Chapter 6
Cooperative SMTs

Cooperative communications create collaboration through distributed transmission/processing by allowing different nodes in a wireless network to share resources. The information for each user is sent out not only by the user but also by other collaborating users. This includes a family of configurations in which the information can be shared among transmitters and relayed to reach final destination in order to improve the systems overall capacity and coverage [209, 216230]. As such, cooperative technologies have made their way toward wireless standards, such as IEEE 802.16 (WiMAX) [231] and long‐term evolution (LTE) [232] and have been incorporated into many modern wireless applications, such as cognitive radio and secret communications.

Driven by the several advantages of space modulation techniques (SMTs) and cooperative communication technologies, cooperative SMTs have been extensively investigated in the past few years. Reported results promise significant enhancements in spectral efficiency and network coverage [98, 206, 233244].

In this chapter, cooperative SMTs are studied and analyzed. In particular, amplify and forward (AF), decode and forward (DF), and two‐way relaying (2WR) will be considered.

6.1 Amplify and Forward (AF) Relaying

In cooperative AF relaying, a source (S) and destination (D) nodes are communicating, and multiple or single AF relays participate in the communication protocol as illustrated in Figure 6.1.

Schematic diagram of the amplifying and Forward (AF) cooperative SMT system model. A system with Nt transmit antennas at the source, Nr receive antennas at the destination, and with R AF relays are considered.

Figure 6.1 AF cooperative SMT system model. A system with c06-i0001 transmit antennas at the source, c06-i0002 receive antennas at the destination, and with c06-i0003 AF relays are considered.

At each particular time instant, c06-i0004 bits are to be transmitted using any of the previously studied SMTs. In the first phase, the source, which is equipped with c06-i0005 transmit antennas, applies an arbitrary SMT scheme and transmits the vector c06-i0006 from the available transmit antennas as discussed in previous chapters. The transmitted signal is received by both the destination and the relay node R. Let c06-i0007 denotes the c06-i0008 multiple‐input multiple‐output (MIMO) channel matrix between the source and the destination. The received signal at the destination is then given by

where depending on the considered SMT, c06-i0009 is the c06-i0010 spatial symbol chosen from the spatial constellation diagram c06-i0011, where c06-i0012 is generated form c06-i0013 as discussed in Chapter 3. Moreover, c06-i0014 is the c06-i0015 signal constellation symbol chosen from the signal constellation diagram c06-i0016. Finally, c06-i0017 is an c06-i0018‐length additive white Gaussian noise (AWGN) vector with zero mean and variance c06-i0019. Note, for simplicity, c06-i0020, and therefore, the signal‐to‐noise‐ratio (SNR) is c06-i0021.

In the depicted scenario in Figure 6.1, c06-i0022 single‐antenna AF relays are assumed. Hence, the signal received at the c06-i0023 relay in the first time slot is given by

where c06-i0024 is the c06-i0025dimensional multiple‐input single‐output (MISO) channel vector between the source and the c06-i0026 relay, which has similar characteristics as c06-i0027, and c06-i0028 denotes the c06-i0029 spatial symbol chosen from the spatial constellation diagram c06-i0030, where c06-i0031 is generated, depending on the used SMT, from c06-i0032. Finally, c06-i0033 is an AWGN with zero mean and variance c06-i0034 seen at the input of the c06-i0035 relay.

In conventional AF relaying, all the relays participate in the second phase by retransmitting the source signal to the destination in a predetermined orthogonal time slots. Therefore, c06-i0036 time slots are needed for each symbol transmission. The relayed signal is an amplified version of the received signal at the relay node. As such, the amplification process is performed in the analog domain without further processing. Hence, the received signal at the destination can be written as

(6.3)images

where c06-i0037 denotes an c06-i0038‐dimensional single–input multiple–output (SIMO) channel vector between the c06-i0039 relay and the destination, c06-i0040 is the amplification factor, and c06-i0041 is a colored Gaussian noise vector, c06-i0042, where c06-i0043, and c06-i0044 and c06-i0045 are an c06-i0046‐length all zeros vector and c06-i0047 dimensional identity square matrix, respectively.

It is assumed that the receiver has full channel state information (CSI). Hence, the optimum maximum–likelihood (ML) detector assuming perfect time synchronization is given by

where the search is over c06-i0048 and c06-i0049, with c06-i0050 and c06-i0051 denoting the number of bits modulated in the spatial and signal domains, respectively.

6.1.1 Average Error Probability Analysis

The ML receiver in (6.4) can be rewritten as

(6.5)images

where c06-i0052 denoting a space containing all possible transmitted vectors.

The pairwise error probability (PEP) of an AF cooperative system is given by

where c06-i0053, and c06-i0054.

Now,

where c06-i0055 is a white Gaussian noise with zero mean and covariance given by

(6.8)images

Plugging (6.7) in (6.6) and following the same steps as in (4.46),

where

(6.11)images

The average PEP is then computed by taking the expectation of (6.9),

(6.12)images

where c06-i0056 and c06-i0057 are the moment‐generation functions (MGFs) of c06-i0058 and c06-i0059, respectively. Note, the upper bound in (6.13) is obtained by using c06-i0060 in the integral in (6.12) [244].

Finally, the average bit error ratio (ABER) of AF cooperative system is

where c06-i0061 is the number of bits in error associated with the corresponding PEP event.

For c06-i0062, then,

(6.15)images

where c06-i0063, c06-i0064, and c06-i0065.

Assuming Rayleigh fading channels, the random variables (RVs) c06-i0066 and c06-i0067 are exponential RVs with means c06-i0068 and c06-i0069, respectively, with c06-i0070 and

The cumulative distribution function (CDF) of the RV c06-i0071, which is the result of the multiplication of the RVs c06-i0072, c06-i0073 and c06-i0074, is given by [123, 245, 246],

and the probability distribution function (PDF) is given by [246]

(6.18)images

where c06-i0075 is the c06-i0076‐order modified Bessel function of the second kind.

Finally, the MGF of c06-i0077 is given by [246]

where c06-i0078 is the exponential integral function.

The MGF of the exponential RV c06-i0079 is given by [123],

The ABER of AF SMTs over MISO Rayleigh fading channels can be calculated by substituting (6.19) and (6.20) in (6.14).

6.1.1.1 Asymptotic Analysis

At asymptotically high SNR values, and using Taylor series, the PDF of c06-i0080 is simplified as

(6.21)images

where c06-i0081 is the digamma function with c06-i0082.

Thus, the average PEP can be computed as [247]

The diversity gain of c06-i0083 is clearly seen in (6.22). Finally, the asymptotic ABER of AF SMTs over MISO Rayleigh fading channels can be computed by plugging (6.22) in (6.14).

Graphical representation of the simulation, analytical, and asymptotic results for AF SSK system with Nt = 2, Nr = 1, and variable R = 1, 3, and 5.

Figure 6.2 Simulation, analytical, and asymptotic results for AF SSK system with c06-i0084, c06-i0085, and variable c06-i0086.

Graphical representation of the analytical, simulation, and asymptotic results for an AF QSM system with Nt = 2, Nr = 1, and 4-QAM modulation while varying R = 2 → 5.

Figure 6.3 Analytical, simulation, and asymptotic results for an AF QSM system with c06-i0087, c06-i0088, and 4‐QAM modulation while varying c06-i0089.

Graphical representation of the AF cooperative QSM and SM Performance comparison with η = 4 bits and with 4-QAM modulation assuming Nt = 2 for QSM and Nt = 8 for SM and Nr = 1 and R = 1.

Figure 6.4 AF cooperative QSM and SM performance comparison with c06-i0090 bits and with 4‐QAM modulation assuming c06-i0091 for QSM and c06-i0092 for SM and c06-i0093 and c06-i0094.

Graphical representation of the AF cooperative QSM and SM Performance comparison with η = 6 bits and with 4-QAM modulation assuming Nt = 4 for QSM and Nt = 16 for SM and Nr = 1.

Figure 6.5 AF cooperative QSM and SM performance comparison with c06-i0095 bits and with 4‐QAM modulation assuming c06-i0096 for QSM and c06-i0097 for SM and c06-i0098.

6.1.1.2 Numerical Results

In the first results shown in Figure 6.2, the performance of AF space shift keying (SSK) system is evaluated through Monte‐Carlo simulations and analytical formulas for c06-i0099, c06-i0100, and variable number of AF relays from c06-i0101. Results reveal that increasing the number of relays significantly enhances the ABER performance. Also, analytical and asymptotic curves are shown to closely match Monte Carlo simulation results for a wide range of SNR values and for the different number of relays. An increase in diversity with the increase of c06-i0102 is also clear from the figure. Such increase in diversity gain is shown to provide about 25 dB gain in SNR at an ABER of c06-i0103. It should be noted, though, that the spectral efficiency is not identical for the depicted curves even though they all transmit the same number of data bits. This is because the number of needed orthogonal time slots for the AF scheme is c06-i0104, which increases with increasing c06-i0105. Therefore, an AF scheme with c06-i0106 requires six time slots to convey source data to destination, whereas a system with c06-i0107 requires only two time slots.

Results for AF quadrature spatial modulation (QSM) system with c06-i0108, c06-i0109, and c06-i0110 while considering 4‐quadrature amplitude modulation (QAM) are shown in Figure 6.3. Again, higher c06-i0111 value results in better error performance, and analytical and simulation results are shown to match closely for a wide range of SNR values. Also and as discussed for the previous results in Figure 6.2, increasing the value of c06-i0112 enhances the performance and degrades the spectral efficiency as c06-i0113 time slots are needed to convey the source information bits. A performance comparison between spatial modulation (SM) and QSM with single AF relay and for a spectral efficiency of c06-i0114 bits is depicted in Figure 6.4 while assuming 4‐QAM modulation. QSM system is implemented with c06-i0115 while SM considers c06-i0116 to achieve the target spectral efficiency. Results show that QSM outperforms SM performance by about 1.7 dB. Similar comparison between SM and QSM but for a spectral efficiency of c06-i0117 bits is shown in Figure 6.5. The target spectral efficiency is achieved by considering 4‐QAM for both schemes and assuming c06-i0118 for QSM and c06-i0119 for SM systems. A single AF cooperative relay is also considered. Again, QSM demonstrates better performance and a gain of about 1.7 dB can be clearly seen from the figure.

6.1.2 Opportunistic AF Relaying

Previous conventional AF relaying scheme requires c06-i0120 time slots to convey the source message to the destination. To enhance the spectral efficiency, opportunistic relaying can be considered, where only the best relay participates in the relaying process. This relay is chosen by selecting the indirect link from the source to the c06-i0121 relay and then to the destination, c06-i0122c06-i0123c06-i0124, that gives the minimum instantaneous error probability.

Following similar steps as in Section 6.1.1, the PEP for the c06-i0125 relay link can be written as

The relay that minimizes (6.23) is selected to participate in the retransmission process. Hence, the chosen relay is formulated as

(6.24)images

Please note that since only a single relay participates in the retransmission process, only two time slots are needed regardless of the number of relays in the network, and proper communication between the relays is assumed.

The ML receiver using opportunistic relaying is then given by

(6.25)images

6.1.2.1 Average Error Probability Analysis

In opportunistic relaying, only one relay is retransmitting. Thus and following similar steps as in Section 6.1.1, the average PEP using opportunistic relaying can be written as

where

(6.27)images

The CDF of c06-i0126 can be written as [248]

(6.28)images

where c06-i0127 is given in (6.17).

Furthermore, the PDF is given by

Thus, the MGF of c06-i0128 is given by

Plugging (6.30) in (6.26) results in the same average PEP as derived for conventional AF SMTs in (6.13). Hence, opportunistic AF relaying offers the same ABER performance as conventional AF relaying, but with an enhanced spectral efficiency.

6.1.2.2 Asymptotic Analysis

At high SNR values, and using Taylor series, the PDF of c06-i0129 in (6.29) can simplified to

(6.31)images

Hence, and using (6.26), the average PEP can be computed as

As in conventional AF relays, a diversity gain of c06-i0130 is clearly seen in (6.32) for opportunistic relaying.

6.2 Decode and Forward (DF) Relaying

In DF relaying, all existing relays process the received signal and decode the transmitted information. It is generally assumed that an error detection mechanism is available and the relay can tell if the decoded bits are correct or not. If the relay decodes the source signal correctly, it participates in the second phase by forwarding the message to the destination. However, if an error is detected in the retrieved data at the relay, it remains silent at this particular time instant. The relays that will participate in the second phase are grouped in a set c06-i0131 and allocated orthogonal slots. Therefore, the spectral efficiency will decay by a factor of c06-i0132, and required synchronization and signaling between relays is needed, with c06-i0133 denoting the number of relays participating in the relaying phase. In practical DF systems, as in IEEE 802.16j standard [231] and other literature [248, 249], error detection techniques [250] are used and the relay participates in the cooperative phase if it detects the whole packet correctly. However, the simplified assumption made here facilitates the derived analysis of the error probability and commonly assumed in the literature (see [235, 237, 239, 248, 249, 251253]). Besides the conducted analysis provides a benchmark for all practical systems.

6.2.1 Multiple single‐antenna DF relays

The source is equipped with multiple antennas and applies a specific SMT. However, in the first scenario considered here, single‐antenna DF relays are considered as shown in Figure 6.1. Similar to previous discussion for AF relaying, c06-i0134 bits are to be transmitted by the source at each particular time instant. The signal received at the destination node through the direct link can be written as in (6.1). Also, the signal received at the c06-i0135 relay in the first time slot is given by (6.2).

The relays apply the specific SMT‐ML decoder to decode the received source signal and retrieve the transmitted bits. If the retrieved bits are correct, the relay participates in the retransmission process. However, the relay is equipped with single antenna. Hence, the participating relay will forward the following message to the destination, c06-i0136. As such, the received signal at the destination from c06-i0137 cooperative relay, c06-i0138, is given by

(6.33)images

At the destination, the ML optimum detector is considered to jointly decode the received signals from the source and the cooperating relays:

(6.34)images

6.2.2 Single DF Relay with Multiple Antennas

The previous DF system considers multiple DF relays each equipped with single transmit and receive antennas. Alternatively, DF relays with multiple transmit antennas can be considered as illustrated in Figure 6.6. In such case, the relay will apply the same SMT considered at the source and transmits identical data. A DF relay with c06-i0139 antennas exists and participates in the retransmission phase if it detects the source signal correctly. The source and the destination nodes are the same as discussed before. The received signal at the destination is the same as given in (6.1). The signal received at the relay is given by

(6.35)images

where c06-i0140 is the c06-i0141 square channel matrix between relay and the source, and c06-i0142 is an c06-i0143‐length AWGN vector with zero‐mean and c06-i0144 variance.

Schematic diagram of the DF cooperative SMTs system model. A system with Nt transmit antennas at the source, Nr receive antennas at the destination, and with Lt transmit and Lr receive antennas at the DF relay are considered.

Figure 6.6 DF cooperative SMTs system model. A system with c06-i0145 transmit antennas at the source, c06-i0146 receive antennas at the destination, and with c06-i0147 transmit and c06-i0148 receive antennas at the DF relays are considered.

The relay decodes the signal using the ML decoder as

(6.36)images

If the signal is decoded correctly, the relay retransmits the decoded symbol vector c06-i0149 using the same SMT used at the transmitter. Therefore, the received signal at the destination in the cooperative phase is given by

(6.37)images

where c06-i0150 is an c06-i0151 fading channel matrix between the relay and the destination, and c06-i0152 is an c06-i0153‐length AWGN vector with zero‐mean and c06-i0154 variance.

The destination node combines the received signals from the direct link and the cooperative link to detect the source signal as

(6.38)images

6.2.3 Average Error Potability Analysis

6.2.3.1 Multiple Single‐Antenna DF Relays

In multiple single‐antenna DF relaying, the transmitted message is received via a direct link and through all relays that detected the transmitted SMT signal correctly. The average PEP that the c06-i0155 relay detects the SMT signal incorrectly, and thus being off, is given by

As defined earlier in Section 6.1.1, c06-i0156 is an exponential RV with a mean c06-i0157 given in (6.16). Hence, and from (4.6),

(6.40)images

In the considered multiple single‐antenna DF relays, different scenarios can be defined. In the first scenario, none of the available relays decoded the source message correctly. Therefore, all relays will be off and will not participate in the relaying phase. As such, the destination has to rely on the direct link signal to decode the message. In the second scenario, part of the relays decoded the signal correctly and are grouped in the set c06-i0158. Finally, all relays can decode the signal correctly and c06-i0159. Let the PEP given that the c06-i0160 scenario occurs be c06-i0161. Then, the average c06-i0162 for all scenarios is given by

In the first scenario, all relays are off and the receiver decodes only the direct link signal. Hence, the PEP of this scenario is

where c06-i0163 is given in (6.10).

In the other two scenarios, c06-i0164 relays detected the signal correctly and retransmitted it to the destination. Hence, the receiver will be receiving the signal from the direct link and from c06-i0165 relaying links. Thus, and following the same steps as in Section 6.1.1, the PEP for the c06-i0166 scenario is given by

where

(6.44)images

where c06-i0167 and c06-i0168 are, as defined earlier in Section 6.1.1, exponential RVs with means c06-i0169 and c06-i0170 given in (6.16).

Plugging (6.42) and (6.43) in (6.41) gives

Taking the expectation of the PEP in (6.45), using (4.6) and following the same steps as in Section 6.1.1,

Note, c06-i0171 is given in (6.20).

From [198], the PDF of c06-i0172, which is the result of the multiplication of two exponential RVs c06-i0173 and c06-i0174, is given by

(6.47)images

The MGF of c06-i0175 is then given by

(6.48)images

where c06-i0176 is the incomplete Gamma function.

6.2.3.2 Single DF Relay with Multiple‐Antennas

From (6.46), the average PEP for a single DF relay is given by

where c06-i0177 is an exponential RV with a mean c06-i0178. Hence, the MGF of c06-i0179 is equal to the MGF of c06-i0180 given in (6.20). Therefore, the average PEP in (6.49) can be rewritten as

(6.50)images

From (6.39), and considering that the relay has c06-i0181 receive antennas, the PEP of the relay detecting the transmitted SMT signal incorrectly and being off is

Different to single‐antenna relay, c06-i0182 in (6.51) is a Chi‐squared RV and from [123, 191], the average PEP in (6.51) is given by

(6.52)images

where c06-i0183.

6.2.3.3 Numerical Results

Simulation results for SM system with DF relays are illustrated in Figure 6.7, where c06-i0184, c06-i0185, and c06-i0186 while considering binary phase shift keying (BPSK) modulation. Similar behavior as noted for AF relaying can be seen here as well. Increasing the number of relays significantly enhances the error performance due to the increase of diversity gain. Increasing c06-i0187 from one to four enhances the SNR performance by about 12 dB at an ABER of c06-i0188. Such gain is achieved while degrading the spectral efficiency, as noted for AF system, since a maximum of c06-i0189 time slots will be needed to convey the source information bits through c06-i0190 DF relays. It is shown in the figure, as well, that analytical and simulation results closely match for a wide range of system parameters.

Graphical representation of the simulation, analytical, and asymptotic results for cooperative DF SM system with Nt = 2, Nr = 1, BPSK modulation and variable R = 1 → 4.

Figure 6.7 Simulation, analytical and asymptotic results for cooperative DF SM system with c06-i0191, c06-i0192, BPSK modulation and variable c06-i0193.

Increasing the modulation order to quadrature phase shift keying (QPSK) is shown to degrade the DF SM performance by about c06-i0194 dB as shown in Figure 6.8. Furthermore, and as in previous results, Figure 6.8 shows that increasing the number of relays enhances the performance. From the figure, compared to no relays, using five relays offers a 20 dB gain in the SNR.

Graphical representation of the simulation, analytical, and asymptotic results for cooperative DF SM system with Nt = 2, Nr = 1, QPSK modulation and variable R = 1 →  4.

Figure 6.8 Simulation, analytical, and asymptotic results for cooperative DF SM system with c06-i0195, c06-i0196, and QPSK modulation and variable c06-i0197.

6.3 Two‐Way Relaying (2WR) SMTs

In all previously discussed cooperative networks, multiple orthogonal time slots are needed to broadcast information from a source node to a destination node. However, an enhanced spectral‐efficiency relaying algorithm that attracted significant interest in literature is 2WR [236, 242, 254256]. In 2WR scheme, two source nodes are allowed to simultaneously transmit their data blocks toward a relay node. The relay node retrieves data bits from both nodes and applies network coding principle on the decoded messages. The new generated coded data block is then forwarded to both nodes. To receive the data from the other node, each node reverses the coding operation applied at the relay node.

A system model for 2WR protocol is illustrated in Figure 6.9, which consists of two source nodes, c06-i0198 and c06-i0199, that exchange information with the aid of a relay node c06-i0200. The number of transmit antennas is, respectively, given by c06-i0201, c06-i0202, and c06-i0203 for c06-i0204, c06-i0205 and c06-i0206 nodes. Similarly, the number of receive antennas is denoted by c06-i0207, c06-i0208, and c06-i0209.

Data transmission is performed in two consecutive phases, namely, transmission phase and relaying phase. In the transmission phase, both c06-i0210 and c06-i0211 nodes concurrently transmit c06-i0212 bits toward the relay node using an SMT. A DF relay decodes the received data from both nodes and obtains an estimate for the c06-i0213 bits. In the relaying phase, the relay precodes the estimated bits to generate a new message with c06-i0214 bits that will be forwarded to both nodes using any of the discussed SMTs.

Schematic diagrams of a two-way relaying system model applying SMTs at any transmitting node. It is assumed that the sources A and B are respectively equipped with NA and NB transmit antennas and LA and LB receive antennas, and the relay has NR transmit and LR receive antennas.

Figure 6.9 A two‐way relaying system model applying SMTs at any transmitting node. It is assumed that the sources c06-i0215 and c06-i0216 are equipped, respectively, with c06-i0217 and c06-i0218 transmit antennas and c06-i0219 and c06-i0220 receive antennas, and the relay has c06-i0221 transmit and c06-i0222 receive antennas.

In what follows, the transmission and reception protocols in both phases are described in detail.

6.3.1 The Transmission Phase

In this phase, both nodes c06-i0223 and c06-i0224 concurrently transmit c06-i0225 bits toward the relay node, c06-i0226, using a specific SMT. In order to keep the same block length (i.e., c06-i0227 bits) from both nodes, different nodes with unequal number of transmit antennas should use different modulation orders.

The transmitted block from c06-i0228 is denoted by c06-i0229, while the transmitted block from c06-i0230 is denoted by c06-i0231. Also, the transmitted vectors from nodes c06-i0232 and c06-i0233 are denoted by c06-i0234 and c06-i0235, respectively, that are generated from their corresponding data blocks (i.e., c06-i0236 and c06-i0237).

The received signal vector at the relay c06-i0238 is denoted by c06-i0239 and is given as follows:

(6.53)images

where c06-i0240 and c06-i0241 are, respectively, the c06-i0242 and c06-i0243 MIMO channel matrices between node c06-i0244 and the relay and node c06-i0245 and the relay. Moreover, the vector c06-i0246 is an c06-i0247‐length AWGN vector with zero mean and c06-i0248 variance.

The ML detector is considered at the relay node to retrieve the transmitted vectors c06-i0249 and c06-i0250 as

(6.54)images

where c06-i0251 and c06-i0252 represent the sets of all possible transmission vectors from nodes c06-i0253 and c06-i0254, respectively. The detected vectors c06-i0255 and c06-i0256 are then, respectively, mapped back to their corresponding data blocks c06-i0257 and c06-i0258.

6.3.2 The Relaying Phase

Upon obtaining the blocks c06-i0259 and c06-i0260, the relay node processes them to obtain a third data block of size c06-i0261 bits, denoted by c06-i0262. This is usually performed through a simple XOR operation as

(6.55)images

The c06-i0263 bits block is then used to obtain the transmission vector c06-i0264, which is transmitted from the relay node in this phase.

The received vectors at nodes c06-i0265 and c06-i0266 are

(6.56)images
(6.57)images

where c06-i0267 and c06-i0268 are, respectively, the c06-i0269 and c06-i0270 MIMO channel matrices between the relay and the nodes c06-i0271 and c06-i0272, and c06-i0273 and c06-i0274 are the c06-i0275 and c06-i0276 length AWGN vectors at c06-i0277 and c06-i0278 nodes, respectively.

At nodes c06-i0279 and c06-i0280, the ML detection is applied on the received signals in order to estimate the detected vectors c06-i0281 as follows:

and

(6.59)images

where c06-i0282 is the set of all possible transmitted vectors from the relay node c06-i0283.

At the end of this phase, each node maps its detected vector into its corresponding bit block. Let c06-i0284 denotes the obtained block at node c06-i0285, and c06-i0286 is the obtained bits block at node c06-i0287. Consequently, each node can extract a corrupted version of the transmitted bits from the other node by performing an XOR operation with its own transmitted bits as

and

(6.61)images

where c06-i0288 being the corrupted version of c06-i0289 obtained at node c06-i0290, and c06-i0291 is the corrupted version of c06-i0292 obtained at node c06-i0293.

At the end of both phases, each node has received c06-i0294 bits from the other node. Compared to the conventional one‐way relaying systems, as discussed previously, 2WR can double the spectral efficiency. However, the cost of the improvement in the spectral efficiency is paid in the overall error performance at both nodes, which will be analyzed and discussed hereinafter.

6.3.3 Average Error Probability Analysis

The error performance at either nodes, c06-i0295 or c06-i0296, is identical. Therefore, only the error performance at node c06-i0297 is considered. Starting from (6.60), the received block c06-i0298 can be expressed in terms of c06-i0299 as follows:

where c06-i0300, c06-i0301, and c06-i0302 are the error vectors in the transmitted block from c06-i0303, c06-i0304, and c06-i0305 nodes, respectively, and all are of c06-i0306 length. A specific bit in an error vector is 1 if the corresponding bit has been received in error. Otherwise, it is 0. For example, a specific bit in c06-i0307 is 1 if the corresponding bits in c06-i0308 and c06-i0309 are not equal. The second line in (6.62) is obtained by substituting c06-i0310, the third line is obtained by substituting c06-i0311, the fourth line is obtained by substituting c06-i0312 and c06-i0313, the fifth line is obtained by substituting c06-i0314 where c06-i0315 is an all zeros vector, and the last line is obtained using c06-i0316.

Now, the last line in (6.62) can be rewritten on a bit base as

(6.63)images

where c06-i0317 is an arbitrary bit in c06-i0318, and c06-i0319, c06-i0320, c06-i0321, and c06-i0322 represent the corresponding bits in c06-i0323, c06-i0324, c06-i0325, and c06-i0326, respectively. The bit c06-i0327 is correct if it is equal to c06-i0328. Hence, the bit error rate at c06-i0329, denoted by c06-i0330, can be expressed as follows:

Notice that c06-i0331 occurs if only one of them is 1 or all of them are 1's. Thus, (6.64) can be expanded to

(6.65)images

which can be further simplified to

where

(6.67)images
(6.68)images
(6.69)images
(6.70)images

and c06-i0332.

Also, it can be easily verified that c06-i0333. Therefore, (6.66) can be further simplified to

The parameter c06-i0334 represents the ABER in the received block at the relay for the source c06-i0335 given that the relay received the block from c06-i0336 correctly. Similarly, c06-i0337 is the average ABER in the received block from c06-i0338 at the relay given that the relay received the c06-i0339 block correctly, and c06-i0340 represents the average ABER in the received block from the relay at node c06-i0341.

The computation of the average ABER can be obtained through the union bound technique. As such, c06-i0342 is formulated as

where c06-i0343 is the hamming distance between the two bit blocks corresponding to c06-i0344 and c06-i0345. The probability c06-i0346 can be expressed using (6.58) as

For a given c06-i0347, and as shown previously and in Chapter 4, the above probability can be expressed by means of c06-i0348‐function as

where c06-i0349.

Assuming Rayleigh fading channels, the probability in (6.74) can be averaged and upper bounded as

where c06-i0350 and c06-i0351 denotes the Kronecker product. A proof of (6.75) can be easily obtained considering the ABER derivations in Chapter 4.

Substituting (6.75) in (6.72), the probability c06-i0352 can be upper bounded as

Following similar steps as used to obtain (6.78), c06-i0353 can be derived and is given by

(6.77)images

where c06-i0354 is the hamming distance between the bit blocks corresponding to c06-i0355 and c06-i0356. Following the same procedure in (6.72), (6.73), (6.74), (6.75), (6.76), the probability c06-i0357 can be upper bounded by

where c06-i0358 and c06-i0359. Please note that both c06-i0360 and c06-i0361 are identical (but not equal) due to the assumption that both c06-i0362 and c06-i0363 have same statistical fading model.

The last probability to be estimated in (6.71) is c06-i0364, which is given by

where c06-i0365 represents the hamming distance between the bit blocks corresponding to c06-i0366 and c06-i0367.

Now using (6.58), and similar to (6.74), the probability c06-i0368 for a given c06-i0369 can be expressed as

where c06-i0370 and (6.80) can be rewritten as

where c06-i0371.

Therefore, c06-i0372 can be computed by substituting (6.81) into (6.79) as follows:

Graphical representation of the simulation and analytical results for the ABER versus the average SNR for 2WR QSM MIMO system. The block length is η = 4 bits per channel use for each transmitting node. The nodes are assumed to have two transmit antennas and using 4-QAM modulation order. The number of received antennas is varied from 1 → 3.

Figure 6.10 Simulation and analytical results for the ABER versus the average SNR for 2WR QSM MIMO system. The block length is c06-i0373 bits per channel use for each transmitting node. The nodes are assumed to have two transmit antennas and using 4‐QAM modulation order. The number of received antennas is varied from c06-i0374.

Finally, the average ABER at node c06-i0375 ( or node c06-i0376) can be expressed by substituting (6.76), (6.78), and (6.82) into (6.71).

6.3.3.1 Numerical Results

The ABER for 2WR QSM system is depicted in Figure 6.10 for c06-i0377, c06-i0378‐QAM, and c06-i0379 is varied from c06-i0380. Considering this setup, each node will deliver c06-i0381 bits by the end of the two phases. As illustrated in the figure, increasing the number of receive antennas at all nodes significantly improves the overall performance and a gain of about 9.2 dB can be noticed at an ABER of c06-i0382 when having three receive antennas instead of two. In addition, depicted analytical and simulation results are shown to match closely and for the different depicted curves with variant number of receive antennas.

Another setup considering c06-i0383, c06-i0384‐QAM and the receive antennas are varied from c06-i0385 is evaluated, and the results are shown in Figure 6.11. With these configurations, the spectral efficiency increases to c06-i0386 bits for each transmitting node. Similar conclusions as drawn in the previous figure can be concluded here as well.

Simulation and analytical results for the ABER versus the average SNR for 2WR QSM MIMO system. The block length is η = 5 bits per channel use from each transmitting node. Each node is assumed to be equipped with two transmit antennas and transmits an 4-QAM symbol. The number of received antennas is varied from 1 → 3.

Figure 6.11 Simulation and analytical results for the ABER versus the average SNR for 2WR QSM MIMO system. The block length is c06-i0387 bits per channel use from each transmitting node. Each node is assumed to be equipped with two transmit antennas and transmits an 4‐QAM symbol. The number of received antennas is varied from c06-i0388.

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