text:
Jointly optimized design of distributed RS-coded spatial modulation by appropriate selection at the relay
Chunli Zhao ORCID: orcid.org/0000-0002-8779-81521,
Fengfan Yang1,
Waheed Ullah2 &
Pengcheng Guo1
EURASIP Journal on Wireless Communications and Networking volume 2023, Article number: 1 (2023) Cite this article
This paper proposes the distributed Reed–Solomon (RS)-coded spatial modulation (DRSC-SM) scheme over quasi-static Rayleigh fading channel. Two different RS codes are used, one having a smaller minimum distance at the source and the other having a larger minimum distance at the relay. In order to obtain an optimized code at the destination, a proper encoding strategy must be adapted at the relay. Thus, an efficient design approach is proposed that suitably selects the symbols at the relay. Using this optimized approach as a benchmark, another design is proposed to achieve reduced complexity. To jointly decode the source and relay RS codes, three different decoding algorithms known as the naive, smart and improved smart algorithms are presented. Monte Carlo simulated results reveal that the proposed DRSC-SM scheme utilizing the improved smart decoding algorithm outperforms its counterpart DRSC-SM scheme employing the naive and smart decoding algorithms under identical conditions. The simulation results also validate the superiority of the proposed DRSC-SM scheme over its corresponding coded noncooperative scheme under the same decoding algorithm.
Multiple-input multiple-output (MIMO) systems have been recognized as a core technique for improving the spectral efficiency and link reliability comparing with single antenna wireless systems [1]. A well-known MIMO technique is the vertical Bell Labs Layered Space-Time (V-BLAST) architecture. However, simultaneously transmitting multiple independent data streams from all the antennas requires inter-antenna synchronization (IAS) [2]. The V-BLAST scheme also suffers from inter-channel interference (ICI) at the receiver. Spatial modulation (SM) is able to avoid these problems [3]. This is because only one active transmit antenna is used by SM to transmit constellation symbol at each time instant. Moreover, the utilization of the transmit antenna index as an information-bearing unit effectively boosts the spectral efficiency.
Recently, coded cooperative diversity has drawn substantial research efforts as an efficient technique to combat channel fading, which is an integration of channel codes and cooperative diversity techniques like compress-and-forward (CF), decode-and-forward (DF) and amplify-and-forward (AF) [4,5,6]. To attain coded cooperative diversity, various distributed channel codes like distributed polar codes [7], distributed low-density parity-check (LDPC) codes [8], distributed Reed–Muller (RM) codes [4] and distributed turbo codes [9] have been proposed. The information theoretical analysis has corroborated that coded cooperative scheme achieves a better bit error rate (BER) performance as compared to its counterpart noncooperative scheme. Many related works about distributed channel codes have been studied to achieve coded cooperation. However, less research in the literature is available on the idea of subcode at the destination to enhance system performance through improving the minimum distance.
The 5G communication will require to support traffic originated by devices (usually sensors and actuators) and machines which are involved in communications such as machine-to-machine (M2M) or device-to-device (D2D) communication [10, 11]. Usually, the information sequences with the short length are required in the 5G communication. The well-known short-length RS codes possess simple encoding and decoding complexity, and are very effective in correcting random symbol errors and random burst errors. As maximum distance separable (MDS) codes [12], the short-length RS codes with simple encoding and decoding complexity are able to obtain the superior error performance as well, which shows that the short RS codes can achieve a trade-off between the complexity and error performance. Therefore, the short-length RS codes may be considered as a powerful candidate in 5G scenarios such as ultra-reliable and low-latency communications (URLLC).
In recent literature works [13,14,15], the short RS codes were distributed effectively to construct the coded cooperative networks, which can offer reasonable BER performance with low encoding and decoding complexity. In [1, 5], the authors have introduced that the amalgamation of existing MIMO techniques with the cooperative techniques promises to enhance the spectral efficiency. Thus, in this paper, an emerging MIMO technique, i.e., SM is incorporated into the distributed RS coding scheme, and the distributed RS-coded SM (DRSC-SM) scheme is proposed.
In our proposed DRSC-SM scheme, the source and relay employ different RS codes. At the relay, the information symbols are chosen out of the source transmitted signal. By the mutual cooperation between source and relay, the jointly constructed code is generated at the destination. The closest work related to our paper is presented in [16] where the relay selects the partial information symbols from the decoded source information symbols by the random selection pattern so that a code resulted by the random selection is generated at the destination. However, our paper uses an efficient approach to get the optimized selection pattern by which partial information symbols are properly selected at the relay in order to achieve the optimized code at the destination. Considering this approach as a benchmark, another reduced complexity approach is employed. Our paper also proposes an improved smart decoding algorithm different from the decoding algorithms in [16] to jointly recover the source information. Moreover, the utilization of SM and the improved smart algorithm makes our proposed scheme be very distinct from the recently published literature [17]. The main contributions of this paper are as follows:
DRSC-SM scheme through quasi-static Rayleigh fading channel is proposed.
An efficient approach for selecting proper symbols at the relay node is proposed to produce optimized code at the destination.
Another approach with reduced complexity for constructing the optimized code at the destination is also developed.
At the destination, the joint RS decoding is used in DRSC-SM system with three decoding algorithms known as the naive algorithm, smart algorithm and improved smart algorithm.
The remaining paper is organized as follows. Section 2 introduces the general design of distributed linear block coded SM by subspace approach. Section 3 presents the system model of DRSC-SM scheme. In Sect. 4, the jointly optimized design for DRSC-SM scheme is presented. Section 5 introduces the joint RS decoding with three different decoding algorithms. Section 6 presents the simulation results. Finally, Sect. 7 concludes this article.
General design of distributed linear block coded SM by subspace approach
In this section, a general design of distributed linear block coded SM by subspace approach is discussed. The following two subsections explain the general design in detail.
Distributed linear block codes by subspace approach
A basic coded cooperative communication scheme includes three nodes such as source node (S), relay node (R) and destination node (D) as illustrated in Fig. 1. The DF relaying protocol is employed in cooperative scheme. It requires two time slots to complete the whole communication transmission. In time slot 1, the source node encodes information sequence \({\textbf {m}}_1\) of length \(K_1\) into codeword \({\textbf {c}}_1\) of length N using the linear block code \(C_1 (N,K_1)\), and broadcasts the modulated sequence toward relay and destination nodes.
General system model of distributed linear block codes. It requires two time slots to complete the whole communication transmission, where solid and dashed lines denote the first and second time slots, respectively. The source and relay nodes have linear block codes \(C_1\) and \(C_2\), respectively, which are distributed to achieve coded cooperation. The resulted code at the destination is \(C=\{|\textbf{c}_1 |\textbf{c}_2 |,\textbf{c}_1\in C_1,\textbf{c} _2\in C_2\}\). Since no additional information is generated at the relay (the information at the relay depends on the information at the source) in our proposed system model, the code C is a subcode of \(\bar{C}=\{|\textbf{c}_1 |\bar{\textbf{c}}_2 |,\textbf{c}_1 \in C_1,\bar{\textbf{c}} _2\in C_2\}\), where \(\bar{\textbf{c}} _2\) corresponds to the information independent on the information at the source
In time slot 2, a part of decoded source transmission sequence \(\widetilde{{\textbf {m}}}_1\), that is, \({\textbf {m}}_2\) of length \(K_2 (K_2<K_1)\) is encoded into \({\textbf {c}}_2\) of length N using \(C_2 (N,K_2)\) at the relay. Note that \(\widetilde{{\textbf {m}}}_1\) may not be equal to \({\textbf {m}}_1\) due to the noisy \(S-R\) channel link. However, without additional statement, it is assumed \(\widetilde{{\textbf {m}}}_1={\textbf {m}}_1\) for notational convenience. The codes \(C_1 (N,K_1)\) and \(C_2 (N,K_2)\) are distributed to achieve coded cooperation, where \(C_1 (N,K_1)\) and \(C_2 (N,K_2)\) have minimum distances denoted as \(d_1\) and \(d_2\), respectively.
If the relay node encodes the additional \(K_2\) information symbols \(\bar{{\textbf {m}}} _2\) independent of information sequence \({\textbf {m}}_1\) into the codeword \(\bar{{\textbf {c}}}_2\in C_2 (N,K_2)\), then by concatenating \({\textbf {c}}_1\) and \(\bar{{\textbf {c}}} _2\) the linear block code \(\bar{C}=\{|{\textbf {c}}_1 |\bar{{\textbf {c}}}_2 |,{\textbf {c}}_1 \in C_1 (N,K_1 ),\bar{{\textbf {c}}} _2\in C_2 (N,K_2)\}\) with the minimum distance \(d_{\mathrm{{min}}}(\bar{C} )=d_1\) is resulted at the destination. However, no extra information symbols are resulted in the relay in our investigated distributed linear block coding scheme since all the information symbols \({\textbf {m}}_2\) depend on \({\textbf {m}}_1\). The resulted code at the destination is \(C=\{|{\textbf {c}}_1 |{\textbf {c}}_2 |,{\textbf {c}}_1\in C_1 (N,K_1 ),{\textbf {c}} _2\in C_2 (N,K_2)\}\), which is the subcode of code \(\bar{C}\), i.e., \(C \in \bar{C}\).
Distributed channel coding combined with SM
Coded cooperative communication takes advantage of broadcast nature of the wireless medium to enhance the performance of transmission, which is a valid way that resists channel impairment and shadowing, and enlarges network coverage. Since future wireless communications have a demand for high spectral efficiency and high data rates, further boosting spectral efficiency and data rates of coded cooperation is very pivotal.
MIMO techniques are promising candidates in wireless networks to enhance data rates. By exploiting multiple antennas to transmit data simultaneously, the spectral efficiency is boosted. Many available studies have also presented that incorporating MIMO techniques into the cooperative communication can enhance spectral efficiency, improve data rates and extend network coverage. SM is a recently developed MIMO technique. At any time instant, only one transmit antenna is activated to transmit the constellation symbol. An important feature is that the transmit antenna index is an added source of information. Thus, there are many advantages using SM. For example, ICI and IAS can be completely avoided. Furthermore, the system complexity can be greatly reduced, which is because only activating one transmit antenna at each time instant implies that only one radio-frequency (RF) chain is required. Also, the use of SM provides the spatial diversity for the system. Moreover, the spectral efficiency can be efficiently enhanced due to the utilization of the transmit antenna index as an information-bearing unit. These attractive characteristics make SM be applied in the distributed channel coding scheme by subspace approach.
The earlier section has discussed that short-length non-binary RS codes are a special type of MDS code. Moreover, short-length RS codes can reach singleton bound and have an ability to correct random burst errors, and are then used for distributed channel codes combined with SM to perform error control. The specific system model of distributed RS-coded SM scheme is presented in the next section.
System model of distributed RS-coded SM scheme
The system model of distributed RS-coded SM (DRSC-SM) scheme is illustrated in Fig. 2, where the source S, relay R and destination D are equipped with \(N_{\mathrm{{t}}}\), \(N_{\mathrm{{t}}}\) and \(N_{\mathrm{{r}}}\) antennas, respectively. Completing an entire communication transmission requires two time slots.
DRSC-SM scheme for cooperative communications. The source S, relay R and destination D are equipped with \(N_{\mathrm{{t}}}\), \(N_{\mathrm{{t}}}\) and \(N_{\mathrm{{r}}}\) antennas, respectively. The RS codes \(RS_1\) and \(RS_2\) are employed at the source and relay nodes, respectively. During time slot 1, the source node encodes the information sequence \(\textbf{m}_{1}\) into codeword \(\textbf{c}_1\), which is given into SM mapper. The generated transmission vector is then sent to relay and destination nodes. During time slot 2, the demodulated codeword by the SM demodulator at the relay node is fed into the \(RS_1\) decoder to generate the estimated information sequence. After the \(RS_2\) encoder, the codeword \(\textbf{c}_2\) is generated and further sent to the SM mapper. Then, the generated transmission vector at the relay is sent to destination node. At the common destination node, the SM demodulator demodulates the received vectors during their time slots. Finally, the joint RS decoder is performed to jointly decode the source and relay signal
In time slot 1, the source utilizes RS code \(RS_1 (N,K_1)\) of dimension \(K_1\) and length N over GF(\(2^{n}\)) (\(n=1, 2,\cdots\)) to encode the non-binary information sequence \({\textbf {m}}_1\), where each element in GF(\(2^n\)) can be denoted as a binary vector with length n. The RS code \(RS_1\) has code rate \(R_1=K_1/N\) and generator polynomial \({\textbf {g}}_1 (X)=(X-\beta )(X-\beta ^{2} )\cdots (X-\beta ^{N-K_1})\), where \(\beta ^{f_1}\in\) GF(\(2^n\)) are the roots of \({\textbf {g}}_1 (X)\) for \(f_1\in \{1,2,\cdots ,N-K_1\}\). Then, the codeword polynomial \({\textbf {c}}_1 (X)\) of systematic codeword \({\textbf {c}}_1\) is expressed as
$$\begin{aligned} {\textbf {c}}_1(X) = X^{N-K_1}{} {\textbf {m}}_1(X)+{\textbf {p}}_1(X) \end{aligned}$$
where the expression \({\textbf {m}}_1(X)=m_0^{(1)}+m_1^{(1)} X+\cdots +m_{K_1-1}^{(1)} X^{K_1-1}\) with \(m_{i_1}^{(1)}\in\) GF(\(2^n\))\((i_1=0,1,\cdots ,K_1-1)\) is the polynomial representation of message sequence \({\textbf {m}}_1\), and \({\textbf {c}}_1 (X)=c_0^{(1)}+c_1^{(1)} X+\cdots +c_{N-1}^{(1)} X^{N-1}\) with \(c_{l_1}^{(1)}\in\) GF(\(2^n\))\((l_1=0,1,\cdots ,N-1)\). Furthermore, \({{\textbf {p}}}_1 (X)=p_0^{(1)}+p_1^{(1)} X+\cdots +p_{N-K_1-1}^{(1)} X^{N-K_1-1}\) having \(p_{\zeta _1}^{(1)}\in\) GF(\(2^n\))\((\zeta _1=0,1,\cdots ,N-K_1-1)\) is the parity polynomial and can be computed using polynomial division in GF(\(2^n\)) such as
$$\begin{aligned} {\textbf {p}}_1 (X)=X^{N-K_1}{} {\textbf {m}}_1 (X)/{\textbf {g}}_1(X) \end{aligned}$$
The codeword polynomial \({\textbf {c}}_1(X)\) can be represented in sequence form, i.e., \({\textbf {c}}_1=[c_0^{(1)},c_1^{(1)},\cdots ,c_{N-1}^{(1)}]=[{\textbf {p}}_1,{\textbf {m}}_1]\), where the expressions \({\textbf {p}}_1=[p_0^{(1)},p_1^{(1)},\cdots ,p_{N-K_1-1}^{(1)}]\) and \({\textbf {m}}_1=[m_0^{(1)},m_1^{(1)},\cdots ,m_{K_1-1}^{(1)}]\) are the parity and message symbol sequences, respectively. Since RS code is a non-binary MDS code, the minimum distance of RS code \(RS_1\) is exactly expressed as \(d_1=N-K_1+1\). In the following, the binary vector \({\textbf {c}}'=[c_0',c_1',\cdots ,c_{n-1}']\) representation for each codeword symbol \(c_{l_1}^{(1)}\) is partitioned into two parts. The first \(n_a=\textrm{log}_2\) (\(N_{\textrm{t}}\)) bits of \({\textbf {c}}'\) are given into the antenna mapper to select one active transmit antenna index labeled by \(a_S (\tau _1)\in \left\{ 1,2,\ldots ,N_{\textrm{t}} \right\}\), where \(\tau _1 \in \{0,1,\cdots ,N-1\}\). The remaining \(n_c=n-n_a={\textrm{log}}_2\)(M) bits are sent to the signal mapper that outputs an M-QAM/PSK modulated symbol \(c_S (\tau _1) \in \varLambda\) with \(\varLambda\) being the signal constellation. Through SM, the modulated symbol \(c_{S} (\tau _1)\) is transmitted at the \(a_S (\tau _1 )\)th antenna, and the transmission vector \({\textbf {x}}_S (\tau _1 )=[\cdots ,0,c_S (\tau _1 ),0,\cdots ]^T\) is generated, where \([\cdot ]^T\) denotes the transpose of a vector or matrix. For illustration, Table 1 exhibits different mapping results of SM scheme with code over GF(\(2^4\)), where the product of \(N_{\mathrm{{t}}}\) and M is equal to 16, i.e., \(MN_{\mathrm{{t}}}=16\). In Table 1, the notation \(\alpha\) is a primitive element of GF(\(2^4\)). Accordingly, \(\alpha ^{-\infty }\triangleq {0},\alpha ^0=1, \alpha ,\cdots ,\alpha ^{14}\) constitute all the elements of GF(\(2^4\)). In fact, various finite fields possess different communication scenarios under various combinations of \(N_{\mathrm{{t}}}\) and M. These communication scenarios with RS code over GF(\(2^3\)), GF(\(2^4\)), GF(\(2^5\)) and GF(\(2^6\)) are given in Table 2, respectively. At the relay and destination nodes, the received signals \({\textbf {y}}_{S,R}(\tau _1)\) and \({\textbf {y}}_{S,D}(\tau _1)\) are separately mathematically written as
$$\begin{aligned} {\textbf {y}}_{S,R}(\tau _1 )&={\textbf {H}}_{S,R}{} {\textbf {x}}_S(\tau _1)+{\textbf {n}}_{S,R}(\tau _1)\nonumber \\&={\textbf {h}}_{S,R}^{a_S (\tau _1)}c_S (\tau _1 )+{\textbf {n}}_{S,R}(\tau _1 ) \end{aligned}$$
$$\begin{aligned} {\textbf {y}}_{S,D}(\tau _1 )&={\textbf {H}}_{S,D}{} {\textbf {x}}_S(\tau _1)+{\textbf {n}}_{S,D}(\tau _1)\nonumber \\&={\textbf {h}}_{S,D}^{a_S (\tau _1)}c_S (\tau _1)+{\textbf {n}}_{S,D}(\tau _1 ) \end{aligned}$$
where \({\textbf {H}}_{S,R}\) and \({\textbf {n}}_{S,R}(\tau _1)\) in (3) separately denote the \(N_{\mathrm{{t}}}\times N_{\mathrm{{t}}}\) channel matrix and \(N_{\mathrm{{t}}}\times 1\) noise vector, and their entries separately follow complex Gaussian distribution CN(0, 1) and \(CN(0,N_0)\) with \(N_0\) being the noise power spectral density. \({\textbf {H}}_{S,D}\) and \({\textbf {n}}_{S,D}(\tau _1)\) in (4) are defined like \({\textbf {H}}_{S,R}\) and \({\textbf {n}}_{S,R}(\tau _1)\) in (3), respectively. Furthermore, \({\textbf {h}}_{S,R}^{a_S (\tau _1)}\) and \({\textbf {h}}_{S,D}^{a_S (\tau _1)}\) stand for the \(a_S (\tau _1)\)th columns of channel matrices \({\textbf {H}}_{S,R}\) and \({\textbf {H}}_{S,D}\), respectively. The relay node then performs the SM demodulation [18] for antenna index \(a_S (\tau _1)\) and modulated signal \(c_S (\tau _1)\).
In time slot 2, the demodulated sequence \(\tilde{{\textbf {c}}} _1\) is given into the \(RS_1\) decoder to generate the estimated information sequence \({\textbf {m}}_1\). The sequence \({\textbf {m}}_2\) is then selected from \({\textbf {m}}_1\) and again encoded by the RS code \(RS_2 (N,K_2)\) of dimension \(K_2\) and length N over GF(\(2^n\)). The RS code \(RS_2\) has the minimum distance \(d_2=N-K_2+1\). After the encoding of RS code \(RS_2\), the generated non-binary systematic codeword \({\textbf {c}}_2=[{\textbf {p}}_2,{\textbf {m}}_2]\) is sent to the SM mapper that provides the transmission vector \({\textbf {x}}_R (\tau _2 )=[\cdots ,0,c_R (\tau _2 ),0,\cdots ]^T\) with \(\tau _2\in \{0,1,\cdots ,N-1\}\), where the modulated symbol \(c_R (\tau _2 )\) is conveyed at the \(a_R (\tau _2)\)th antenna. The received signal \({\textbf {y}}_{R,D} (\tau _2)\) at the destination node is mathematically modeled as
$$\begin{aligned} {\textbf {y}}_{R,D}(\tau _2 )&={\textbf {H}}_{R,D}{} {\textbf {x}}_R(\tau _2)+{\textbf {n}}_{R,D}(\tau _2)\nonumber \\&={\textbf {h}}_{R,D}^{a_R(\tau _2)}c_R(\tau _2)+{\textbf {n}}_{R,D}(\tau _2) \end{aligned}$$
where the definitions of \({\textbf {H}}_{R,D}\) and \({\textbf {n}}_{R,D}(\tau _2)\) are similar to \({\textbf {H}}_{S,R}\) and \({\textbf {n}}_{S,R}(\tau _1)\) in (3), respectively. \({\textbf {h}}_{R,D}^{a_R(\tau _2)}\) denotes the \(a_R(\tau _2)\)th column of the channel matrix \({\textbf {H}}_{R,D}\). Afterward, the spatial demodulation is utilized to detect the received sequences during their respective time slots. Finally, the joint decoding is performed to generate the estimated information sequence \(\bar{{\textbf {m}}}_1\) of \({\textbf {m}}_1\) transmitted at the source node.
As mentioned earlier, the generated codeword belongs to the subcode of C of \(\bar{C}\). It should be noted that the codeword with weight \(d_1\) is likely to occur. This is due to the worst scenario that zero weight codeword is generated at the relay when the codeword with weight \(d_1\) is produced at the source node. Therefore, the design criterion of an optimized subcode is to avoid large number of worst scenarios, which implies \({\textbf {m}}_2\) must be suitably selected from \({\textbf {m}}_1\) at the relay. The detailed optimized subcode design approaches by properly selecting information at the relay are given in Sect. 4.
Table 1 Different mapping results of SM scheme with code over GF(\(\varvec{2^{4}}\))
Table 2 Different communication scenarios with RS code over various Galois fields
In this section, two design approaches for optimizing the proposed DRSC-SM scheme are proposed, which makes the destination node generate an optimized code with a better weight distribution.
Two jointly optimized design approaches for DRSC-SM scheme
In the proposed DRSC-SM scheme, each selection in relay yields the subcode C of code \(\bar{C}\). Therefore, a proper symbol selection at the relay is very pivotal in order to obtain an optimized subcode at the destination. The complexity analysis of the two optimized approaches is also presented in this section.
Approach 1: brute-force search over all candidates
An efficient Approach 1 for constructing an optimized subcode is proposed in this subsection. In this approach, we perform the brute-force search over all candidates (i.e., all selection patterns) at the relay to find an optimized pattern by which \(K_2\) information symbols are selected from the decoded \(K_1\) source information symbols. During the search, all information symbol sequences that may generate the codeword of weight \({\textrm{wt}}({\textbf {c}}_1)= d_1\) at the source are considered. In order to construct an optimized code with better weight distribution at the destination, three scenarios are considered. The considered event in the first scenario contains those codewords with low weight \(d_1\) in destination, which are jointly built by the codeword of minimum weight \({\textrm{wt}}({\textbf {c}}_1)= d_1\) at the source and the codeword with weight \({\textrm{wt}}({\textbf {c}}_2 )=0\) at the relay. In the second scenario, the destination node includes all codewords of low weight \(d_1+d_2\), where the codewords of minimum weights \(d_1\) and \(d_2\) are generated at the source and relay, respectively. The third scenario considers those codewords having low weight greater than \(d_1+d_2\) for \({\textrm{wt}}({\textbf {c}}_1 )= d_1\) at the source and \({\textrm{wt}}({\textbf {c}}_2 )> d_2\) at the relay. Assume that \(J_1\), \(J_2\) and \(J_3\) represent the number of occurrences for the first, second and third scenarios, respectively. The design steps of Approach 1 are given as follows:
Determine \(\varOmega =\{{\textbf {e}}_q\}, (q=1,2,\cdots ,K_b)\) that is the set of all information symbol sequences generating the codewords of weight \({\textrm{wt}}({\textbf {c}}_1 )=d _1\) at the source, where \(K_b\) denotes the number of codewords associated with weight \(d_1\) [10].
Determine the set of information selection patterns \(A=\{\varvec{\upeta }_g\}\) with \(\varvec{\upeta }_g=[\eta _{0},\eta _{1},\cdots ,\eta _{K_2-1}]\) and \(g=1,2,\cdots ,\varPhi\), where \(\eta _l \in \{1,2,\cdots ,K_1\}\) for \(l=0,1,\cdots ,K_2-1\) denotes the position of the selected symbols in the \(K_1\) symbols, and \(\varPhi\) is expressed as
$$\begin{aligned} \varPhi = \left( {\begin{array}{c}K_1\\ K_2\end{array}}\right) = \frac{K_1!}{K_2!(K_1-K_2)!} \end{aligned}$$
For the first worst scenario, find out \(J_1\) for all \({\textbf {e}}_q\in {\varOmega }\) by keeping each information selection pattern \(\varvec{\upeta }_g\in A\) fixed at the relay.
Pick up selection patterns \(\varvec{\upeta }_g\) resulting in the minimum value of \(J_1,\) i.e., min\((J_1)\) and store them in the set B. If B has cardinality \(|B| = 1\), go to step 9 else move to next step.
For the second worst scenario, find out \(J_2\) for all \({\textbf {e}}_q\in {\varOmega }\) by keeping each information selection pattern \(\varvec{\upeta }_g\in B\) fixed at the relay.
Pick up selection patterns \(\varvec{\upeta }_g\) that result in min\((J_2)\) and store them in the set C. If |C| = 1, go to step 9 else move to next step.
For the third worst scenario, find out \(J_3\) for all \({\textbf {e}}_q\in {\varOmega }\) by keeping each information selection pattern \(\varvec{\upeta }_g\in C\) fixed at the relay.
Pick up selection patterns \(\varvec{\upeta }_g\) that result in min\((J_3)\) and store them in the set E. If |E| = 1, go to step 9. If not, E is used as the update of the set C, increase the codeword weight \({\textrm{wt}}({\textbf {c}}_2)\) by 1 at the relay, and then go back to step 7.
Pick up the unique information selection pattern \(\varvec{\upeta }^{(1)}=\varvec{\upeta }_g\) and finish Approach 1.
Approach 2: local search over partial candidates
As presented in Sect. 4.1.1, Approach 1 can effectively choose the optimized selection pattern from the total \(\varPhi\) selection patterns, during which all the information symbol sequences generating the codewords with weight \({\textrm{wt}}({\textbf {c}}_1)=\) \(d_1\) at the source are considered. Therefore, for the larger block length code, the search complexity of Approach 1 is relatively high. To low the computational complexity of Approach 1, we propose the local search Approach 2. Different from Approach 1, Approach 2 searches for the optimized selection pattern from the partial candidates of the total \(\varPhi\) selection patterns, during which we only consider the partial information symbol sequences resulting in the codeword weight \({\textrm{wt}}({\textbf {c}}_1)=\) \(d_1\) at the source. The following steps show the search process:
Determine the set \(\varOmega ^{'}\) of partial information symbol sequences resulting in the codeword weight \({\textrm{wt}}({\textbf {c}}_1)=d _1\) at the source. Since the source information sequence \({\textbf {e}}_q\) generating \({\textrm{wt}}({\textbf {c}}_1)=d _1\) can be searched from those information sequences of weight \(0<{\textrm{wt}}({\textbf {e}}_q )\le d _1\), it implies that \({\textbf {e}}_q\) has at least \(\mu =K_1-{\textrm{min}}(K _1, d _1)\) zeros. The set \(\varOmega ^{'}\) can be obtained by the following method:
Divide the information sequence \({\textbf {e}}_q\) of length \(K_1\) into two parts. Scenario (a): the first \(\lceil K_1/2\rceil\) symbols form the first part and the remaining \(K_1-\lceil K_1/2\rceil\) symbols form the last part, where \(\lceil \cdot \rceil\) denotes ceil operation. Scenario (b): the first \(K_1-\lceil K_1/2\rceil\) symbols constitute the first part and the remaining \(\lceil K_1/2\rceil\) symbols constitute the last part. The symbol distribution of the information sequence \({\textbf {e}}_q\) is shown in Fig. 3.
Determine the positions of \(\mu\) zeros in the information sequence \({\textbf {e}}_q\). Scenario (a): the \(i\quad (\lceil \mu /2\rceil \le i\le \textrm{min}\) \((\mu ,\lceil K_1/2\rceil ))\) zeros are randomly distributed in the first part of Fig. 3a and other \(\mu -i\) zeros are uniquely assigned in the second part of Fig. 3a. Scenario (b): the i zeros are randomly distributed in the last part of Fig. 3b and the other \(\mu -i\) zeros are uniquely assigned in the first part of Fig. 3b.
Based on the above two substeps, we get the information symbol sequence set \(\varOmega '=\{{\textbf {e}}_{\bar{q}} '\}\) with \(\bar{q}=1,2,\cdots ,|\varOmega '|\), where \(\varOmega '\) is the subset of \(\varOmega\).
Determine the set \(A^{'}\) of partial selection patterns.
Select \(K_2\) symbols out of \(K_1\) source message symbols. Scenario (a): randomly select t \((\lceil K_2/2\rceil \le t\le\)min(\(K_2,\lceil K_1/2\rceil ))\) symbols from the first part of Fig. 3a, and the other \(K_2-t\) symbols are uniquely selected from the remaining part of Fig. 3a. Scenario (b): randomly select t symbols from the last part of Fig. 3b, and the other \(K_2-t\) symbols are uniquely selected from the first part of Fig. 3b.
Based on the positions of the selected \(K_2\) symbols in the above substep, we get the set \(A^{'}=\{\varvec{\upeta }_{\bar{g}}^{'}\}\) of selection patterns \(\varvec{\upeta }_{\bar{g}}'=[\eta _0', \eta _1',\cdots ,\eta _{K_2-1}']\), where \(\bar{g}=1, 2,\cdots ,|A'|\) and \(A^{'}\) is the subset of A.
The other steps follow steps 3–9 of the brute-force search method. Lastly, pick up this unique selected pattern \(\varvec{\upeta }^{(2)}=\varvec{\upeta }_{\bar{g}}^{'}\) and finish Approach 2.
The determined \(\varvec{\upeta }^{(2)}\) is fixed in the relay and \(K_2\) symbols are selected from \(K_1\) source information symbols for further encoding by the RS code \(RS_2 (N,K_2)\). At the destination, an optimized code with a better weight distribution is generated.
Information sequence with more symbols in a first part, b last part. For information sequence with length \(K_1\) at the source node, its symbols are divided into two cases. One case is the first part has more symbols, and the other case is the last part has more symbols. a denotes the symbol distribution diagram of information sequence with more symbols in the first part. b denotes the symbol distribution diagram of information sequence with more symbols in the last part
Design example for two approaches
In order to better understand the above two approaches, a simple example is listed here. In our example, we consider the distributed RS codes employing \(RS_1 (N,K_1 )=RS_1 (15,13)\) and \(RS_2 (N,K_2 )=RS_2 (15,7)\) over GF(\(2^4\)) that is constructed using polynomial \(1+X+X^4\), where \(RS_1 (15,13)\) and \(RS_2 (15,7)\) are placed at the source and relay nodes, respectively. At the relay, only \(K_2 = 7\) symbols are selected out of \(K_1 = 13\) recovered symbols. First, Approach 1 is used to determine the selection pattern at the relay as follows:
Determine the set \(\varOmega =\{{\textbf {e}}_q\}\) of information sequences generating the codeword of weigh three for \(RS_1 (15,13)\) at the source, where \(|\varOmega |\) = 6825 with the help of computer simulations or theoretical analysis in [19].
Determine the set \(A=\{\varvec{\upeta }_g\}\) that consists of all information selection patterns, i.e., 1716 patterns \(\varvec{\upeta }_g = [\eta _0,\eta _1,\cdots ,\eta _6]\).
Determine the parameter \(J_1\) of the first worst scenario by keeping each \(\varvec{\upeta }_g\in A\) fixed at the relay and transmitting all \({\textbf {e}}_q\in \varOmega\), and find out min(\(J_1\)) = 840.
Select the patterns \(\varvec{\upeta }_g\) resulting in min(\(J_1\)) = 840 and store them in B. Since \(|B|=1716 >1\), then we proceed to the next step. Table 3 presents five patterns out of 1716 patterns causing min(\(J_1\)) = 840.
Determine the parameter \(J_2\) of the second worst scenario by keeping each \(\varvec{\upeta }_g\in B\) fixed at the relay and transmitting all \({\textbf {e}}_q\in \varOmega\), and find out min(\(J_2\)) = 4140, also presented in Table 3.
Select the patterns \(\varvec{\upeta }_g\) resulting in min(\(J_2\)) = 4140 and store them in C. Now, |C| = 1; therefore, we move to step 9 and terminate Approach 1. Then, the unique selection pattern is defined such as \(\varvec{\upeta }^{(1)}=\varvec{\upeta }_g=[1,3,5,8,9,11,12]\).
Now, we employ Approach 2 to determine the selection pattern at the relay. Since the minimum weight of codewords generated by \(RS_1\) (15,13) is three, the corresponding information sequences of length thirteen have at least ten zeros. The detailed process that obtains the selection pattern is presented as follows:
Determine the positions of ten zeros of information sequence at the source. Scenario (a): the \(i\quad (5\le i\le 7)\) zeros 10 zeros are randomly distributed in the first part (7 symbols) of Fig. 4a and the other \(10-i\) zeros are uniquely assigned in the remaining part (6 symbols) of Fig. 4a. Scenario (b): the i zeros of 10 zeros are randomly distributed in the last part (7 symbols) of Fig. 4b and the other \(10-i\) zeros are uniquely assigned in the remaining part (6 symbols) of Fig. 4b. According to this way, we obtain the set \(\varOmega '=\{{\textbf {e}}_{\bar{q}}'\}\) with \(|\varOmega ' |=1965\), which includes partial information sequences generating codewords of weight three.
7 symbols are selected from 13 symbols. Scenario (a): randomly select \(t\quad (4\le t \le 7)\) symbols from the first 7 symbols of Fig. 4a but other \(7-t\) symbols are uniquely selected from other 6 symbols of Fig 4a. Scenario (b): randomly select t symbols from the last 7 symbols of Fig. 4b but other \(7-t\) symbols are uniquely selected from those remaining 6 symbols of Fig. 4b. Based on this method, we determine the set \(A'=\{\varvec{\upeta }_{\bar{g}}'\}\) of information selection patterns \(\varvec{\upeta }_{\bar{g}}'=[\eta _0',\eta _1',\cdots ,\eta _6']\) and find out \(|A' |\) = 128.
Determine the parameter \(J_1\) of the first worst scenario by keeping each \(\varvec{\upeta }_{\bar{g}}'\in A'\) fixed at the relay and transmitting all \({\textbf {e}}_{\bar{q}}'\in \varOmega '\), and find out min(\(J_1\)) = 195.
Select the patterns \(\varvec{\upeta }_{\bar{g}}'\) resulting in min(\(J_1\)) = 195 and store them in \(B'\). Since \(|B'|=128>1\), then we proceed to the next step. Table 4 presents five patterns out of 128 patterns causing min(\(J_1\)) = 195.
Determine the parameter \(J_2\) of the second worst scenario by keeping each \(\varvec{\upeta }_{\bar{g}}'\in B'\) fixed at the relay and transmitting all \({\textbf {e}}_{\bar{q}}'\in \varOmega '\), and find out min(\(J_2\))=68, also presented in Table 4.
Select the patterns \(\varvec{\upeta }_{\bar{g}}'\) resulting in min(\(J_2\))=68 and store them in \(C'\). Now \(|C'|=1\); therefore, we move to step 9 and terminate Approach 2. Accordingly, the selection pattern is chosen as \(\varvec{\upeta }^{(2)}=\varvec{\upeta }_{\bar{g}}'=[3,5,7,9,10,11,12]\).
Information sequence with 7 symbols in a first part, b last part. For information sequence with length \(K_1=13\) at the source node, its symbols are divided into two cases. One case is the first part has more symbols (7 symbols), and the other case is that the last part has more symbols (7 symbols). a denotes the symbol diagram of information sequence with more symbols (7 symbols) in the first part. b denotes the symbol diagram of information sequence with more symbols (7 symbols) in the last part
Table 3 Five information selection patterns from 1716 patterns that result in min\(\varvec{(J_1)}\) and min\(\varvec{(J_2)}\)
Table 4 Five information selection patterns from 128 patterns that result in min\(\varvec{(J_1)}\) and min\(\varvec{(J_2)}\)
The complexity of both proposed design approaches is compared in terms of multiplication and addition operations in this subsection. At the source, encoding one information sequence with length \(K_1\) requires \(\varpi _S^{\times }=K_1 (N-K_1)\) multiplication operations and \(\varpi _S^{+}=K_1 (N-K_1)\) addition operations. Therefore, the total \(\varpi _S=\varpi _S^{\times }+\varpi _S^{+}=2K_1 (N-K_1)\) elementary operations are involved in encoding one information sequence. Similarly, encoding one information sequence with length \(K_2\) at the relay requires the total \(\varpi _R=\varpi _R^{\times }+\varpi _R^{+}=2K_2 (N-K_2)\) elementary operations, where \(\varpi _R^{\times }=K_2 (N-K_2)\) and \(\varpi _R^{+}=K_2 (N-K_2)\).
In Approach 1, \(|\varOmega |\) information sequences generating the codewords with \({\textrm{wt}}({\textbf {c}}_1)=\) \(d_1\) at the source are considered. Thus, the total number of elementary operations at the source is \(\varpi _{S,\varOmega }=|\varOmega | \varpi _S=2K_1 |\varOmega |(N-K_1)\). At the relay, the \(K_2\) symbols are selected from the decoded \(K_1\) source symbols for further encoding, and therefore, encoding \(|\varOmega |\) information blocks needs to perform \(\varpi _{R,\varOmega }=2K_2 |\varOmega |(N-K_2)\) for each selection pattern. Let the number of considered selection patterns be \(M_{d_1}^{(1)}\), \(M_{d_1+d_2}^{(1)}\), \(M_{d_1+d_2+1}^{(1)}\),..., \(M_{2N}^{(1)}\), respectively, when finding out the number of codeword weight \(d_1\), \({d_1+d_2}\), \({d_1+d_2+1}\),..., 2N at the destination. If the optimized selection pattern \(\varvec{\upeta }^{(1)}\) is determined in finding out the number of codeword weight w at the destination, the computational complexity of Approach 1 is then expressed as
$$\begin{aligned} {\varpi _{\mathrm{{Approach 1}}}=\varpi _{S,\varOmega }+\varpi _{R,\varOmega }(M_{d_1}^{(1)}+M_{d_1+d_2}^{(1)}+M_{d_1+d_2+1}^{(1)}+\ldots +M_{w}^{(1)})} \end{aligned}$$
In Approach 2, we consider \(|\varOmega ' |\) information sequences resulting in the codewords with \({\textrm{wt}}({\textbf {c}}_1)=\) \(d_1\) in the source, and the overall elementary operation number is then denoted as \(\varpi _{S,\varOmega '}=|\varOmega ' | \varpi _S=2K_1 |\varOmega ' |(N-K_1)\). Then, for each selection pattern, encoding \(|\varOmega ' |\) information sequences at the relay requires \(\varpi _{R,\varOmega ' }=|\varOmega ' | \varpi _R=2K_2 |\varOmega ' |(N-K_2)\) elementary operations. If the optimized selection pattern \(\varvec{\upeta }^{(2)}\) is determined in finding out the number of codeword weight \(w'\) at the destination, the complexity of Approach 2 is represented as
$$\begin{aligned} \varpi _{\mathrm{{Approach 2}}}&=\varpi _{S,\varOmega '}+\varpi _{R,\varOmega '}(M_{d_1}^{(2)}+M_{d_1+d_2}^{(2)}+M_{d_1+d_2+1}^{(2)}+\ldots +M_{w'}^{(2)}) \end{aligned}$$
where \(M_{d_1}^{(2)}\), \(M_{d_1+d_2}^{(2)}\), \(M_{d_1+d_2+1}^{(2)}\),..., \(M_{w'}^{(2)}\) are the number of considered selection patterns, respectively, in finding out the number of codeword weight \(d_1\), \({d_1+d_2}\), \({d_1+d_2+1}\),..., \({w'}\) at the destination. Based on Eqs. (7) and (8), we can get the complexity of Approach 1 and Approach 2. As presented in Sect. 4.1.3, the parameters \(|\varOmega ' |=1965\), \(M_{d_1}^{(2)}=|A' |=128\) and \(M_{d_1+d_2}^{(2)}=|B' |=128\) in Approach 2 are less than the parameters \(|\varOmega |=6825\), \(M_{d_1}^{(1)}=|A |=1716\) and \(M_{d_1+d_2}^{(1)}=|B|=1716\) in Approach 1, respectively. Thus, Approach 2 has a reduced complexity over Approach 1.
Table 5 Relationship between different distributed RS codes and optimized selection patterns
Error performance comparisons of two approaches
We present the BER performance curves of ideal DRSC-SM scheme (\(\lambda _{S,R}=\infty\)) with \(RS_1\) (15,13) and \(RS_2\) (15,7) based on two design approaches as presented in Fig. 5. In the simulation, the smart decoding algorithm discussed in Sect. 5, 4-QAM and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\) are used. Through the simulated results, we notice that the DRSC-SM scheme under Approach 1 and Approach 2 perform nearly identical performance at low signal-to-noise ratio (SNR). Moreover, at high SNR, the DRSC-SM scheme employing Approach 2 is only 0.15 dB worse than that of Approach 1 at BER \(\approx 2.2\times 10^{-6}\). This phenomenon reveals the validity of our proposed Approach 2 performing partial search as compared to Approach 1 performing brute-force search. Similarly, we investigate the other distributed RS-coded SM schemes in this paper, whose distributed RS codes and the corresponding optimized selection patterns are presented in Table 5.
Performance of DRSC-SM scheme (4-QAM) with \(RS_1(15,13)\) and \(RS_2(15,7)\) employing Approach 1 and Approach 2. The dashed line with circle mark denotes the BER performance of DRSC-SM scheme under Approach 2, and the solid line with square mark denotes the BER performance of DRSC-SM scheme under Approach 1
In Sect. 4.1.2, the general description of Approach 2 has been presented. Based on substep (i) in step 2, multiple reasonable selections of \(K_2\) information symbols from the \(K_1\) information symbols at the relay can be obtained. If we apply the selection way to the selection process of t symbols from \(\lceil K_1/2\rceil\) symbols in scenarios (a) and (b) of substep (i) in step 2, Approach 2 will be further modified into an approach with lower complexity. In Sect. 6, we will analyze the system performance under the modified Approach 2.
It should be noted that our presented search approaches determining the optimized selection pattern are the general approaches, where the selection can be binary bit selection or non-binary symbol selection. In our paper, we have already applied these methods to the symbol selection of RS codes which are non-binary.
Joint RS decoding of DRSC-SM scheme
Joint decoding at the destination is one of the key features of coded cooperative systems. In this section, three decoding algorithms, i.e., naive algorithm, smart algorithm and improved smart algorithm, are presented to jointly decode the transmitted sequences.
Naive decoding algorithm
In this subsection, the naive decoding algorithm is presented as follows:
Give the first and second parts of demodulated \(|\hat{{\textbf {c}}}_1|\hat{{\textbf {c}}}_2 |\) into the \(RS_1\) decoder and \(RS_2\) decoder, respectively. The Euclidean algorithm [19] which is based on the greatest common divisor (GCD) is employed in the M-ary \(RS_k (k=1, 2)\) decoder. Then, the estimated non-binary information sequences \(\hat{{\textbf {m}}}_1\) and \(\hat{{\textbf {m}}}_2\) are obtained.
Select a threshold value \(\delta\), which is taken as the SNR where the BER performance (individual performance) of \(RS_1\) and \(RS_2\) codes crosses each other or meets each other.
If SNR\(<\delta\), then the estimated sequence of \({\textbf {m}}_1\) is taken as \(\bar{{\textbf {m}}}_1=\hat{{\textbf {m}}}_1\). If SNR\(\ge \delta\), those selected \(K_2\) symbols of sequence \(\hat{{\textbf {m}}}_1\) are replaced with sequence \(\hat{{\textbf {m}}}_2\), then the new generated sequence \(\bar{{\textbf {m}}}_1\) of \(\hat{{\textbf {m}}}_1\) is eventually viewed as the estimated sequence of \({\textbf {m}}_1\).
In channel coding, the BER performance of code with more error correcting capability is worse than the code with less error correcting capability for low SNR regime, while it outperforms the less error correcting capability code over high SNR regime. Thus, because of large \(d_2\) of RS code \(RS_2\), for high SNR all \(K_2\) symbols of \(\hat{{\textbf {m}}}_2\) are more reliable than those selected \(K_2\) symbols of \(\hat{{\textbf {m}}}_1\), but vice versa for low SNR. This is the reason for applying the judgment criteria in step 3.
Smart decoding algorithm
We use the naive decoding algorithm as a benchmark and propose another decoding algorithm, i.e., smart decoding algorithm. The steps of the smart decoding algorithm are shown as follows:
Firstly feed the second part \(\hat{{\textbf {c}}}_2\) of demodulated \(|\hat{{\textbf {c}}}_1|\hat{{\textbf {c}}}_2 |\) into the \(RS_2\) decoder, which generates the estimated systematic non-binary information symbol sequence \(\hat{{\textbf {m}}}_2\).
Replace those selected \(K_2\) symbols of the sequence \(\hat{{\textbf {c}}}_1=[\hat{{\textbf {p}}}_1,\hat{{\textbf {m}}}_1']\) comprised of the parity symbol sequence \(\hat{{\textbf {p}}}_1\) and information symbol sequence \(\hat{{\textbf {m}}}_1'\) by using the sequence \(\hat{{\textbf {m}}}_2\). The joint source relay sequence \(\bar{{\textbf {c}}}_1=[\hat{{\textbf {p}}}_1,\hat{{\textbf {m}}}_1'']\) of length N is then produced.
Send the generated joint source relay sequence \(\bar{{\textbf {c}}}_1\) to the \(RS_1\) decoder. Then, we obtain an estimate \(\bar{{\textbf {m}}}_1\) of information symbol sequence \({\textbf {m}}_1\).
Improved smart decoding algorithm
In this subsection, we propose a new decoding algorithm, i.e., the improved smart decoding algorithm. The decoding process is realized by the following steps:
The \(RS_2\) decoder receives the second part \(\hat{{\textbf {c}}}_2\) of the demodulated sequence \(|\hat{{\textbf {c}}}_1|\hat{{\textbf {c}}}_2 |\) to yield the estimation \(\hat{{\textbf {m}}}_2\) of the information symbol sequence \({\textbf {m}}_2\) of the \(RS_2\) encoder.
The estimated information symbol sequence \(\hat{{\textbf {m}}}_2\) and the first part \(\hat{{\textbf {c}}}_1=[\hat{{\textbf {p}}}_1,\hat{{\textbf {m}}}_1']\) of \(|\hat{{\textbf {c}}}_1|\hat{{\textbf {c}}}_2 |\) are given into the decision combiner block.
Choose a threshold value \(\delta\) that is the SNR where the error performance of the codes \(RS_1\) and \(RS_2\) meets each other. Since the minimum distance of the \(RS_2\) code is larger than that of the \(RS_1\) code, the \(RS_2\) code has a better error correcting capability. Based on the channel coding theory [19], the \(RS_1\) code performs better performance than the \(RS_2\) code for SNR\(<\delta\), but the latter outperforms the former for SNR\(\ge \delta\).
If SNR\(<\delta\), the decision combiner directly uses the sequence \(\hat{{\textbf {c}}}_1\) as the output \(\bar{{\textbf {c}}}_1\), i.e., \(\bar{{\textbf {c}}}_1=\hat{{\textbf {c}}}_1\).
If SNR\(\ge \delta\), \(\hat{{\textbf {m}}}_2\) is more reliable than the \(K_2\) symbols (placed in the selected \(K_2\) positions) in \(\hat{{\textbf {m}}}_1'\). Thus, we replace the selected \(K_2\) symbols of \(\hat{{\textbf {m}}}_1'\) by using the estimated information symbol sequence \(\hat{{\textbf {m}}}_2\), i.e., the output of the decision combiner is \(\bar{{\textbf {c}}}_1=[\hat{{\textbf {p}}}_1,\hat{{\textbf {m}}}_1'']\).
Finally, the \(RS_1\) decoder accepts the sequence \(\bar{{\textbf {c}}}_1\) from the decision combiner and generates the estimation \(\bar{{\textbf {m}}}_1\) of the source information symbol sequence \({\textbf {m}}_1\).
The improved smart algorithm cleverly incorporates the idea of threshold selection, and flexibly selects more reliable symbols to enter the \(RS_1\) decoder according to the change of SNR. However, the smart algorithm directly uses the decoded \(K_2\) symbols of the \(RS_2\) decoder as the input of the \(RS_1\) decoder over the whole SNR. Therefore, the proposed improved smart algorithm can effectively ameliorate the performance of the smart algorithm under low SNR, as presented in Sect. 6.
The BER performance of DRSC-SM scheme and reference schemes through quasi-static Rayleigh fading channel is presented in this section. Three cases of distrusted RS codes presented in Table 5 are mainly considered to generalize our proposed DRSC-SM scheme. For the first case, we employ the RS codes over GF(\(2^4\)) constructed based on the polynomial \(1+X+X^4\). Furthermore, the RS codes over GF(\(2^5\)) and GF(\(2^6\)) which are constructed using polynomials \(1+X^2+X^5\) and \(1+X+X^6\) are considered in the second and third cases, respectively. The modulation schemes such as 4-QAM, 8-QAM and 16-QAM are applied in the first, second and third cases, respectively. The SNR per bit between the \(S-D\) link is represented as \(\lambda _{S,D}\). Similarly, \(\lambda _{S,R}\) and \(\lambda _{R,D}\) denote the SNR per bit for the \(S-R\) and \(R-D\) links, respectively. It is assumed that the relay node has a 2 dB SNR gain over the source node, i.e., \(\lambda _{R,D}=\lambda _{S,D}+2\) dB. The Euclidean decoding algorithm is used for overall simulation. Furthermore, we assume that all corresponding receivers have perfect channel state information.
BER performance of DRSC-SM scheme under different subcode design approaches
In this section, the BER performance of DRSC-SM scheme under different subcode design approaches for the first case is illustrated in Fig. 6. The ideal source-to-relay channel (\(\lambda _{S,R}=\infty\)), \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\), maximum likelihood (ML) detection and joint RS decoding (smart algorithm) are used. The simulated results show the benefits of Approach 1 and Approach 2 as compared with the randomly selected approach, which is due to the fact that Approach 1 and Approach 2 effectively reduce the number of minimum weight codewords generated at the destination. The phenomenon illustrates the advantageous effect of proper symbol selection of our presented approaches on the performance of DRSC-SM scheme. In addition, Fig. 6 shows that the system performance under Approach 1 is very near to that under Approach 2 for low SNR. Also, for high SNR, the performance degradation is negligible. This is because the optimized code generated at the destination under the two search approaches have the same minimum distance (i.e., 3). Thus, this fully demonstrates the effectiveness of the local search Approach 2 over the brute-force search Approach 1. Furthermore, the system performance under (\(\lambda _{S,R}=\infty\) ) employing the smart decoding algorithm with the randomly selected approach and Approach 2 under ML detection and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\) for the second and third cases is presented in Figs. 7 and 8. It is seen that the DRSC-SM scheme with Approach 2 achieves better performance than that with the randomly selected approach. It can be explained that Approach 2 generates an optimized code with a larger minimum distance (i.e., 32 for the second case and 30 for the third case) at the destination.
Performance of DRSC-SM scheme (4-QAM) with \(RS_1 (15,13)\) and \(RS_2(15,7)\) under different approaches. The solid line with diamond mark denotes the BER performance of DRSC-SM scheme under random approach. The dashed line with circle mark denotes the BER performance of DRSC-SM scheme under Approach 2. The dashed line with right triangle mark denotes the BER performance of DRSC-SM scheme under modified Approach 2. The solid line with square mark denotes the BER performance of DRSC-SM scheme under Approach 1
Moreover, Figs. 6, 7 and 8 also show the system performance under the modified Approach 2 that has lower complexity than Approach 2. The simulation results demonstrate that the performance difference between the modified Approach 2 and Approach 2 is relatively small. For example, at SNR=14 dB in Fig. 6, the DRSC-SM scheme under Approach 2 and the modified Approach 2 achieves the approximate performances, i.e., \(3.2\times 10^{-6}\) and \(2.2\times 10^{-6}\). The phenomenon reflects the rationality of the modified Approach 2 with the reduced complexity.
Performance of DRSC-SM scheme (8-QAM) with \(RS_1(31,25)\) and \(RS_2(31,13)\) under different approaches. The solid line with diamond mark denotes the BER performance of DRSC-SM scheme under random approach. The solid line with square mark denotes the BER performance of DRSC-SM scheme under Approach 2. The dashed line with square mark denotes the BER performance of DRSC-SM scheme under modified Approach 2
Performance of DRSC-SM scheme (16-QAM) with \(RS_1(63,61)\) and \(RS_2(63,41)\) under different approaches. The solid line with diamond mark denotes the BER performance of DRSC-SM scheme under random approach. The solid line with square mark denotes the BER performance of DRSC-SM scheme under Approach 2. The dashed line with square mark denotes the BER performance of DRSC-SM scheme under modified Approach 2
Performance of DRSC-SM scheme employing different decoding algorithms
The performance comparison of DRSC-SM employing joint decoding algorithms (naive algorithm, smart algorithm and improved smart algorithm) for the first case is shown in Fig. 9. The ideal source-to-relay channel condition, i.e., \(\lambda _{S,R}=\infty\), ML detection and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4,\) are supposed. As shown in Fig. 9a, for SNR\(\ge 3\), the DRSC-SM system using the smart algorithm outperforms that using the naive one when the code resulted by Approach 1 is constructed at the destination. The reason behind the performance gains is that the more reliable decoded symbols of the \(RS_2\) decoder are used as the input to the \(RS_1\) decoder at SNR\(\ge 3\). However, in the case of SNR\(<3\), the smart algorithm lags behind the naive algorithm. The poor performance can be ameliorated by adopting the improved smart algorithm as illustrated in Fig. 9b, which is explained that the improved smart algorithm can flexibly select more reliable symbols based on the change of SNR so that an additional advantage is given to the improved smart DRSC-SM scheme. Furthermore, Figs. 10 and 11 analyze the BER performance under different decoding algorithms for the second and third cases with \(\lambda _{S,R}=\infty\), ML detection and \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\). The simulation results confirm the effectiveness of our proposed improved smart algorithm again.
Performance of DRSC-SM scheme (4-QAM) with \(RS_1(15,13)\) and \(RS_2(15,7)\) employing different algorithms. In Fig. 9a, the solid line with diamond mark denotes the BER performance of DRSC-SM scheme employing naive algorithm. The solid line with square mark denotes the BER performance of DRSC-SM scheme employing smart algorithm. In Fig. 9b, the solid line with circle mark denotes the BER performance of DRSC-SM scheme employing improved smart algorithm. The solid line with square mark denotes the BER performance of DRSC-SM scheme employing smart algorithm
Performance of DRSC-SM scheme (8-QAM) with \(RS_1(31,25)\) and \(RS_2 (31,13)\) employing different algorithms. In Fig. 10a, the solid line with diamond mark denotes the BER performance of DRSC-SM scheme employing naive algorithm. The solid line with square mark denotes the BER performance of DRSC-SM scheme employing smart algorithm. In Fig. 10b, the solid line with circle mark denotes the BER performance of DRSC-SM scheme employing improved smart algorithm. The solid line with square mark denotes the BER performance of DRSC-SM scheme employing smart algorithm
Performance of DRSC-SM scheme (16-QAM) with \(RS_1(63,61)\) and \(RS_2 (63,41)\) employing different algorithms. In Fig. 11a, the solid line with diamond mark denotes the BER performance of DRSC-SM scheme employing naive algorithm. The solid line with square mark denotes the BER performance of DRSC-SM scheme employing smart algorithm. In Fig. 11b, the solid line with circle mark denotes the BER performance of DRSC-SM scheme employing improved smart algorithm. The solid line with square mark denotes the BER performance of DRSC-SM scheme employing smart algorithm
BER performance of DRSC-SM scheme over non-ideal and ideal source-to-relay channels and noncooperative scheme
Next, the BER performance of DRSC-SM over more practical scenarios (\(\lambda _{S,R}\ne \infty\)) and noncooperative scheme for the first, second and third cases is analyzed as shown in Figs. 12, 13 and 14, respectively. Approach 1 is applied in the first case, but Approach 2 is applied in the other two cases. Furthermore, the joint decoding (smart decoding algorithm), \(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\) and ML detection are employed in the three cases. For a fair comparison, the DRSC-SM scheme and its corresponding noncooperative scheme have an identical code rate from the destination point of view. It is noticed from Fig. 12 that the DRSC-SM scheme under \(\lambda _{S,R}=\infty\) excels its noncooperative scheme under the same code rate, i.e., 13/30. At BER\(=3\times 10^{-5}\), the DRSC-SM scheme obtains a BER performance gain of approximately 2.7 dB over the noncooperative scheme. It can be explained that the cooperation gives the performance gains to the system. Furthermore, the performance of DRSC-SM scheme under \(\lambda _{S,R}=14\) dB is near to that of DRSC-SM scheme under \(\lambda _{S,R}=\infty\). However, the BER performance becomes worse when the SNR between the source S and relay R becomes poor, i.e., \(\lambda _{S,R}=8\) dB. For such a poor link, an error floor is exhibited at BER \(\approx 6\times 10^{-4}\). This is because the erroneous decoding at the relay node incurs the error propagation of destination node. The cyclic redundancy check (CRC) technique [9] can be applied to control the error propagation to some extent. However, the details are beyond the scope of this manuscript. Similarly, from Figs. 13 and 14, we not only discover the superiority of ideal DRSC-SM scheme (\(\lambda _{S,R}=\infty\)) over noncooperative scheme, but also observe the performance of DRSC-SM under \(\lambda _{S,R}=14\) dB and \(\lambda _{S,R}=19\) dB is near to that of ideal DRSC-SM scheme. However, when \(\lambda _{S,R}\) is taken as a poor value, i.e., 9 dB and 11 dB, the BER curves of Figs. 13 and 14 will be flat at high SNR regime.
Performance of DRSC-SM and noncooperative schemes (4-QAM) employing \(RS_1(15,13)\) and \(RS_2(15,7)\). The solid line with upper triangle mark demonstrates the BER performance of DRSC-SM under non-ideal source-to-relay channel (\(\lambda _{S,R}= 8\) dB). The solid line with diamond mark demonstrates the BER performance of noncooperative scheme. The dashed line with right triangle mark demonstrates the BER performance of DRSC-SM under non-ideal source-to-relay channel (\(\lambda _{S,R}= 14\) dB). The solid line with circle mark demonstrates the BER performance of DRSC-SM under ideal source-to-relay channel (\(\lambda _{S,R}=\infty\))
Performance of DRSC-SM and noncooperative schemes (8-QAM) employing \(RS_1(31,25)\) and \(RS_2(31,13)\). The solid line with upper triangle mark demonstrates the BER performance of DRSC-SM under non-ideal source-to-relay channel (\(\lambda _{S,R}= 9\) dB). The solid line with diamond mark demonstrates the BER performance of noncooperative scheme. The dashed line with right triangle mark demonstrates the BER performance of DRSC-SM under non-ideal source-to-relay channel (\(\lambda _{S,R}= 14\) dB). The solid line with circle mark demonstrates the BER performance of DRSC-SM under ideal source-to-relay channel (\(\lambda _{S,R}=\infty\))
Performance of DRSC-SM and noncooperative schemes (16-QAM) employing \(RS_1(63,61)\) and \(RS_2(63,41)\). The solid line with upper triangle mark demonstrates the BER performance of DRSC-SM under non-ideal source-to-relay channel (\(\lambda _{S,R}= 11\) dB). The solid line with square mark demonstrates the BER performance of noncooperative scheme. The dashed line with right triangle mark demonstrates the BER performance of DRSC-SM under non-ideal source-to-relay channel (\(\lambda _{S,R}= 19\) dB). The solid line with circle mark demonstrates the BER performance of DRSC-SM under ideal source-to-relay channel (\(\lambda _{S,R}=\infty\))
BER performance of DRSC-SM scheme with various number of receive antennas
Figures 15 and 16 discuss the BER performance of DRSC-SM scheme with various number of receive antennas for the second and third cases. In Monte Carlo simulations, the joint decoding (smart algorithm), Approach 2, ML detection and \(\lambda _{S,R}=\infty\) are assumed. The simulated results demonstrate that incresing \(N_{\mathrm{{r}}}\) improves the performance. In Fig. 15, at SNR=10 dB, the DRSC-SM scheme with \(N_{\mathrm{{r}}}=2\) receiving antennas gets BER= \(2.4\times 10^{-2}\). When the destination uses \(N_{\mathrm{{r}}}=3, 4\) and 6 receive antennas, the performances \(1.9\times 10^{-3}\), \(1.1\times 10^{-4}\) and \(2.4\times 10^{-7}\) are separately obtained at the same SNR. Furthermore, Fig. 16 presents that the DRSC-SM scheme under \(N_{\mathrm{{r}}}=2, 3, 4\) and 6 obtains BER= \(8.7\times 10^{-3}\), \(6.4\times 10^{-4}\), \(5\times 10^{-5}\) and \(4.1\times 10^{-7}\), respectively, at \(\lambda _{S,R}=13\) dB. This phenomenon can be explained that augmenting its value of \(N_{\mathrm{{r}}}\) adds the spatial diversity in DRSC-SM scheme that will eventually enhance the overall performance of the communication system.
Performance of DRSC-SM scheme (8-QAM) with various \(N_{\mathrm{{r}}}\) employing \(RS_1(31,25)\) and \(RS_2(31,13)\). The solid line with square mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=2\). The solid line with circle mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=3\). The solid line with diamond mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=4\). The solid line with right triangle mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=6\)
Performance of DRSC-SM scheme (16-QAM) with various \(N_{\mathrm{{r}}}\) employing \(RS_1(63,61)\) and \(RS_2(63,41)\). The solid line with square mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=2\). The solid line with circle mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=3\). The solid line with diamond mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=4\). The solid line with right triangle mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=6\)
Performance of the proposed DRSC-SM scheme and the existing schemes
This section compares the performance of the proposed DRSC-SM scheme and the existing schemes, i.e., RS-coded cooperative SM (RSCC-SM) [16] and distributed RS coding (DRSC) [17] under \(\lambda _{S,R}=\infty\).
Figure 17 shows the simulation results of the proposed DRSC-SM scheme and the RSCC-SM scheme [16]. The simulation conditions such as \(N_{\mathrm{{t}}}=4\), 16-QAM, ML detection and smart decoding algorithm are used in the two schemes. Moreover, our proposed scheme adopts the optimized Approach 2 at the relay to get the optimized selection pattern by which partial symbols are selected from the source information symbols for further encoding. However, in the existing RSCC-SM scheme, the relay selects the partial symbols from the source information symbols for further encoding based on the random selection pattern. From Fig. 17, we observe that our proposed DRSC-SM outperforms the existing RSCC-SM. For example, at BER \(=1.4\times 10^{-4}\), the DRSC-SM scheme with \(N_{\mathrm{{r}}}=4\) is 0.9 dB better than the RSCC-SM with \(N_{\mathrm{{r}}}=4\). At the same BER, under \(N_{\mathrm{{r}}}=6\), the DRSC-SM scheme is 1 dB better than its counterpart. The main reason for such an attractive performance gain is that our proposed DRSC-SM scheme enables to construct a code with a larger minimum distance (i.e., 48) at the destination by appropriately selecting the partial information symbols at the relay.
BER performance of DRSC-SM and RSCC-SM schemes employing \(RS_1(63,51)\) and \(RS_2(63,31)\). The solid line with diamond mark demonstrates the BER performance of RSCC-SM scheme with random selection pattern under \(N_{\mathrm{{r}}}=4\). The solid line with circle mark demonstrates the BER performance of RSCC-SM scheme with random selection pattern under \(N_{\mathrm{{r}}}=6\). The solid line with square mark demonstrates the BER performance of DRSC-SM scheme with Approach 2 under \(N_{\mathrm{{r}}}=4\). The solid line with right triangle mark demonstrates the BER performance of DRSC-SM scheme with Approach 2 under \(N_{\mathrm{{r}}}=6\)
BER performance of DRSC-SM (\(N_{\mathrm{{t}}}=N_{\mathrm{{r}}}=4\) and 8-QAM) and DRSC (32-QAM). The solid line with square mark demonstrates the BER performance of DRSC-SM scheme. The solid line with circle mark demonstrates the BER performance of DRSC scheme
Figure 18 shows the performance comparison between the proposed DRSC-SM scheme and the existing DRSC scheme [17]. The same conditions such as \(RS_1(31,27)\), \(RS_2(31,17)\), Approach 2, ML detection and smart decoding algorithm are adopted in the two compared schemes. The simulated results in Fig. 18 reflect the superiority of our proposed scheme. For instance, at SNR \(=13\) dB, the DRSC-SM scheme achieves a very promising performance (i.e., \(4.3\times 10^{-6}\)) but the DRSC scheme obtains a poor performance (i.e., \(2.6\times 10^{-3}\)). The performance enhancement is mainly because the adopted novel SM technique provides spatial diversity for the proposed DRSC-SM scheme.
Performance comparison between DRSC-SM and DRSC-V-BLAST schemes
In Sect. 6.5, Fig. 18 exhibits the performance advantages of our proposed DRSC-SM scheme over the DRSC (without SM). To further confirm the performance superiority of our proposed DRSC-SM scheme, we compare it with the distributed RS-coded V-BLAST (DRSC-V-BLAST) scheme under an identical spectral efficiency. The condition \(\lambda _{S,R}=\infty\) is supposed. In our proposed DRSC-SM scheme, \(N_{\mathrm{{t}}}=4\) and 4-QAM are used. However, \(N_{\mathrm{{t}}}=2\) and 4-QAM are used in the DRSC-V-BLAST scheme. At the destination, the smart decoding algorithm is used to jointly recover the source information. Moreover, the DRSC-SM scheme utilizes the maximum ratio combining (MRC) reception [20], and the minimum mean squared error (MMSE) detection [20] is adopted in the DRSC-V-BLAST scheme. The significant performance gains of the DRSC-SM scheme over the DRSC-V-BLAST scheme are shown in Fig. 19. For instance, under \(N_{\mathrm{{r}}}=4\) and 6, 0.5 dB and 0.4 dB gains are obtained by our system at BER=\(10^{-4}\). The main reason behind the attractive gains is that our presented DRSC-SM scheme can completely avoid the drawback (ICI) of the DRSC-V-BLAST scheme so that the ability to recover the source information is improved.
Performance comparison between the DRSC-SM and DRSC-V-BLAST schemes with \(RS_1(15,13)\) and \(RS_2(15,7)\). The dashed line with square mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=4\). The dashed line with circle mark demonstrates the BER performance of DRSC-SM scheme under \(N_{\mathrm{{r}}}=6\). The solid line with square mark demonstrates the BER performance of DRSC-V-BLAST scheme under \(N_{\mathrm{{r}}}=4\). The solid line with circle mark demonstrates the BER performance of DRSC-V-BLAST scheme under \(N_{\mathrm{{r}}}=6\)
A novel DRSC-SM scheme is proposed. The optimized design approaches are employed for appropriate selection of symbols at the relay node. The numerical results show the effectiveness of the proposed approaches. The destination performs joint RS decoding, which is based on three different algorithms called naive algorithm, smart algorithm and improved smart algorithm. The BER performance curves show that under the same decoding algorithm, the proposed DRSC-SM scheme has an excellent performance over the corresponding coded noncooperative scheme. Also, the DRSC-SM scheme utilizing the improved smart decoding algorithm outperforms the counterpart DRSC-SM scheme using the naive and smart decoding algorithms under identical conditions.
Reed–Solomon
DRSC-SM:
Distributed RS-coded spatial modulation
MIMO:
Multiple-input multiple-output
V-BLAST:
Vertical Bell Labs Layered Space-Time
IAS:
Inter-antenna synchronization
ICI:
Inter-channel interference
Spatial modulation
Compress-and-forward
DF:
Decode-and-forward
Amplify-and-forward
LDPC:
Low-density parity-check
RM:
Reed–Muller
BER:
Bit error rate
M2M:
Machine-to-machine
D2D:
Device-to-device
MDS:
Maximum distance separable
ML:
Maximum likelihood
SNR:
GCD:
CRC:
C. Zhao, F. Yang, R. Umar, Two-source asymmetric turbo coded cooperative spatial modulation scheme with code matched interleaver. Electronics. 9(1) (2020)
R. Govender, N. Pillay, H. Xu, Soft-output space-time block coded spatial modulation. IET Commun. 9(8), 2786–2796 (2014)
R.Y. Mesleh, H. Haas, S. Sinanovic, Spatial modulation. IEEE Trans. Veh. Technol. 57(4), 2228–2241 (2008)
S. Ejaz, F. Yang, Jointly optimized ReedMuller codes for multilevel multirelay coded-cooperative VANETS. IEEE Trans. Veh. Technol. 66(5), 4017–4028 (2017)
S. Mughal, F. Yang, H. Xu, R. Umar, Coded cooperative spatial modulation based on multi-level construction of polar code. Telecommun. Syst. 70(3), 435–446 (2018)
R. Umar, F. Yang, S. Mughal, Multiple relay based Reed Muller network coded cooperation for wireless communication systems. IET Commun. 13(13), 2034–2044 (2019)
S. Ejaz, F. Yang, T.H. Soliman, Multi-level construction of polar codes for half-duplex wireless coded-cooperative networks. Frequenz 69(11–12), 509–517 (2015)
S. Zhang, F. Yang, L. Tang, Joint design of QC-LDPC codes for coded cooperation system with joint iterative decoding. Int. J. Electron. 103(3), 384–405 (2015)
S. Mughal, F. Yang, S. Ejaz, Asymmetric turbo code for coded-cooperative wireless communication based on matched interleaver with channel estimation and multi-receive antennas at the destination. Radioengineering 26(3), 878–889 (2017)
S. Mughal, F. Yang, R. Umar, Reed–Muller network coded-cooperation with joint decoding. IEEE Commun. Lett. 23(1), 24–27 (2019)
J. Kim, J. Lee, J. Kim, J. Yun, M2M service platforms: survey, issues and enabling technologies. IEEE Commun. Surv. Tutor. 16(1), 61–76 (2014)
J. Qiu, L. Chen, S. Liu, A novel concatenated coding scheme: RS-SC-LDPC codes. IEEE Commun. Lett. 24(10), 2092–2095 (2020)
A.H. Almawgani, M.F. Salleh, RS coded cooperation with adaptive cooperation level scheme over multipath Rayleigh fading channel, in Proceedings of IEEE 9th Malaysia International Conference on Communications (MICC) (IEEE Kuala Lumpur, Malaysia, 2009), pp. 480–484
A.H. Almawgani, M. Salleh, Coded cooperation using Reed Solomon codes in slow fading channel. IEICE Electron. Express 7(1), 27–32 (2010)
Y.M. Al-moliki, M.A. Aldhaeebi, G.A. Almwald, M.A. Shaobi, The performance of RS and RSCC coded cooperation systems using higher order modulation schemes, in Proceedings of 6th International Conference on Intelligent Systems, Modelling and Simulation. (IEEE Kuala Lumpur, Malaysia, 2015), pp. 211–214
C. Zhao, F. Yang, D.K. Waweru, Reed–Solomon coded cooperative spatial modulation based on nested construction for wireless communication. Radioengineering 30(1), 172–183 (2021)
P. Guo, F. Yang, C. Zhao, W. Ullah, Jointly optimized design of distributed Reed-Solomon codes by proper selection in relay. Telecommun. Syst. 78(3), 391–403 (2021)
M. Koca, H. Sari, Bit-interleaved coded spatial modulation, in IEEE 23rd international symposium on personal indoor and mobile radio communications (IEEE Sydney, Australia, 2012), pp. 1949–1954
L. Shu, S. Lin, D.J. Costello, Error Control Coding (Pearson Education India, 2004)
J.R. Barry, E.A. Lee, D.G. Messerschmitt, Digital Communication (Springer, New York, 2004)
The authors would like to thank the National Natural Science Foundation of China under the contract no. 61771241.
The work is supported by the National Natural Science Foundation of China under the contract no. 61771241.
College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
Chunli Zhao, Fengfan Yang & Pengcheng Guo
School of Electrical and Information Engineering, University of the Witwatersrand, Johannesburg, 2000, South Africa
Waheed Ullah
Chunli Zhao
Fengfan Yang
Pengcheng Guo
CZ wrote this manuscript. All the authors read and approved the final manuscript.
Correspondence to Chunli Zhao.
Zhao, C., Yang, F., Ullah, W. et al. Jointly optimized design of distributed RS-coded spatial modulation by appropriate selection at the relay. J Wireless Com Network 2023, 1 (2023). https://doi.org/10.1186/s13638-023-02211-0
RS codes
Coded cooperation
Joint RS decoding


Stella And Dot Business Cards stella and dot business cards stella and dot business card design 7 download. stella and dot business cards stella and dot business card design 1 ideas. stella and dot business cards stella and dot business card design 5 printable. Stella And Dot Business Cards stella and dot business cards stella and dot business cards download. stella and dot business cards stella and dot business card design 3 ideas. stella and dot business cards stella and dot business card design 4. Stella And Dot Business Cards stella and dot business cards stella and dot business cards download.
Stella And Dot Business Cards stella and dot business cards stella and dot business card design 1 ideas. stella and dot business cards stella and dot business card design 5 printable. stella and dot business cards stella and dot business cards download. stella and dot business cards stella and dot business card design 3 ideas. stella and dot business cards stella and dot business card design 4.


text:
No Companies have been
added to compare yet
Top 10 SoundCloud Growth Hacks
7 Ways to Promote Your SoundCloud Music
Creating a Marketing Plan for SoundCloud
How To Make Awesome Music
How To Avoid Getting Scammed
How to Master Social Media Music Marketing
How to Distribute Your Music From SoundCloud to Other Platforms
SoundCloud is the go-to platform for music discovery. Other platforms like Spotify and YouTube have music from mainstream music artists. But only on SoundCloud can you find individuals who just launched their music careers.
When a person likes an independent artist's music so much, they can't help but root for them. It's in human nature. So, they want to support the artists as much as they can.
One of the best things a fan of an independent artist on SoundCloud can do is share their tracks so that more people find them. Big news: SoundCloud increasing real plays has released a new feature that makes it easier. This new feature allows people to easily share music on other social media platforms, namely Snapchat and Facebook Stories.
It is not an entirely new feature. SoundCloud already allows users to share tracks on Instagram Stories before this update.
That's also great news for artists on the platform. The ability to share tracks is not exclusive to listeners. They, too, can share their music on other platforms.
With that, an artist on SoundCloud can easily increase their online presence. Facebook is the biggest social platform in the world, and Snapchat's user base is not negligible. And more people discovering an artist's tracks increases the chances of that artist becoming popular.
It's not just single tracks you can share too. You can create playlists and share that on your Snapchat or Facebook Stories. That way, you can support many of your favorite indie artists in a few taps.
How To Share Music From SoundCloud On Other Platforms
First, make sure the SoundCloud app on your phone is in the latest version. If it is not, update it from the App Store or Google Play first.
Then, look for the track, playlist, album, or artist profile you want to share. Tap the share icon, then tap the logo of Facebook, Instagram, or Snapchat. Some devices may have the options listed as "Share to Facebook Stories," "Share to Instagram Stories," or "Share to Snapchat" instead.
Before making the post visible to others, you can add personal touches like captions and Giphy stickers. Finish uploading the Story as you normally would.
Why Bother Spreading Out Your Plays?
First and foremost, distributing your music to other platforms allows for greater visibility and reach. Much like a tree in a forest, your music may get lost in the vast expanse of SoundCloud, but by branching out to other platforms like Spotify and Apple Music, your music can be discovered by a wider audience. The more platforms your music is available on, the more opportunities you have to connect with new listeners and grow your fanbase.
Additionally, distributing your music to other platforms also allows for more monetization opportunities. Soundcloud does not have a built-in mechanism for direct monetary compensation for artists. By distributing your music to platforms such as Spotify and Apple Music, you open yourself up to potential royalties and streaming revenue.
Furthermore, distributing your music to other platforms allows you to understand better and engage with your audience. By looking at these platforms' data and analytics, you can gain valuable insights into which tracks resonate with listeners, where your audience is located, and what type of content they respond to best. This knowledge can be used to inform your future creative decisions and optimize your marketing strategy, allowing you to make more targeted, effective decisions, much like a chef adapting his recipe to suit the tastes of his diners.
Finally, distributing your music to other platforms is a great way to establish yourself as a professional and credible artist. Having your music on multiple platforms demonstrates your commitment to your craft and your willingness to invest in your career. This can help you gain the attention of industry professionals and increase your chances of being signed to a record label or landing gigs, much like a store window display on a busy street could help a business attract more customers.
https://mobile.twitter.com/verge/status/1098088863199870976
Distributing Tracks to Stores
Sharing your music tracks on other social media platforms will help you gain more fans. But your end goal is still to distribute on stores like Spotify and Apple Music. These will make you more known in the music world. So, you must also learn the best practices when distributing music on these platforms.
Admittedly, that's an intimidating task, especially for first-timers. Stores are more strict than platforms like YouTube or SoundCloud. They have specific metadata, audio, and artwork guidelines. Not following these guidelines will delay you from sharing your music on these platforms.
It is best to study the dos and don'ts to avoid that inconvenience. SoundCloud revealed the common errors its artists make that delay approval of delivering tracks to stores.
One of the most common mistakes is using artwork that is not original. It causes a problem with intellectual property rights. Thus, ensure that you control all the rights in the artwork that you use. If an image is copyrighted, but you have permission to use it, send the documentation to Repost's support.
Inappropriate artwork is not allowed. It should not contain nudity, excessive violence, overly blurry content, etc.
Generic artwork can also delay approval for your release's delivery or make the stores outright reject it. So avoid using direct pictures or single colors. It's best to put effort into it and personalize it.
Also, make sure your artwork is not misleading. It should not mimic famous album covers, movie posters, or other popular reference points. Furthermore, the text on the artwork should match your metadata. The artist, title, label, etc., should be the same.
Here are some things to remember when filling out the metadata for your release.
First and most importantly, do not misspell – proper casing included – your Artist Name. Ensure you write it exactly the same way it is written on Stores such as Apple Music or Spotify. Otherwise, your release may appear on another artist's page rather than on your page.
Please also remember not to add the Artist Name on the Release and Track titles. Contributors should also go only to the Contributors field. Speaking of that, please remember to add contributors individually. If you put them all in one contributor field, your release may go to a different artist's page.
In the Composer Names field, use only the legal names of composers. That means only full names are accepted.
What not to add? You should not add misleading metadata. That includes adding names of popular artists, song names, or generic SEO terms. Also, do not put your social handles in the field for Artist Name.
If your release has explicit content, don't forget to flag them. Some stores will hide incorrectly labeled releases. So, instead of not telling them your tracks contain profanity, drug references, etc., it is better to admit it.
SoundCloud tells users to refrain from attempting to distribute content where they don't control all aspects of the release. Unofficial remixes and content containing uncleared samples are some examples. Again, if you have the right to distribute, you can send the documentation to support@repostnetwork.com.
How Important Is It To Share Your Music To Other Platforms?
SoundCloud allows you to upload your tracks so your fans can listen to them. It is where you can start and grow your music career. But you should not limit yourself to this platform. Your music should be on stores like Spotify for you to be successful in the music industry. And sharing your music on platforms like Facebook and Instagram helps make more people aware of your existence as an artist.
That said, sharing your music on other platforms is extremely important. It is how you can take your music career to the next level.
Date: January 10, 2023 / Categories: Getting Discovered, / Author: Rich Drees
SoundCloud Reviews © 2023 - All Rights Reserved. This website is in no way affiliated with SoundCloud.

redpajama_set_name: RedPajamaCommonCrawl


text:
A Chelsea Pensioner, or In-Pensioner, is a resident at the Royal Hospital Chelsea, a retirement home and nursing home for former members of the British Army located in Chelsea, London. The Royal Hospital Chelsea is home to 300 retired British soldiers, male and female (the latter since 2009), and is located on Royal Hospital Road. Historically, the term "Chelsea Pensioner" applied more widely, referring to both In-Pensioners and Out-Pensioners who live elsewhere.

In- and Out-Pensioners

The Royal Hospital was founded by King Charles II in 1682 as a retreat for veterans. The provision of a hostel rather than the payment of pensions was inspired by Les Invalides in Paris.

During the reign of King William III and Queen Mary II, the Royal Hospital was still under construction so they introduced a system for distribution of army pensions in 1689. The pension was to be made available to all former soldiers who had been injured in service, or who had served for more than 20 years.

By the time the Royal Hospital was completed, there were more pensioners than places available in the Royal Hospital. Eligible ex-soldiers who could not be housed in the Hospital were termed Out-Pensioners, receiving their pension from the Royal Hospital but living outside it. In-Pensioners, by contrast, surrender their army pension and live within the Royal Hospital.  A number of veterans were incorporated into the Corps of Invalids from 1688 to 1802, receiving the equivalent of the out-pension in exchange for performing garrison duty.

In 1703 there were 51 Out-Pensioners. By 1815 this had risen to 36,757. The Royal Hospital remained responsible for distributing army pensions until 1955, following which the phrase "Out-Pensioner" became less common, and "Chelsea Pensioner" was used largely to refer to "In-Pensioners".

Conditions for admission as a Pensioner
To be eligible for admission as a Chelsea Pensioner a candidate must be a former soldier or non-commissioned officer of the British Army (including National Service) or a former officer of the British Army who served in the ranks or was awarded a disablement pension while serving in the ranks. They must be of pensionable age, be able to live independently on arrival at the Royal Hospital Chelsea and be free of any financial obligation to support a spouse or family.

Life of In-Pensioners

Applicants for the Royal Hospital Chelsea (RHC) are invited for a four-day stay during which they get a taste of what life is like for a Chelsea Pensioner. If they enjoy their stay and it is felt that they will fit in they are invited to become a Chelsea Pensioner. Upon arrival at the Royal Hospital, each In-Pensioner is  measured up for their Blues (day-to-day uniform) and Scarlets (the famous uniform that they wear on parade). They are given their own room, or "berth" in a ward, and are allocated to a Company. Up until refurbishment works carried out in the 1950s, these rooms measured 6 × 6 feet. Work completed in 2015 has ensured that all In-Pensioner berths include en suite facilities, a writing desk, and natural light.

On entry, In-Pensioners surrender their army pension, in return receiving board, lodging, clothing and full medical care.

In-Pensioners also have their own club with a bar, lounges and restaurant; there are allotments for those who enjoy gardening and facilities for lawn bowls. If an In-Pensioner becomes unwell they are moved into the Infirmary, which is a care home with nursing and a G.P. medical centre. The Royal Hospital Burial Ground is at Brookwood Cemetery in Surrey.

The Chelsea Pensioners were among the first to be included in the UK's rollout of COVID-19 vaccinations, having received their first dose on 23 December 2020.

Admission of women

Until 2009, only male candidates were admitted. It was announced in 2007 that female ex-service personnel would be admitted on the completion of modernisation of the long wards. In March 2009 the first women in the Hospital's 317-year history were admitted as In-Pensioners:  Dorothy Hughes (aged 85) and Winifred Phillips (aged 82).

Winifred Phillips (1926–2016) trained as a nurse and joined the Auxiliary Territorial Service in 1948 and enlisted in the Women's Royal Army Corps in 1949 while serving in Egypt. For the next 22 years she served in Singapore, Cyprus and Egypt reaching the rank of Warrant Officer Class 2. She wrote two books about becoming one of the first female Chelsea Pensioners: My Journey to Becoming the First Lady Chelsea Pensioner (2010), and Mum's Army: Love and Adventure from the NAAFI to Civvy Street (2013). She never married.

Dorothy Hughes joined the British Army in 1941, later working as part of 450 Heavy Anti Aircraft Battery in the London Division. In 1945 the Battery was deployed near Dover to defend against V-1 flying bomb attacks. She later worked with the Army Operational Research Group developing fuses in shells used against V-2 rockets and was discharged from the Army in 1946 with the rank of Sergeant.

The records at Royal Hospital show that another woman, Christian Davies, was admitted to Chelsea Hospital circa 1717, and was awarded a pension for her service in the army and the wounds she received in the service of Queen Anne. She died in 1739 and was buried in the Royal Hospital Chelsea with full military honours.

Clothing

Chelsea Pensioners are entitled to come and go from the Royal Hospital as they please, and are permitted to wear civilian clothing wherever they travel. However, within the Hospital, and in the surrounding area, they are encouraged to wear a blue uniform. If they travel further from the Royal Hospital, they should wear the distinctive scarlet coats instead of the blue uniform. The scarlet coats are also worn for ceremonial occasions, accompanied by tricorne hats. At other times a peaked hat, known as a shako, is generally worn.

In uniform, the pensioners wear their medal ribbons and the insignia of the rank they reached while serving in the armed forces. They may also wear other insignia they earned during their service and this has included parachute jump wings and SAS jump wings.

Memorials and burial sites

Several cemeteries have memorials to and graves of Chelsea Pensioners. Burials were initially at Old Burial Ground, Royal Hospital Chelsea and then Brompton Cemetery, where the Chelsea Pensioners' Monument was erected in 1901. Since 1893, burials have taken place at Brookwood Cemetery in Surrey where the care of the graves was transferred in 1961 to the Commonwealth War Graves Commission.

Men In Scarlet album
Seven Chelsea Pensioners at the Royal Hospital Chelsea released an album on 8 November 2010 in order to raise money for the Chelsea Pensioners' Appeal.  Featuring Dame Vera Lynn, Katherine Jenkins, The Soldiers and Janey Cutler, the album is full of well known wartime songs and includes their traditional march, "The Old Brigade".

Chelsea Football Club
Nearby Chelsea Football Club has been affiliated with the Chelsea Pensioners for many years; the club's first nickname was the Pensioners and until the 1950s the club crest featured a Chelsea pensioner. Residents of the Hospital can be seen attending Chelsea's home games at Stamford Bridge.

When Chelsea won the Premier League title in 2005, Chelsea Pensioners formed a guard of honour as the players and management came out for the trophy presentation. This was repeated when Chelsea won the title in 2010. In tribute to the trademark scarlet coats worn by the pensioners, Chelsea's kit for the 2010–11 season featured a red trim on the collars.

See also
 Chelsea Pensioners reading the Waterloo Dispatch

References

External links
Chelsea Pensioners Website
Selected service records of Chelsea pensioners available from the UK National Archives' website

1682 establishments in England
British Army
British ceremonial units
Chelsea, London
Retirement in the United Kingdom
Veterans' affairs in the United Kingdom

redpajama_set_name: RedPajamaWikipedia


text:
Q: Either zero or 2 or more [DropdownMenuItem]s were detected with the same value Failed assertion: line 882 pos 15: 'items == null || items.isEmpty I tried the solutions provided on the same question, but each question has its own case, so in my case i am trying to create drop down of countries list and before use it in the Scaffold widget i want to save it to an instance of type DropdownButton(), here is the code:
 List<String> provinces = [
    'عمان',
    'إربد',
    'الزرقاء',
    'المفرق',
    'الطفيلة',
    'عجلون',
    'معان',
    'الكرك',
    'مادبا',
    'العقبة',
    'البلقاء',
    'جرش'
  ];

String dropDownValue = "";

final provincesField = DropdownButton<String>(
  value: dropDownValue,
  icon: Icon(Icons.keyboard_arrow_down),
  items: provinces.map((String items) {
    return DropdownMenuItem(value: items, child: Text(items));
  }).toList(),
  onChanged: (String? newValue) {
    setState(() {
      dropDownValue = newValue!;
    });
  },
);

Am not sure why its not accessing the list and consider it empty. Here is the full error:
The following assertion was thrown building RegistrationScreen(dirty, state: _RegistrationScreenState#b01ae):
There should be exactly one item with [DropdownButton]'s value: .
Either zero or 2 or more [DropdownMenuItem]s were detected with the same value
'package:flutter/src/material/dropdown.dart':
package:flutter/…/material/dropdown.dart:1
Failed assertion: line 882 pos 15: 'items == null || items.isEmpty || value == null ||
              items.where((DropdownMenuItem<T> item) {
                return item.value == value;
              }).length == 1'

A: The error:
There should be exactly one item with [DropdownButton]'s value: one.
Either zero or 2 or more [DropdownMenuItem]s were detected with the same value

Indicates that in your list provinces it should contain the same value as your dropDownValue.
Which means, that your file should look like this:
String dropDownValue = "one";
  List<String> provinces = [
    "one", // <-- Notice we're adding the word "one" to the list since that's the value of `dropDownValue`
    'عمان',
    'إربد',
    'الزرقاء',
    'المفرق',
    'الطفيلة',
    'عجلون',
    'معان',
    'الكرك',
    'مادبا',
    'العقبة',
    'البلقاء',
    'جرش'
  ];

Here is a complete example:
class Test extends StatefulWidget {
  const Test({Key? key}) : super(key: key);

  @override
  State<Test> createState() => _TestState();
}

class _TestState extends State<Test> {
  List<String> provinces = [
    "one", // <-- Notice wer're adding the word "one" to the list
    'عمان',
    'إربد',
    'الزرقاء',
    'المفرق',
    'الطفيلة',
    'عجلون',
    'معان',
    'الكرك',
    'مادبا',
    'العقبة',
    'البلقاء',
    'جرش'
  ];

  @override
  Widget build(BuildContext context) {

    // make sure this is this value is in your list
    String dropDownValue = "one";

    final provincesField = DropdownButton<String>(
      value: dropDownValue,
      icon: Icon(Icons.keyboard_arrow_down),
      items: provinces.map((String items) {
        return DropdownMenuItem(value: items, child: Text(items));
      }).toList(),
      onChanged: (String? newValue) {
        setState(() {
          dropDownValue = newValue!;
        });
      },
    );

    return Container();
  }
}

A: Try and replace the code instead of String dropDownValue.
var dropDownValue;

And items should have a value other than null.

A: in constructor (){
   dropDownValue = provinces.first;
}

make sure a value is in your list

redpajama_set_name: RedPajamaStackExchange


On December 15th Creative Mentorship community celebrated the finalization of another program cycle in the Museum of Contemporary Arts in Belgrade.
On Friday, November 9th the team of "Creative Mentorship" organized a visit to Kosjerić.
Creative Mentorship, the first long term nonprofit mentoring program for perspective professionals in culture, arts, creative industries and cultural heritage from throughout Serbia is expanding its team!
HOW FEMININE IS CULTURAL HERITAGE?


Frederick Field was born in Hounslow Middlesex in 1890. His father, Frederick (1855-1932) was a gardener and his mother, Letitia Ann (nee Still, 1851-1940) was a washing servant. They married in Sevenoaks, Kent in 1889. Frederick had one sister, Francis Harriet (1892-1992).
In 1891 Frederick and Letitia were living with their eight-month-old baby Frederick in Bath Road, Heston, Hounslow.
They moved to Mickleham about 1901, now also with their daughter Frances, where they lived at Laundry Cottage, 6 Elm Cottages. Elm Cottages were known as Praed's Cottages after the wealthy Mackworth Praed family of Mickleham Downs who owned them. During the time the Field family lived in Mickleham, Robert Mackworth Praed was Lord of the Manor of Mickleham.
According to the Mickleham School Logbook on 17th April 1905 Frederick went to Dorking for Confirmation, and on 2nd August 1905 he left school. He was described as a 'good and diligent scholar'.
By 1915, Frederick's parents moved to Arthur's Bridge Road, Horsell near Woking, with Frances, who continued to live in the house with her husband Ernest Vidler after the death of her parents. The Commonwealth War Grave Commission entry for Frederick Richard lists the name of the house as 'Mickleham', presumably chosen by his parents to remember their time at Laundry Cottage. Frances was worked as a teacher.
Before enlisting in 1915, Frederick worked as a Millers clerk.
Frederick enlisted in 1st/13th Battalion of the London (Princess Louise's) Regiment. The regimental headquarters were in Iveagh Gardens Kensington, and most of the recruits were drawn from that area.
In June 1916 the battalion was mobilised for war, landing at Le Havre before being sent into the line in the Vimy sector. In November the battalion was deployed to Salonika and then sent to Alexandria in July 1917 as part of the Egyptian Expeditionary Force in the Palestine Campaign.
Corporal Field died on 8th December 1917 during the capture of Jerusalem, the day before the city surrendered.
The action on that day was as follows: Ottoman artillery began firing on a road junction, which Mott's Detachment had to negotiate. Unable to advance or retaliate against the accurate shell fire from an Ottoman battery near Bethlehem, the detachment waited. At around noon, Chetwode, the corps' commander, ordered the detachment to get moving. Mott finally attacked his main objective at Beit Jala at 16:00, but the Ottoman Army had already retired. It was not until the evening that they continued their advance to find the way completely clear of Ottoman defenders. At the crucial moment, Mott's Detachment was unable to cover the southern flank of the 60th (London) Division, forcing the Londoners to pause during daylight, as enfilading fire would have made any advance extremely costly. Three days later General Allenby entered Jerusalem on, foot. Lloyd George described the capture of the city as an early Christmas Present.
Frederick Field was aged 27 at the time of his death and was buried in Jerusalem Cemetery. He is commemorated on the Jerusalem Memorial and also on the Horsell Village War Memorial in Surrey.


Fromagination provides box / bag lunches that are personalized, and made fresh daily with the best ingredients. They can be delivered in an attractive box, or bag, with the name of the sandwich and/or name of the person for ease of distribution at your event.
Decide if you want them bagged or in a Fromagination Gable Box. See prices and options below.
Call us at 608-255-2430 to place your order.
For more advice for group events, call the Business Office at 608-665-3363.
LOVE Fromagination group box lunches for team meetings. We order these whenever we get the chance. The right mix of salt, sweet and substance.


text:
290: Godless America
Share a clip
Two stories about people who suddenly realize they're the only ones around who value the separation of church and state. Paul Williams, a city councilman in Janesville, Wisconsin, wants to make sure a Salvation Army built with public money doesn't proselytize. Soon he's getting attacked in the local press...and by President Bush. And a Georgia teacher finds that by teaching what she's supposed to teach—evolution—she turns her school against her. (9 minutes)
785: Through the Looking Glass
Act One: Flies, Meet Honey
An elections administrator in Texas, Heider Garcia, uses a radical approach to try to win over election skeptics: tactical patience.
781: Watching the Watchers
Act One: Magazona
Washington Post reporter, Isaac Arnsdorf, and producer, Zoe Chace, continue the story about the takeover of the Republican party.
776: I Work Better on Deadline
Act One: A Funny Thing Happened on the Way to the Quorum
Reporter Sarah Gibson tells the story of a huge political divide in the tiny town of Croydon, NH – population 800.
More by Ira Glass
Act Two: I'll Be Watching You
A group of 6th grade boys are bothered by their teacher's behavior.
777: Name. Age. Detail.
Margus Morrison
Margus Morrison was a school bus aide, father of six.
774: The Pink House at the Center of the World
Act Four: Next
Host Ira Glass walks through possible next steps with a pro-life activist who worked on the Texas SB8 bill, that set a precedent for enforcement of abortion bans throughtout the country.


The year is not yet over, and we have had a full year of serving our community and spending time with one another. Check out the video below to relive some of the highlights of 2016 so far, and we'll see you Sunday morning for the first Sunday of Advent.
Newer Post"I Just Came By"