Delaytolerant distributed spacetime coding with feedback for cooperative MIMO relaying systems
 Tong Peng^{1} and
 Rodrigo C. de Lamare^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1363801710202
© The Author(s) 2018
Received: 26 April 2017
Accepted: 28 December 2017
Published: 18 January 2018
Abstract
An adaptive delaytolerant distributed spacetime coding (DSTC) scheme equipped with a feedback channel is proposed for twohop cooperative multipleinput multipleoutput (MIMO) networks. A maximum likelihood receiver and adjustable code matrices subject to a power constraint with a decodeandforward cooperation strategy are considered with different DSTC antenna configurations. In the proposed delaytolerant DSTC schemes, an adjustable code matrix is employed to transform the spacetime coded matrices at the relay nodes. Stochastic gradient and least squares algorithms are also developed with reduced computational complexity. The proposed algorithms are then extended to a cooperative MIMO system using amplifyandforward strategy and opportunistic relaying algorithms in order to develop a delaytolerant coding scheme combined with optimal relay selection strategies. An upper bound on the pairwise error probability and a rank criterion analysis are derived which indicate the advantage of the proposed algorithms. Simulation results show that the proposed algorithms obtain significant performance gains and address the delay issue in cooperative MIMO systems as compared to existing DSTC schemes.
Keywords
1 Introduction
Cooperative multipleinput multipleoutput (MIMO) systems can obtain diversity gains by providing copies of transmitted signals with the help of relays to improve the reliability of wireless communications [1–5] and in applications to spectrum sensing [6, 7]. The idea behind cooperative relaying systems is to employ multiple relay nodes between the source node and the destination node to form a distributed antenna array, which can provide significant advantages in terms of diversity gains. Several cooperation strategies that exploit the links between relay nodes and the destination node such as amplifyandforward (AF), decodeandforward (DF), compressandforward (CF) [1], and various distributed spacetime coding (DSTC) schemes [2, 3, 8, 9] have been extensively studied in the literature. A key problem that arises in cooperative MIMO systems and which degrades the performance of such systems is the existence of delays between the signals that are spacetime coded at the relays and decoded at the destination.
The deployment of distributed spacetime coding (DSTC) schemes at relay nodes in a cooperative network can provide the system diversity and coding gains to mitigate the interference by adding more copies of the transmitted signals at the destination. However, a serious issue for distributed MIMO systems using a DSTC scheme is the asynchronous transmission from the relays or the delayed reception of the DSTC scheme at the destination. The delayed code matrices and data forwarded from the relays shift the DSTC structure which results in the collapse of the rank design criterion and leads to degradation of decoding performance. In order to address the delay problem, various delaytolerant distributed spacetime coding (DTDSTC) schemes [10–14] have been recently reported for distributed MIMO systems. Extending the distributed threaded algebraic spacetime (TAST) codes [10], Damen and Hammons designed a delaytolerant coding scheme in [11] by the extension of the Galois field employed in the coding scheme to achieve full diversity and full rate. A further optimization which ensures that the codes in [11] obtain full diversity with the minimum length and lower decoding complexity has been presented in [12]. The authors of [13] have proposed delaytolerant linear convolutional DSTC schemes which can maintain the full diversity property under any delay situation among the relays. In [14], a transmit processing technique based on linear constellation precoder (LCP) design has been employed to construct an optimal DTDSTC scheme to achieve the upper bound on the error probability.
In the literature, two basic configurations of DSTC schemes have been reported: one in which the coding is performed independently at the relays [15, 16], denoted multipleantenna system (MAS) configuration, and another in which coding is performed across the relays [11, 14], called singleantenna system (SAS) configuration. The received symbol matrix at the destination node in the MAS configuration is a sum of fully encoded DSTC schemes from each relay node, while in the SAS configuration, the received symbol matrix contains only one DSTC scheme encoded across all the relays. Considering delays between the relay nodes, the sum of the DSTC schemes will be affected by imperfect asynchronous superposition of signals in the MAS configuration, while the structure of the received DSTC schemes in the SAS configuration will be switched. It is not clear the key advantages of these schemes and their suitability for situations with delays. In addition, the work on delaytolerant DSTC has focused on encoding using a fixed scheme, such as TAST codes in [10], which are difficult to implement with other DSTC schemes. In the literature, the DSTC designs in [17–23] can achieve full diversity order and high coding gains but are vulnerable to delays. In order to improve the existing DSTC schemes and overcome the delay issues, a general delaytolerant encoding design and algorithm is needed. Moreover, the DTDSTC schemes in the literature do not employ feedback or optimal relay selection algorithms to improve the performance of the systems. The advantage of employing feedback to optimize DSTC schemes have been studied in [16, 24] with various DSTC schemes. However, strategies to exploit feedback and improve the design of DTDSTC schemes remain unexplored. Opportunistic cooperation algorithms are reported in [18] and [19]. The authors in [18] provide three relay selection algorithms for distributed systems which can achieve a lower bit error rate (BER) performance compared to the traditional equal power allocation schemes, and an opportunistic resource allocation optimization design which maximizes the delaylimited capacity and minimizes the outage probability is given in [19]. However, the singleantenna relays are employed in the designs so that if a DSTC scheme is employed at the relay node, it will be severely affected after selecting the optimal relay or allocating the optimal relay to the best position when delays are considered among the relays.
In this paper, we propose an adaptive delaytolerant DSTC scheme and algorithms for cooperative MIMO relaying systems equipped with a feedback channel in both MAS and SAS scenarios. We first propose a delaytolerant adjustable code matrices optimization (DTACMO) algorithm based on the maximum likelihood (ML) criterion subject to constraints on transmitted power at the relays for different cooperative systems. Unlike [10–14] which are assessed in the SAS scenario, the proposed algorithms are designed and investigated in both MAS and SAS scenarios. Specifically, adaptive optimization algorithms using stochastic gradient (SG) and recursive least square (RLS) estimation methods are developed for the DTACMO algorithm in order to release the destination from the high computational complexity of the optimization procedure. We study how the adjustable code matrices affect the DSTC scheme during the encoding process and how to optimize the adjustable code matrices by employing an ML detector. Then, we analyze the differences in terms of the rank criterion and pairwise error probabilities (PEP) of the DSTBC schemes in MAS and SAS configurations with the same number of antennas and delay profiles. We focus on how the different system configurations affect the delay tolerance of the DSTBC schemes. Moreover, we extend our design to cooperative systems exploiting the AF protocol and the opportunistic relaying algorithms in [18] in order to devise a delaytolerant adjustable code matrices opportunistic relaying optimization (DTACMORO) algorithm which can address the delay issue when optimal power allocation and relay selection are employed. The proposed delaytolerant designs can be implemented with different types of DSTC schemes in cooperative MIMO relaying systems with DF or AF protocols.
Unlike our prior work [16], here, we study delaytolerant DSTBC in different system configurations and with different protocols. The issue of asynchronous reception at the destination node can be addressed by the proposed DTACMO algorithms, while the BER performance is improved by the optimization algorithms. An analysis of the proposed and existing designs show the advantages of the proposed designs in cooperative MIMO relaying systems with asynchronous relaying. Moreover, the combination of the proposed DTACMO algorithms and the opportunistic schemes in [18] results in a delaytolerant DSTC design with optimal relay selection. All the detection errors, including errors in DF protocol and feedback errors at each relay node, have been considered in the algorithm design in order to show the potential of implementation of the proposed algorithms in practical circumstances.
The paper is organized as follows. Section 2 describes the twohop cooperative MIMO systems with multiple relays considered in this work, the DF strategy, and the DSTC configurations. In Section 3, the proposed DTDSTC scheme and optimization algorithms for the adjustable code matrix are presented. The proposed DTACMO algorithm is extended to the AF protocol and combined with the opportunistic relaying algorithms in Section 4. The rank criterion and the PEP of the delayed DSTBC schemes employed in these two types of systems are analyzed in Section 5. The results of the simulations are given in Section 6. Section 7 gives the conclusions of the work.
Notation: the italic, the bold lowercase, and the bold uppercase letters denote scalars, vectors, and matrices, respectively. The operator E[·] stands for expected value, and \(\parallel {\boldsymbol {X}}\parallel _{F}=\sqrt {\text {{Tr}}(\boldsymbol {X}^{H}\cdot {\boldsymbol {X}})}=\sqrt {\text {Tr}(\boldsymbol {X}\cdot {\boldsymbol {X}}^{{H}})}\) is the Frobenius norm. Tr(·) stands for the trace of a matrix, and (·)^{ † } for pseudoinverse. The N×N identity matrix is written as I_{ N }.
2 Cooperative MIMO system model

1. 0≤δ_{1}≤δ_{2}≤...≤δ_{max}.

2. The source and the relay nodes do not know the delay profile Δ. However, the destination node knows Δ perfectly. In practice, a designer can set Δ as an upper bound obtained by experimentation.

3. The destination node knows the channels between the relays and the destination perfectly, and the relays know the channels between the source node and themselves perfectly.

4. The DSTC scheme used for the two system models is the same in order to provide a fair comparison.

5. The feedback channel is errorfree and delayfree for simplicity.
We consider only one user at the source node in our system that operates in a spatial multiplexing configuration. Let s[i] denotes the transmitted information symbol vector at the source node, which contains N parameters, s[i]=[s_{1}[i],s_{2}[i],…,s_{ N }[i]], and has a covariance matrix \(E\big [\boldsymbol {s}[i]\boldsymbol {s}^{H}[i]\big ] = \sigma _{s}^{2}{\boldsymbol {I}}_{N}\), where \(\sigma _{s}^{2}\) denotes the signal power. The source node broadcasts s[i] to n_{ r } relay nodes as well as to the destination in the first hop. After the reception, each relay performs detection and subsequently encodes the data. Alternatively, the relays can employ cyclic redundancy check (CRC), which allows the relay node to know if the detection is successful, and automatic repeat request (ARQ) schemes.
where the N(δ_{max}+T)×NT delay profile matrix \(\phantom {\dot {i}\!}{\boldsymbol {\Delta }}_{k} = [{\boldsymbol {0}}_{\delta _{k} \times N} ; {\boldsymbol {I}}_{N} ; {\boldsymbol {0}}_{(\delta _{\text {max}}\delta _{k}) \times N}]\) at the kth relay nodes is considered. The block diagonal NT×NT matrix \({\boldsymbol {\Phi }}_{{eq_{k}}_{\text {MAS}}}[i]\) denotes the equivalent adjustable code matrix, and the NT×N matrix \({\boldsymbol {G}}_{{eq_{k}}_{\text {MAS}}}[i]\) stands for the equivalent channel matrix which is the combination of the DSTC scheme \({\boldsymbol {M}}_{R_{k}D_{\text {MAS}}}[i]\) in (1) and the channel matrix \({\boldsymbol {G}}_{R_{k}D_{\text {MAS}}}[i]\). The N×1 vector \({\tilde {\boldsymbol {s}}}[i]\) stands for the data processed by the relay node with the DF protocol whereas the N(δ_{max}+T)×1 vector \({\boldsymbol {N}}_{R_{k}D}[i]\) generated at the destination node contains noise samples.
where the block diagonal N×N matrix \({\boldsymbol \Phi }_{{eq_{k}}_{\text {SAS}}}[i]\) denotes the diagonal equivalent adjustable code matrix and the block diagonal NT×N matrix \({\boldsymbol G}_{{eq_{k}}_{\text {SAS}}}[i]\) stands for the equivalent channel matrix. The N(δ_{max}+T)×1 equivalent noise vector \({\boldsymbol N}_{R_{k}D}[i]\) generated at the destination node contains the noise parameters.
The use of an adjustable code matrix or a randomized matrix \({\boldsymbol \Phi }_{eq_{k}}[i]\) which achieves the full diversity order and provides a lower error probability has been studied in [15]. The uniform sphere randomized matrix which achieves the lowest BER of the analyzed schemes and contains elements that are uniformly distributed on the surface of a complex hypersphere of radius ρ is used in our system. The proposed adaptive algorithms detailed in the next section optimize the code matrices employed at the relay nodes in order to achieve a lower BER. At the destination node, the adjustable code matrices are normalized and then transmitted back to the relay nodes so that no increase in the energy is introduced at the relay nodes and the comparison between different schemes is fair.
where H_{SD}[i] denotes the channel matrix between the source and the destination and the N(δ_{max}+T+1)×N matrices H[i] in MAS and SAS denote the channel gain matrix of all the links in the network. We assume that the coefficients in all channel matrices are independent and remain constant over the transmission. The N(δ_{max}+T+1)×N noise vector n_{ D }[i] contains the equivalent received noise vector at the destination, which can be modeled as complex Gaussian random variables with zero mean and variance \(\sigma ^{2}_{d}\). The system considered here needs a feedback channel for transmitting the optimized code matrices and the information of the optimal relay node back to the relays. We have studied the effect of using a binary symmetric channel (BSC) as the feedback channel in [16] and concluded that imperfect feedback channel with different error probabilities and different numbers of bits in quantization lead to different detection errors in the optimized code matrices at the relays and also cause degradation of the BER performance at the destination.
3 Delaytolerant adjustable code matrix optimization for delayed DSTC schemes
In this section, we propose a delaytolerant adjustable code matrix optimization (DTACMO) strategy which employs an ML receiver at the destination for designing DSTC schemes in both MAS and SAS configurations. Adaptive RLS and SG algorithms [25] for determining the parameters of the adjustable code matrix with reduced computational complexity are also devised. The DSTC scheme encoded at each relay node employs an MLbased adjustable code matrix, which is computed at the destination node and obtained by a feedback channel in order to process the data symbols prior to transmission to the destination. We assume that the functions used in the optimization problems in this section are continuously differentiable and the proof is given in Section 5. It is worth to mention that the code matrices will be sent back to the relays through a feedback channel after the optimization and they are only used at the relays so the direct link from the source node to the destination node is not considered for simplicity.
3.1 DTACMO algorithm for MAS
We also note that the algorithm presented in (9)–(12) belongs to the class of stochastic gradient algorithms [32], which employ instantaneous values of the gradient. Therefore, the learning of the proposed approach suffers from noisy estimates and often exhibits some excess mean squared error in relation to the optimal MMSE solution.
3.1.1 RLS code matrix estimation algorithm
The RLS estimation algorithm for the code matrix \({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{\text {MAS}}}[i]\) is derived in this subsection. The superior convergence behavior of LS type algorithms when the size of the adjustable code matrix is large is the reason for its utilization. The computational complexity is reduced from cubic to square by employing the RLS algorithm.
Summary of the DTACMO RLS algorithm
1:  Initialize: \({\boldsymbol {P}}[0]=\delta ^{1}{\boldsymbol {I}}_{N(\lambda _{max}+T) \times N(\lambda _{max}+T)}\), 
\({\boldsymbol {Z}}\left [0\right ] = {\boldsymbol {I}}_{N(\lambda _{max}+T) \times N(\lambda _{max}+T)}\),  
2:  Generate Φ[0] randomly with the power constraint 
\({\text {Tr}}\left ({\boldsymbol {\Phi }}_{{eq_{k}}_{MAS}}[i]\boldsymbol {\Phi }^{{H}}_{{eq_{k}}_{MAS}}[i]\right) \leq {\mathrm {P}_{\mathrm {R}}}\).  
3:  For each instant of time, i=1, 2, …, compute 
4:  \({\boldsymbol {k}}[i]=\frac {\lambda ^{1}{\boldsymbol {P}}[i1] {\boldsymbol {r}}_{k_{MAS}}[i]}{1+\lambda ^{1}{\boldsymbol {r}}_{k_{MAS}} ^{H}[i]{\boldsymbol {\Psi }}^{1}[i1] \boldsymbol {r}_{k_{MAS}}[i]}\), 
5:  \(\boldsymbol {\Phi }^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] = {\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i1]+\lambda ^{1}({\boldsymbol {r}}_{e_{MAS}}[i]{\boldsymbol {Z}}[i1]{\boldsymbol {k}} [i])\) 
\({\boldsymbol {r}}_{k_{MAS}}^{{H}}[i]{\boldsymbol {P}}[i1]\),  
6:  Update: \({\boldsymbol {P}}[i]=\lambda ^{1}{\boldsymbol {P}}[i1]\lambda ^{1}{\boldsymbol {k}}[i]{\boldsymbol {r}}_{k_{MAS}} ^{{H}}[i]{\boldsymbol {P}}[i1]\), 
7:  Update: \({\boldsymbol {Z}}[i] = \lambda {\boldsymbol {Z}}[i1]+{\boldsymbol {r}}_{e_{MAS}}[i] {\boldsymbol {r}}_{k_{MAS}}^{{H}}[i]\). 
8:  Normalization: \({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] = \frac {{\sqrt {\mathrm {P}_{\mathrm {R}}}}{\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]} {\sqrt {\sum _{k=1}^{n_{r}}{\text {Tr}}\left ({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] \left ({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]\right)^{{H}}\right)}}\). 
3.1.2 SG code matrix estimation algorithm
The SG estimation algorithm for the code matrix \({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{\text {MAS}}}[i]\) is derived in this section. As shown in (11), the inversion of the matrices is required and the computational complexity will be increased by employing a large number of relay nodes or number of antennas.
where β denotes the step size in the recursion.
Summary of the DTACMO SG algorithm in MAS
1:  Generate Φ[0] randomly with the power constraint. 
2:  For each instant of time, i=1, 2, …, compute 
3:  
4:  where \({\boldsymbol {r}}_{e}[i]={\boldsymbol {r}}_{MAS}[i]  \sum _{k=1}^{n_{r}}{\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]{\boldsymbol {G}}_{{eq_{k}}_{MAS}}[i]\hat {\boldsymbol {s}}[i]\). 
5:  Update \({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]\) by 
6:  
7:  Normalization: \({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i] = \frac {{\sqrt {\mathrm {P}_{\mathrm {R}}}}{\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]}{\sqrt {\sum _{k=1}^{n_{r}}{\text {Tr}} \left ({\boldsymbol {\Phi }} ^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]\left ({\boldsymbol {\Phi }}^{\boldsymbol {\Delta }}_{{eq_{k}}_{MAS}}[i]\right)^{{H}}\right)}}\). 
3.2 DTACMO algorithm for SAS
where the block diagonal N×N matrix \({\boldsymbol {\Phi }}_{{eq_{k}}_{\text {SAS}}}[i]\) denotes the equivalent adjustable code matrix for the kth antenna and the block diagonal NT×N matrix \({\boldsymbol {G}}_{{eq_{k}}_{\text {SAS}}}[i]\) stands for the equivalent channel matrix combined with the DSTC scheme and the 1×N channel vector between the kth antenna and the destination.
where \({\boldsymbol {\Theta }}[i]={\boldsymbol {r}}_{\text {SAS}}[i]\hat {\boldsymbol {s}}^{{H}}[i]{\boldsymbol {G}}^{{H}}_{{eq_{k}}_{\text {SAS}}} [i]{\boldsymbol {\Delta }}^{H}_{k}  \left (\sum _{l=1,l \neq k}^{n_{r}N} {\boldsymbol {\Delta }}_{l}{\boldsymbol {G}}_{{eq_{l}}_{\text {SAS}}}[i]{\boldsymbol {\Phi }}_{{eq_{l}}_{\text {SAS}}}[i]\hat {\boldsymbol {s}}[i]\vphantom {\left (\sum _{l=1,l \neq k}^{n_{r}N}\right.}\right) \hat {\boldsymbol {s}}^{H}[i]{\boldsymbol {G}}^{H}_{{eq_{k}}_{\text {SAS}}}[i]{\boldsymbol {\Delta }}^{H}_{k}\). The detailed derivation is shown in the Appendix. Note that the last term in (24) which contains the Lagrange multiplier λ is not considered in the optimization algorithm. The value of λ can be obtained by substituting (25) into (24), or this procedure can be avoided by employing the normalization to achieve the power constraint as derived in (12) in order to reduce the computational complexity.
3.2.1 RLS code matrix estimation algorithm
By using the algorithm in Table 1, we can obtain the optimal code matrices, and the only differences are the initialized matrices P[0]=δ^{−1}I_{N×N}, and Z[0]=I_{N×N} for SAS.
3.2.2 SG code matrix estimation algorithm
The SG estimation algorithm for the code matrix \({\boldsymbol {\Phi }}_{{eq_{k}}_{\text {SAS}}}[i]\) is derived in this section. As shown in (25), the inversion of the matrices is required and the computational complexity will be increased by employing a large number of relay nodes or number of antennas.
By following the steps in Table 2, we have the DTACMO SG algorithm in SAS.
4 DTACMO algorithm with opportunistic DSTCs

1. Opportunistic AF scheme: using DSTC schemes at the source node in the first phase and selecting the best relay nodes in the second phase for forwarding what the relay received

2. Opportunistic source scheme: selecting the best antenna at the source node and using DSTCs at the relay nodes in the second phase

3. Fully opportunistic scheme: optimizing the transmitting power both at the source node and at the relay nodes.
A feedback channel is employed which allows the destination node to inform the optimal relay and to send back the optimized code matrices to the relays. The delay profiles for different relay nodes are considered in the design described in this section. For simplicity, we show the design of the system model with AF protocol and DTACMORO algorithms for MAS.
where F_{ k } denotes the N×N channel matrix between the source node and the kth relay node and N_{ k } is the matrix whose entries are complex zeromean complex random variables with variance \(\sigma ^{2}_{n_{1}}\).
According to the opportunistic relaying algorithms [18], the relays which can achieve the highest SNR_{ ins } are chosen to forward the symbols with transmission power P_{2}. The DTACMORO can be implemented by firstly calculating the optimized adjustable code matrices using the DTACMO algorithms proposed in Tables 1 or 2, followed by choosing the optimal relay node using the SNR expressed in (43).
5 Analysis of the proposed DSTBC schemes and the algorithms in MAS and SAS
In this section, we will develop an analysis that explains the differences between the delayed DSTBCs in SAS and MAS in terms of the rank criterion and the error probability. The convergence properties of the proposed algorithms are briefly derived in this section as well. We also describe the computational complexity of the proposed and existing algorithms. The ML criterion is employed in the proposed DTACMO algorithms, and different DSTC schemes can be implemented. Although the fundamental diversity order is not changed, further coding gains can be achieved and the delay tolerance of the SAS and MAS configurations is different.
5.1 Rank criterion
where s_{1} and s_{2} are symbols from a desired PSK or QAM constellation. The first antenna sends the first row in C, s_{1} and \(s^{*}_{2}\), to the destination node, and the second antenna sends the symbols s_{2} and \(s^{*}_{1}\) during the transmission interval. It is important that the determinant of the code matrix is different from zero unless both s_{1}=0 and s_{2}=0.
Unless C_{1}=C_{2}, the rank of the difference matrix ΔC is full. According to the rank criterion in [34] for the STCs, the Alamouti scheme achieves full diversity.
which suffers a rank reduction. Hence, the Alamouti scheme in SAS cannot achieve the full diversity when a time delay is introduced.
which achieves the same rank as (48) even when Δs_{1}=0 or Δs_{2}=0. Therefore, the delay tolerance of a DSTC scheme with MAS depends on the size of the code matrix.
5.2 Error probability
where S is the set of all possible combinations of the symbol vectors.
where r=rank(R_{SAS}) denotes the rank of the autocorrelation matrix and λ_{ n } denotes the nth eigenvalue of R_{SAS}. The diversity order of the code matrix is given by rN and is determined by the minimum rank of the correlated code matrix. According to (47), the rank reduction affects the diversity order of the code matrix so that the Alamouti matrix code in SAS cannot achieve the full diversity and is not a delaytolerant code.
where r=rank(R_{MAS}) stands for the rank of the autocorrelation matrix and λ_{ n } is the nth eigenvalue of R_{MAS}. The diversity order depends on the minimum rank r and according to (49) that is r=N which guarantees that the full diversity order can be achieved. According to [34], the rank and diversity analysis can be considered as the design criterion of the delaytolerant DSTC schemes in cooperative communication systems in a high SNR scenario. The optimization algorithm derived in the previous section ensures that the design of the STC scheme is delaytolerant for any SNR and achieves a high coding gain. The ML detection algorithm is employed in order to achieve the full receive diversity. The upper bound derived in this section is relatively loose and becomes tighter at high SNR values but allows a designer to infer that the diversity order obtained with DTACMORO algorithms will remain the same as those of existing techniques but there will be coding gains.
The design of the DTACMORO algorithms achieves a lower BER performance by employing the proposed DTACMORO algorithms in cooperative systems with opportunistic relaying methods in [18]. As a result, the PEP analysis of the DTACMORO algorithms in MAS and SAS is the same as that of the DTACMO algorithms. The only difference is the channel matrix G_{RD}[i] in (50) and (54) denotes the N×N channel matrix between the optimal relay node and the destination node in SAS and MAS, respectively.
5.3 Convergence analysis
As shown in the previous sections, the proposed MLbased DTACMO algorithms allow the optimization of the code matrix Φ[i] adaptively. In this section, we briefly study the differentiability of the cost functions that are optimized by the proposed algorithms and the convergence of the DTACMO RLS and SG algorithms.
5.3.1 Differentiability of the cost functions
In Section 3, we compute an instantaneous gradient of the Lagrangian (9) to derive the proposed DTACMO algorithms, while in this subsection, we show the proof of differentiability of (9).
As shown in (60), the limit of (9) is equal to 0 when the change of the code matrix approaches 0. The differentiability of the optimization problem is proved. The same procedure could be used for the SAS scheme.
5.3.2 Convergence analysis of DTACMO algorithms in MAS
where the first term s[i]s^{ H }[i] is a positive matrix and the rest of the terms denotes the multiplication of the equivalent channel vectors which is a positivedefinite matrix and the problem is convex. Therefore, the Hessian matrix of the Lagrangian cost function is a positivedefinite matrix, which ensures the RLSbased DTACMO algorithms converge to the global optimum under the usual assumptions used to prove the convergence of these algorithms for convex problems [25].
5.3.3 Convergence analysis of DTACMO algorithms in SAS
As shown in the previous section, the Hessian matrix of the Lagrangian cost function is a positivedefinite matrix so that the MLbased SG DTACMO algorithm in SAS converges to the global optimum under the usual assumptions used to prove the convergence of these algorithms for convex problems.
5.4 Computational complexity
Computational complexity
Algorithms  Multiplications 

DAlamouti  \({\mathcal {O}}(N (\lambda _{\text {max}}+T))\) 
RAlamouti  \({\mathcal {O}}\left (N (\lambda _{\text {max}}+T)^{2}\right)\) 
DTACMOSG  \({\mathcal {O}}\left ((N(\lambda _{\text {max}}+T)^{2}\right)\) 
DTACMORLS  \({\mathcal {O}}\left ((4N(\lambda _{\text {max}}+T)^{2}\right)\) 
DTACMORO  \({\mathcal {O}}\left (TN^{2}\right)\) 
6 Simulations
The simulation results are provided in this section to assess the proposed schemes and algorithms. The cooperative MIMO system considered employs AF and DF protocols with the distributed Alamouti (DAlamouti) STBC scheme in [9], randomized Alamouti (RAlamouti) in [15], and linear dispersion code (LDC) in [35] using BPSK and QPSK modulation in a quasistatic block fading channel with AWGN. The effect of the direct link is also considered. It is possible to employ different DSTC schemes with a simple modification and to incorporate the proposed algorithms. The cooperative MAS configuration equipped with n_{ r }=1,2 relay nodes and N=2 antennas at each node, while the cooperative SAS scheme employed N=1,2 antennas at the source node and the destination node and n_{ r }=1,2 relay nodes with a single antenna. In the simulations, we set the symbol power \(\sigma ^{2}_{s}\) as equal to 1, the delay profiles are perfectly known at the destination, and the power of the adjustable code matrix in the DTACMO algorithm is normalized. The SNR in the simulations is the received SNR which is calculated by \(\text {SNR} = \frac {\parallel {\boldsymbol {D}}_{D}[i]\parallel ^{2}_{F}}{\parallel {\boldsymbol {n}}_{D}[i]{\boldsymbol {n}}^{H}_{D}[i]\parallel ^{2}_{F}}\).
7 Conclusions
We have proposed a delaytolerant adjustable code matrix optimization (DTACMO) scheme and algorithms for cooperative MIMO systems with a feedback channel using an ML receiver at the destination node to mitigate the effect of the delay associated with DSTCs from relay nodes. The MAS and SAS configurations have been analyzed by comparing the pairwise error probability of the delayed DSTCs and the rank criterion. In order to achieve a better BER performance and eliminate the computation complexity at the destination node, we have introduced the SG and RLS algorithms in the proposed design. We have extended the proposed algorithms to cooperative MIMO systems with the AF strategy and opportunistic relaying algorithms in order to devise delaytolerant adjustable code matrix opportunistic relaying optimization (DTACMORO) algorithms. The simulation results have illustrated the performance of the two different cooperative systems with the same number of transmit and receive antennas and the advantage of the proposed DTACMO and DTACMORO algorithms by comparing them with the cooperative network employing the traditional DSTC scheme and the RSTC scheme. The proposed algorithms can be used with different DSTC schemes and can also be extended to cooperative systems with any number of antennas.
8 Appendix
Declarations
Funding
The authors would like to thank CNPq and FAPERJ from partially funding this work.
Authors’ contributions
TP has contributed towards the development of the system model, the algorithms, the analysis, and the simulations. In particular, he has written the code that resulted in the curves shown in the simulations section. RL has contributed towards the development of the system mode, the algorithms, and the analysis. As the supervisor of TP, he has proofread the paper several times and provided guidance throughout the whole preparation of the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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