} = 4) = ℛ(2, 4) code and the relay employs C _{2}(n _{2} = 16, k _{2} = 5, d _{2} = 8) = ℛ(1, 4) code, and their joint coded cooperation results in a new code \( {C}_3\subseteq {\tilde{C}}_3=\mathcal{R}\left(2,\kern0.5em 5\right) \) at the destination. The source can encode k _{1} = 11 message bits, and the relay can encode only k _{2} = 5 message bits out of k _{1} = 11 message bits (recovered at the relay). The selection procedure of k _{2} = 5 out of k _{1} = 11 message bits according to the proposed algorithm is explained as follows.
Pool ‘ C ’, unique combinations out of 462 combinations, which result in least number of K _{1} and K _{2} cases \( {C}_3\subseteq \left\{{\tilde{C}}_3\right\} \)
Serial number | Pool ‘ C ’ | K _{ 1 } | K _{ 2 } |
---|---|---|---|
Sequence of bit positions | |||
1 | 6,7,8,9,10 | 4 | 30 |
2 | 6,7,8,9,11 | 4 | 24 |
3 | 6,7,810,11 | 4 | 20 |
4 | 6,7,9,10,11 | 4 | 36 |
5 | 6,8,9,10,11 | 4 | 24 |
6 | 7,8,9,10,11 | 4 | 24 |
The cardinality of the set C is |C| = 6. Since, |C| > 1 we proceed to step 5. In the fifth step, K _{2} is determined ∀ λ _{ g } ∈ C by keeping each λ _{ g } intermediately fixed at the relay and transmitting all m _{ κ } ∈ A. The unique combination λ _{ g } → min(K _{2}) = 20 also shown in Table 1 is stored in the set D, where |D| = 1, therefore, we proceed to step 9, and the optimum combination λ ^{ o } is chosen as λ ^{ o } = λ _{ g } = {6, 7, 8, 10, 11} → min(K _{2}) = 20 and the algorithm search is terminated.
6 Average error probability
6.1 AWGN channel
6.2 Rayleigh fading channel
From (18), it can be seen that the diversity order of the coded-cooperative scheme is equal to the Hamming weight w = w _{1} + w _{2}, which is similar to the diversity order of the non-cooperative scheme. Moreover, in the case of the fast Rayleigh fading if γ _{ RD } = γ _{ SD }, then the BER performances of the coded-cooperative and non-cooperative schemes are identical. In the fast Rayleigh fading, the relay node can provide extra benefit only if γ _{ RD } > γ _{ SD }. However, in the case of the slow Rayleigh fading, the true significance of the relay node (or coded cooperation), i.e., the spatial diversity, is observed even if γ _{ RD } = γ _{ SD }, and the scenarios such as γ _{ RD } > γ _{ SD } can be beneficial but not mandatory. These facts are also supported with the help of Monte-Carlo simulations presented in the next section.
7 Simulation results and observations
For the simulations, we consider two different cases to generalize our proposed RM coded-cooperative scheme and the code design algorithm at the relay. In the first case, ℛ(2, 4) and ℛ(1, 4) codes are considered for encoding, and in the second case, we use ℛ(2, 5) and ℛ(1, 5) codes for the encoding. The Rayleigh fading channel is considered among all the communication nodes with perfect CSI known at all corresponding receivers. All transmitting nodes transmit at an equal power, and BPSK modulation is used for the transmission of a codeword over radio frequency (RF) channel. BER simulations are reported in terms of SNR per information bit, defined as γ _{ SD }/R _{ c }, where γ _{ SD } is the SNR per code bit between the S-D link and R _{ c } is the code rate at which the source encodes message bits.
7.1 Case I
The first case employs C _{1} = ℛ(2, 4), and C _{2} = ℛ(1, 4) codes for encoding. In the case of coded cooperation, C _{1} = ℛ(2, 4) code is used at the source and C _{2} = ℛ(1, 4) code is used at the relay, and they jointly construct a new code C _{3} at the destination, where \( {C}_3\subseteq {\tilde{C}}_3=\mathcal{R}\left(2,\kern0.5em 5\right) \). The source encodes k _{1} = 11 message bits, and the relay encodes only k _{2} = 5 bits, which are selected from k _{1} bits (decoded at the relay). The optimum bit selection rule was determined in Example 1, i.e., λ ^{ o } = {6, 7, 8, 10, 11}. The code rate at the source is R _{ c } = 11/16, and the code rate of the overall distributed code is \( {R}_c^0=11/32 \). For case I, the source node of non-cooperative scheme consists of two encoders C _{1} = ℛ(2, 4) and C _{2} = ℛ(1, 4). The information bit selection for encoding by the second encoder C _{2} = ℛ(1, 4) is similar to the coded-cooperative scheme. The direct sum of the two codewords u + v generated by each encoder is determined and then concatenated with the codeword u ∈ ℛ(2, 4) resulting in the codeword |u|u + v| ∈ C _{3}, which is then BPSK-modulated and sent to the destination. Both coded-cooperative and non-cooperative schemes are compared under the condition of an equal rate (11/32) for a fair comparison. BER performance simulations for case I are presented for six different scenarios.
This degradation is due to the uncontrolled error propagation at the relay. The most common remedy to control this error propagation is the utilization of cyclic redundancy check (CRC) at the relay as suggested in [10]. Based on the CRC, the relay decides whether to participate in the cooperation or not; however, error control at the relay is beyond the scope of this paper, and interested reader is referred to [39]. It is observed that at γ _{ SR } = 15 dB the performance of the coded-cooperative scheme significantly improves and approaches the performance of the coded-cooperative scheme with ideal S-R link, i.e., γ _{ SR } = ∞.
The source transmission power P _{ s } is used as a proxy to the SNR per bit γ _{ SD } and changed independently. In Figure 11, BER curves are plotted for different levels of relay transmission power P _{ r } = 14, 16.8, and 19 dBm. It is observed that to achieve BER ≈ 10^{−3} the non-cooperative scheme consumes P _{total} = P _{ s } = 56.2 mW, where P _{ total } is the total transmission power and the coded-cooperative scheme consumed only P _{total} = P _{ s } + P _{ r } = 28.2 mW, where P _{ s } = 3.1 mW, P _{ r } = 25.1 mW. Moreover, under a total power constraint, i.e., P _{total} = 56.2 mW, the coded-cooperative scheme achieves BER ≈ 1.4 × 10^{−4}. The results clearly show that the coded-cooperative scheme outperforms the non-cooperative scheme based on power consumption. Further, it is observed that the overall BER performance of the coded-cooperative scheme improves when P _{ r } is increased at the relay even if the P _{ s } is kept fixed.
7.2 Case II
Since |D| > 1, therefore we proceed to step 7 of the design algorithm. In the seventh step, we observed that again the two combinations λ _{1} and λ _{2} result in equal and minimum number of K _{3} worst cases. Consequently, |E| = 2 > 1 therefore, we select arbitrarily λ _{1} as the optimum combination, i.e., λ ^{ o } = λ _{1}. The code rate at the source is R _{ c } = 1/2, and the overall code rate of the coded-cooperative scheme is \( {R}_c^0=1/4 \).
8 Conclusions
We presented a novel RM coded-cooperative diversity scheme for half-duplex wireless relay networks. The RM coded-cooperative scheme clearly outperforms the RM non-cooperative scheme, and comprehensive BER performance gains are observed for the SD-MLD (≥10 dB) and for the majority logic decoding (≥5 dB) relative to the non-cooperative schemes in Rayleigh fading channels.
The BER performance gains achieved are due to the following factors. At first, it is the joint construction of two channel codes such as ℛ(r + 1, m) code (of small minimum Hamming distance d _{1}) used at the source and ℛ(r, m) code used at the relay to construct a ℛ(r + 1, m + 1) code (of large minimum Hamming distance d _{3}) at the destination. The second factor is the design criterion and an efficient algorithm to ensure that the jointly constructed code C _{3} has the better weight distribution among all the possible subcodes of the code \( {\tilde{C}}_3 \). Thirdly, the joint decoding of the two signals received in two different time slots, transmitted at the source and the relay, provides additional coding gain. Finally, it is the path diversity provided by the relay node that contributes to improve the overall BER performance of the RM coded-cooperative scheme. The performance gains achieved due to the RM coded-cooperative scheme are shown with the help of numerical simulations and validated with the theoretical bounds.
In this paper, we presented only two RM codes at the source. However, the proposed methodology of coded cooperation can be extended in general to other longer length and higher order RM codes as well as to any other family of binary linear block codes. The proposed encoding scheme and code design algorithm at the relay can easily be extended to the multi-relay/user and multi-hop systems; we leave this as our future research work. Moreover, further work is required in performing the sub-optimum decoding of the jointly constructed code particularly of longer lengths.
Declarations
Acknowledgements
The authors are thankful to the editor and the anonymous reviewers for their technical comments and suggestions to improve the overall quality of this manuscript
Authors’ Affiliations
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