An adaptive transmission protocol for exploiting diversity and multiplexing gains in wireless relaying networks
 Mohammed Taha O El Astal^{1}Email author,
 Ammar M AbuHudrouss^{2},
 Brian P Salmon^{1} and
 Jan Corné Olivier^{1}
https://doi.org/10.1186/s1363801502669
© El Astal et al.; licensee Springer. 2015
Received: 22 September 2014
Accepted: 21 January 2015
Published: 21 February 2015
Abstract
Wireless relaying networks with distributed spacetime block codes have been shown to provide high link reliability. This is because of the space diversity gain from multiple transmitting relays, which improves by adding more relays. The drawback of this approach is the overall reduction in throughput of the network. In this paper, we propose a method to construct a distributed spacetime block code that is combined with spatial modulation to find a flexible tradeoff between reliability and throughput. This proposed method is not restricted to a specific number of relays and can be constructed as necessary. The constructed code also uses a novel adaptive transmission protocol to achieve higher space diversity, even with relays equipped with a single antenna. This protocol assumes use of coherent detection, meaning that a perfect channel estimation is available at the destination. Lastly, a new decoder is proposed that offers significant reduction in complexity to maintain high data throughput. All claims in this work are supported with theoretical analysis and backed up with empirical results.
Keywords
1 Introduction
In recent years, the spatial diversity of wireless relaying networks (WRNs) has been used to improve signal quality in wireless transmission [15]. A WRN constructs a virtual multipleinput multipleoutput (MIMO) system using multiple neighbouring client apparatuses as relaying nodes. It has been suggested that distributed spacetime block codes (DSTBC) be used in WRNs to allow simultaneous transmission among relaying nodes to increase the spectral efficiency of the network [69]. It is known that the diversity is further improved when the number of relays in the WRN is increased. Unfortunately, this has the drawback of reducing the overall throughput of the network as the number of relays increase [10]. The throughput can be retained by increasing the code rate of the DSTBC at the cost of decoding complexity [11]. It has been shown that an orthogonal design of fullrate codes is possible only with two relays [12]. The orthogonal designs lead to a low decoding complexity at the destination[11].
To summarise, the major challenge in WRNs is to coordinate an arbitrary number of relay transmissions with high rate while maintaining both full diversity and low decoding complexity.
1.1 Prior work
The overall interest in this field of research lies in the construction of a highrate DSTBC that has the ability to utilise any arbitrary number of relays [13,14]. It would appear that research has had limited success in designing highrate codes while maintaining a singlesymbol decodability at the destination. In [15], the authors proposed systematic construction steps of a rowmonomial DSTBC with a code rate bounded by 2/(2+N), where N denotes the number of relays. In [16], a new code class called semiorthogonal precoded distributed singlesymbol decodable STBC (SemiPDSSDSTBCs) was proposed. It has the advantage of performing precoding on the information symbols, which in turn doubles the achieved code rate. Although these codes were designed for an arbitrary number of relays, they may be preferable to be used only in WRNs with a few number of relays. This is because the code rate decreases dramatically as the number of relays increases. In [17], a coding scheme, with a code rate of \(\frac {1}{4}\), was proposed to operate in WRNs of high number of relays. However, this scheme has a high decoding delay. This is as the required transmission time increases exponentially with the number of relays. In [18], an adjustable fullrate STBC matrix was designed, but it requires a feedback channel to adjust the code. This creates additional network overhead that inherently reduces the achieved diversity gain. In [1921], the generalised ABBA code (GABBA) of [22] was adapted for amplifyandforward (AF) WRNs to offer a fullrate while maintaining singlesymbol decodability. However, the resulting DGABBA scheme has some constraints: (1) the number of symbols per block (T) should be expressible as a power of two, and (2) the number of available relays (N) should be smaller or equal to T (N≤T). These constraints were resolved by utilising global knowledge of channel state information (CSI) of the entire WRN at the relays [10].
Spatial modulation (SM) was developed to improve the throughput of MIMO systems and was initially extended for WRNs in [2325]. The intention was to add the spatial dimension to the modulated signal constellation; this allowed information to be transmitted not only using amplitude/phase modulation (APM) but also using the relay index. This offers higher throughput due to data multiplexing via different relays [24,26]. However, the overall achieved diversity gain was limited to the number of receive antennas, as only one relay was active for any given transmission. This was partially solved when employing SM specifically at the source node and leaving the full diversity gain to be achieved in the link between the relays and the destination [25,27,28]. The effective use of SM was still bounded by the number of antennas available at the source node. Due to size, cost and hardware limitations, the availability of multiple antenna may not be feasible in many systems [29]. In [30,31], transmit diversity gain can be achieved at the expense of orthogonal channel resources.
In conclusion, the current WRN uses its relays either to offer high throughput only (using SM) or to achieve transmit diversity gain (using DSTBC).
1.2 Our contribution

An adaptive transmission protocol to accommodate N relays in decodeandforward (DF) WRNs, where \(N \in \mathbb {N}^{+}\). Unlike [3236], this protocol offers a transmit diversity gain of N _{0} (N≤N _{0}≤1) because the possibility of activating N _{0} number of relays simultaneously during transmission. Given some control information, the protocol can be adapted to transmit through any given number of relays (N _{0}). This means the network can work with a varying number of N _{0}, which is a desirable feature for networks that experience imperfect time synchronisation [37,38].

An algorithm to construct DSTBCSM codes. Unlike [15,16,1921], this algorithm offers a rate that increases with the number of relays. The rate for this code is \(r_{0} \text {log}_{2} M_{2}+\frac {1}{T_{2}}\text {log}_{2} c\) bpcu instead of r _{0}log_{2} M _{2} bpcu, where \(c=\left \lfloor \left (\begin {array}{c} N\\N_{0}\end {array}\right)\right \rfloor _{2^{p}}\) is the number of possible relay combinations, r _{0} is the code rate of the used STBC, M _{2} is the order of modulation used by the relays and ⌊.⌋ represents the floor integer operator. The increase in throughput uses the same total average transmitting energy. This is as only N _{0} of the N available participating relays are active at any given time. The proposed algorithm uses multilevel optimisation processes, and hence the constructed code outperforms the few existing STBCSM codes.

An optimal and suboptimal reducedcomplexity decoder to work for the proposed protocol. The orthogonal property of the STBC allows a decoding complexity to reduce from \(\mathcal { O}\left (cM_{2}^{n_{s}}\right)\) to \(\mathcal { O}(cn_{s}M_{2})\), where n _{ s } is the number of symbols per information block of transmission. Using the suboptimal decoder, this decoding complexity can be further reduced to \(\mathcal { O}(c+n_{s}M_{2})\) with some performance penalty. This makes it suitable to acquire higher throughput.

Theoretical analysis of diversity gain, coding gain and required decoding complexity of the proposed protocol. Also, these claims are backed up by numerical simulations.
This paper is organised as follows: First, the network model is described followed by an outline of the proposed transmission protocol. Designing the DSTBCSM code is then discussed. This is followed by a theoretical analysis of performance and complexity. Last, empirical results to back up the claims are presented.
1.3 Notations
Hereafter, small letters, bold small letters and bold capital letters will designate scalars, vectors and matrices, respectively. If A is a matrix, then A ^{ H }, A ^{∗} and A ^{ T } denote the hermitian, conjugate and the transpose of A, respectively.
2 Network model
3 The proposed transmission protocol
In this section, an adaptive protocol that uses an efficient combination of DSTBC and SM is proposed to obtain better cooperative diversity gain and higher overall throughput. This will be supported by a mathematical evaluation and an empirical comparison to existing protocols on bit error rate (BER) graphs.
3.1 Protocol description
For a network configuration of \(\left (\hat {N},N_{r}\right)\), the transmission through the network is conducted in two phases:
Broadcasting phase:
where g _{ n } (\(g_{n} \sim \mathcal { C} \mathcal { N}(0,1)\)) is the channel coefficient for the link between and R _{ n }. The vector v _{ n } is the noise at R _{ n } with entries \(v_{n} \sim \mathcal { C} \mathcal { N}\left (0,\sqrt {\frac {2}{P}}\right)\).
Multiplexing and relaying phase:
 1.The K decoded bits are divided into two groups. The first group contains K _{1} bits, and it is multiplexed to determine the relays that will be used to transmit the second group of K _{2} bits. These K _{2} bits are modulated and then encoded with the DSTBC. If K≠K _{1}+K _{2}, the excess bits must be buffered. The values of K _{1} and K _{2} are chosen according to Section 2.
 2.
The binary sequence of the first group is converted to a decimal value ℓ. This value is used to determine if the relay is allowed to transmit and what it should transmit. The second group of K _{2} bits is modulated to a symbol vector y _{2}(i)=[y _{2}(1,i),…,y _{2}(n _{ s },i)]^{ T } using a twotier star M _{2}ary QAM modulator (see Section 2).
 3.Given ℓ, the relay R _{ n } encodes its second group of K _{2} bits with a column of the DSTBC. The column index is obtained from the entries given in the codemapping table. This encoding process is expressed as$$ \underset{L \times 1}{\mathbf{t}_{n}(i)}=\underset{L \times n_{s}}{\mathbf{A}_{n_{\ell}}}\underset{n_{s}\times 1}{\mathbf{y}_{2}(i)}+\underset{L \times n_{s}}{\mathbf{B}_{n_{\ell}}}\underset{n_{s} \times 1}{\mathbf{y}_{2}^{*}(i)}, $$(2)
where L=n _{ s }/r _{0} and r _{0} is the code rate of the code used. n _{ ℓ } is the logic index used to identify the encoding matrices that should be used at relay R _{ n }. Both \(\mathbf {A}_{n_{\ell }}\) and \(\mathbf {B}_{n_{\ell }}\) are the encoding matrices responsible for generating the column n _{ ℓ } of the DSTBC matrix. All of \(\mathbf {A}_{n_{\ell }}\) and \(\mathbf {B}_{n_{\ell }} \left (n_{\ell }=1\ldots N_{0}\right)\) are characterized by the DSTBC matrix and are used to construct the code in a distributive manner (see Appendix Appendix 1: Example STBC encoding matrices) [11,29]. It is worth noting that many existing DSTBC matrix can be used, given the number of relays and the extent of the diversity gain required (N _{0}). This will be discussed in detail in Section 2.
 4.
The vector t _{ n }(i) is then phaserotated to ensure maximum diversity and coding gain. If the rotation is not used, then rank deficiency occurs due to interfering codewords and the achieved diversity gain is reduced. This results in the transmitted vector given as \(\mathbf {t}^{\theta }_{n}(i)=\text {exp}(j\theta _{i})\mathbf {t}_{n}(i)\), where θ _{ i }, θ _{ i }∈[0,π] are provided by the codemapping table.
where \(\mathbf {h}_{n}=\left [h_{n,1},\ldots,h_{n,N_{r}}\right ]^{T}\) is the channel coefficient vector from the relay R _{ n } to the destination with entries \(h_{n,j} \sim \mathcal { C} \mathcal { N}(0,1)\). The matrix η is the noise at the destination with entries \(\eta _{\textit {ij}} \sim \mathcal { C} \mathcal { N}\left (0,\sqrt {\frac {2}{P}}\right)\).
where \(\boldsymbol {\mathcal { H}}_{\ell }\) is the equivalent channel matrix that encapsulates both the rotation employed and the channel coefficients of the relay set ℓ used for transmission.
3.2 DSTBCSM system design and optimisation
3.2.1 Review of conventional STBCSM
Recently, the design of STBCSM codes has drawn attention because of the promising improvements shown [3236]. In this section, background and some required definitions of STBCSM that are needed to work for DSTBCSM are shown.
Definition 1.
where the columns represent the encoded L timeslot sequences that should be transmitted by the N transmitting antennas of the source. If n _{ s } denotes the number of encoded symbols, then the code rate r of the used STBC is defined as \(r=\frac {n_{s}}{L}\). ■
Definition 2.
Let X denote the STBC matrix (see Definition 1) and the STBCSM code word denoted by X _{ i,j } [32]. An STBCSM codebook \(\mathcal { X}_{d}\) is defined as a set of n _{ X } STBCSM codewords. The STBCSM code is, here, formally defined as a set of \(n_{\mathcal { X}}\) codebooks. ■
Definition 3.
The variables l,k=1…n _{ X }, \(\mathbf {H}_{\lambda }\left (\mathbf {X}_{\textit {ik}}, \hat {\mathbf {X}}_{\textit {jl}}\right)\) are the harmonic mean of the nonzero eigenvalues of \(\mathbf {A}\left (\mathbf {X}_{\textit {ik}}, \hat {\mathbf {X}}_{\textit {jl}}\right)= \Delta \left (\mathbf {X}_{\textit {ik}}, \hat {\mathbf {X}}_{\textit {jl}}\right)\Delta ^{H}\left (\mathbf {X}_{\textit {ik}}, \hat {\mathbf {X}}_{\textit {jl}}\right)\), and \(\Delta \left (\mathbf {X}, \hat {\mathbf {X}}\right)=\mathbf {X}\hat {\mathbf {X}}\). ■
In this STBCSM design, the first 2 bits are used to decide on the codeword X _{ ij }. These bits define the set of antennas to transmit the remaining encoded bits (y(1) and y(2)). It should be noted that each codeword X _{ ij } uses a unique antenna mapping pattern. The phase rotation θ _{ i } is set to avoid rank deficiency among codebooks to mitigate diversity gain loss. The designs of a more complex STBCSM in a MIMO system have shown promising improvements and recently have started to attract attention [3236].
3.2.2 DSTBCSM code construction
In a conventional MIMO system, the number of antennas at both the destination and source is fixed. In contrast, the number of transmitting antennas is determined by the number of available relays in WRNs. The number of relays can differ for each initiated transmission, a variation that equates to the need for a DSTBCSM code to operate over a varying distributed network. In this section, an algorithm is defined to construct DSTBCSM codes to work in WRNs. Two examples of constructing a DSTBCSM code for given WRNs are also provided.
The construction of an optimal DSTBCSM code relies on the proper selection of a relay index pattern that maximises the CGD. To reduce the search space for the optimised codebooks, the inner and mutual CGDs are formally defined:
Definition 4.
respectively. ■
Another two important metrics in the construction of the code are how the relay patterns are chosen, namely the indexing distance and the Hamming distance. Both are used in the shown algorithm and are formally defined as
Definition 5.
Let {R _{ i }} and {R _{ j }}, \(i \text {and }j \in \mathbb {N}\), denote two sets of relays. Then, \(\rho : \mathbb {N} \rightarrow (\{R_{i}\}\cap \{R_{j}\})\) defines the indexing distance. This computes the size of the intersection between these two sets. ■
Definition 6.
Let {R _{ i }} and {R _{ j }}, \(i \text {and }j \in \mathbb {N}\), denote two sets of relays with similar cardinality. Then, \(d_{\text {min}}: \mathbb {N} \rightarrow (\{R_{i}\},\{R_{j}\})\) defines the Hamming distance between the two sets of relays. ■
To maximise the BER performance, the proposed algorithm is designed to construct codewords with minimal indexing distance ρ and maximum Hamming distance d _{min}. This design can operate on the common modulation schemes, but an additional improvement was observed using star QAM. The star QAM relies on two parameters that are also optimised in the algorithm as an additional step and, for completeness, are defined in Definition 7.
Definition 7.
Both a and ϕ should be optimised to maximise codeword differences. \(b=\sqrt {2a^{2}}\) should hold to ensure a unity average transmitted power. ■
where r and r _{0} are the code rate of the constructed DSTBCSM and the STBC used, respectively.
The code rate for a different number of relays
N _{ 0 }  r _{ 0 }  N  c  r 

2  1  3  2  0.5+log_{2} M _{2} 
4  4  1+log_{2} M _{2}  
5  8  1.5+log_{2} M _{2}  
3  \(\frac {3}{4}\)  4  4  \(0.5 + \frac {3}{4}\text {log}_{2}M_{2} \) 
5  8  \(\frac {3}{4}+\frac {3}{4}\text {log}_{2}M_{2}\)  
6  16  \(1+\frac {3}{4}\text {log}_{2}M_{2}\) 
Example 1.
The codemapping table for Example 1
ℓ  R _{ 1 }  R _{ 2 }  R _{ 3 }  R _{ 4 }  θ  

\(\mathcal { X}_{{1}}\)  0 (00)  1  2  0  0  0 
1 (01)  0  0  1  2  
\(\mathcal {{X}}_{{2}}\)  2 (10)  0  1  2  0  0.96 
3 (11)  2  0  0  1 
Example 2.
The codemapping table for Example 2
ℓ  R _{ 1 }  R _{ 2 }  R _{ 3 }  R _{ 4 }  R _{ 5 }  R _{ 6 }  θ  

\(\mathcal { X}_{{1}}\)  0 (000)  1  0  0  0  3  2  0 
1 (001)  0  3  2  1  0  0  
\(\mathcal { X}_{{2}}\)  2 (010)  0  1  0  0  2  3  0.6 
3 (011)  3  0  1  2  0  0  
\(\mathcal { X}_{{3}}\)  4 (100)  0  2  3  0  1  0  0.35 
5 (101)  2  0  0  3  0  1  
\(\mathcal { X}_{{4}}\)  6 (110)  0  2  1  0  3  0  1.1 
7 (111)  1  0  0  2  0  3 
3.3 Decoding methods
The growing demand for more complex WRN transmissions requires more complex decoders at the receiving node. In this section, an optimal maximum likelihood (ML) decoder for the transmission protocol is described, followed by a proposal of a reducedcomplexity decoder.
3.3.1 Optimal ML decoder
where \(\mathcal { S}_{2}\) is the M _{2}ary starQAM modulation used by the relays.
where \(\mathbf {u}= \left [ \mathbf {u}(1), \mathbf {u}(2), \ldots, \mathbf {u}(n_{S})\right ]^{T}=\boldsymbol {\mathcal { H}}_{\ell }^{H}\mathbf {y}_{d}\) and \(\boldsymbol {\mathcal { H}}_{\ell }^{H}\boldsymbol {\mathcal { H}}_{\ell }=\beta \mathbf {I}_{n_{s}}\).
This simplification based on the orthogonality property reduces the complexity of the decoder from \(cM_{2}^{n_{s}}\) to n _{ s } c M _{2}.
3.3.2 Proposed reducedcomplexity decoder
where \(\mathbf {u}=\boldsymbol {\mathcal { H}}_{\hat {\ell }}^{H}\mathbf {z}\), and \(\boldsymbol {\mathcal { H}}_{\hat {\ell }}^{H}\boldsymbol {\mathcal { H}}_{\ell }=\beta \mathbf {I}_{n_{s}}\) and \(\mathbf {I}_{n_{s}}\) is n _{ s }×n _{ s } identity matrix.
Thus, it can be observed that the number of all possible combinations for an ML search reduces from c n _{ s } M _{2} to (c+n _{ s } M _{2}).
4 Performance analysis
In this section, the proposed transmission protocol is evaluated theoretically in terms of diversity gain, coding gain and the decoding complexity.
4.1 Diversity analysis
The diversity gain of a DF WRN is determined by the broadcasting or relaying phase that offers the lowest diversity gain individually. The scope of this work is focused on the diversity gain in the relaying phase. This section shows analysis of the diversity gain in the case of the optimal ML decoder and then in the case of the RC decoder.
4.1.1 ML decoder diversity analysis
Lemma 1.
If the diversity gain of an STBC code is N _{0}×N _{ r }, then constructing the DSTBCSM based on Algorithm 1 using this STBC will also have a diversity gain of N _{0}×N _{ r }, if the ML decoder of Section 2 is used at the destination.
Proof.
It is known that a wireless communication system achieves a space diversity gain of d if the average error probability is upper bounded in the high SNR range by \(\mathbb {P}\leq \alpha \gamma ^{d}\), where \(\alpha \in \mathbb {R}^{+}\) and it is independent of γ. γ is the SNR.
where h is the relaysdestination channel vector.
where \(Q\left (x\right)=\int _{x}^{\infty }{e^{y^{2}/2}\text {dy}}\). The error matrix is denoted by \(\Delta \mathbf {X} =\mathbf {X}\hat {\mathbf {X}}\).
where \(\Lambda =\text {diag}\left ({\lambda _{1}^{2}},\ldots,{\lambda _{N}^{2}}\right)\) and λ _{ n } denotes the singular values of Δ X. V is a unitary matrix and \(\tilde {\mathbf {h}}=\mathbf {V} \mathbf {h}\). It is worth noting that there is no rank deficiency in Λ due to using of the phase rotation (see Appendix Appendix 3: Rank of the constructed code).
This concludes that a transmit diversity gain of N _{0} is achieved. As the equivalent channel matrix is a concatenation of the equivalent channel matrices of each receive antenna, it suffices to check the diversity gain for only one receiving antenna. This results in a diversity gain of N _{0}×N _{ r } for a code employed with an N _{ r }antenna destination.
4.1.2 Diversity analysis of RC decoder
From (28), it can be concluded that the twostep decoder can achieve only the fullreceive diversity gain N _{ r }. Therefore, the complexity reduction is at the expense of transmit diversity gain.
4.2 Coding gain analysis
The coding gain of the proposed DSTBCSM code is improved by maximising the CGD. The inner and mutual CGD of the code designed in Example 1 is compared with the code of [32] under the assumption of using an 8ary modulation.
4.3 Complexity analysis
5 Simulation results
Simulation parameters used in Figure 5
Code  Network  Broadcast phase  Relaying phase 

(2^{2},4)  J=2 (16QAM)  n _{ S }=2 (16QAM)  
\(\frac {3}{4}\)OSTBC [11]  (4^{4},4)  J=3 (32QAM)  n _{ S }=4 (32QAM) 
ABBA code [21]  (4^{4},4)  J=4 (16QAM)  n _{ S }=4 (16QAM) 
(4^{1},4)  J=1 (16QAM)  n _{ S }=1 (4QAM)  
DSTBCSM (Example 1)  (4^{2},4)  J=2 (16QAM)  n _{ S }=2 (8*QAM) 
In Figure 5, the proposed transmission protocol (using the code in Example 1 on a (4^{2},4) network) is compared to Alamouti’s STBC, the \(\frac {3}{4}\)OSTBC, the ABBA code and the traditional SM. Alamouti’s STBC is limited to only two relays and can achieve a transmit diversity gain of 2 (N _{0}=2), while both the \(\frac {3}{4}\)OSTBC and the ABBA code constantly utilise all four relays and achieve a transmit diversity gain of N _{0}=N=4. However, they lose the throughput that can be offered by SM. The traditional SM offers better throughput at the cost of not using an STBC, which results in a loss of transmit diversity gain and coding gain. For a fair comparison in this experiment, no constraint was imposed on the modulation order used and the resulting system was required to provide a bpcu of 2. With reference to Figure 5, it is observed that the proposed protocol reports an improved BER performance. Specifically, it results in performance gains of 1.3, 2.3 and 3.6 dB over networks employing Alamouti’s STBC, \(\frac {3}{4}\)OSTBC and ABBA code, respectively. In addition, a gain of 1 dB was achieved over the STBCSM of [32] because of the unique selection of the c codewords and the use of optimised starQAM modulation. It should be noted that this STBCSM code of [32] was originally designed for a conventional MIMO system and was extended here to operate on the WRN for the purpose of this comparison.
The proposed protocol was simulated in Figure 6 for another network configuration of (6^{3},4) to investigate the ability to achieve higher diversity gain and to offer higher throughput. It was compared to an ABBA system which offers a transmit diversity gain of N _{0}=N=6 by using all the available relays but without throughput improvement. A simulation of the traditional SM was included again to illustrate how the throughput can be maximised. It was observed that the proposed protocol had the lowest BER in this experiment with a diversity gain of (N _{0}=3)×(N _{ r }=4). It provides an SNR gain of 1.2 and 2.8 dB over networks employing the ABBA code and SM, respectively.
In the next experiment, the loss in performance in the case of using the RC decoder was investigated in Figure 7. It was compared to the optimal ML decoder in WRNs of (3^{2},4), (5^{2},4) and (8^{2},4). It should be noted that a reduction in complexity is ideal in certain applications when higher throughput is needed. In this experiment, it was observed that the optimal ML decoder achieved a diversity gain of (2×N _{ r }=8), while the RC decoder had a reduced diversity gain of N _{ r }=4.
6 Conclusions
Motivated by the developments in spatial modulation, an adaptive transmission protocol was proposed for WRNs to exploit the potential space diversity gain while obtaining higher spectral efficiency. Unlike the existing literature, this protocol can accommodate an arbitrary number of relays while improving the overall throughput and maintaining the same achieved space diversity gain. In addition, an algorithm to generate DSTBCSM codes to be used by this protocol was shown. The achieved performance of the resulting codes is better than that of many existing codes because the criteria are designed to choose the codewords and to employ a multilevel optimization process. Also, a further reducedcomplexity decoder is proposed. All of these claims are accompanied by numerical and theoretical evaluations.
7 Appendices
7.1 Appendix 1: Example STBC encoding matrices
with encoding matrices: A _{1}=I _{2},B _{1}=0 _{2},A _{2}=0 _{2}, and \(\mathbf {B}_{2}=\left [ \begin {array} {cc} 0&1\\ 1&0\\ \end {array} \right ]\).
7.2 Appendix 2: Optimising a codemapping table
7.3 Appendix 3: Rank of the constructed code
Following the steps of Algorithm 1, the constructed DSTBCSM code based an STBC ( ), is with rank of N _{0}, has the same rank of .
Proof.
[t]̱ It can be noticed that \(\mathbf {A}(\mathbf {X},\hat {\mathbf {X}})\) of the constructed DSTBCSM is equal to the original STBC used in the construction of the code. Therefore, it has a rank of N _{0} if the used STBC has this rank.
It should be noted that if the phase θ is set to zero, then the code is rank deficient over several symbols, e.g. when \(\{\mathbf {y}(1)=\hat {\mathbf {y}}(2),\mathbf {y}(2)=\hat {\mathbf {y}}(1)\}\) and \(\{\mathbf {y}(1)=\mathbf {y}(2)=\hat {\mathbf {y}}(1)=\hat {\mathbf {y}}(2)\}\). However, if θ is selected to mitigate this effect, then the rank is preserved. ■
Declarations
Authors’ Affiliations
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