Distributed Space Shift Keying in Cooperative Wireless Networks: Capacity Analysis and Error Performance

. In this study, a distributed fashion of space shift keying as an e ﬃ cient cooperative framework is proposed and studied to address the main challenges in future cellular networks by providing high energy e ﬃ ciency and meeting the demand growth with the lowest complexity and the highest performance. A comprehensive mathematical analysis has been performed to ﬁ nd a closed form for the channel capacity of distributed space shift keying and to depict the in ﬂ uence of the number of distributed antennas on the achievable channel capacity. In addition, a simulation study of the error performance is presented using two detection methods: maximum-likelihood detection based on channel state information and blind detection. It was shown that as the number of cooperating relays increases, the error performance of the detectors converges, although they incur di ﬀ erent costs and computational complexities. A comparison study in terms of error performance and capacity is applied with other cooperative spatial modulation to portray the outperformance of the system presented in this work.


Introduction
Cooperative communication technologies have been proven to be an efficient strategy for improving network coverage, combating shadowing and multipath fading, as well as for leveraging channel utilization [1]. The research community has investigated distributed cooperative techniques through exploiting the antennas of surrounding users/relays as members of a virtual multiple-input-multiple-output (MIMO) system [2]. The distributed antenna array provides seamless connectivity, higher diversity gain, and higher channel capacity and allows a single antenna terminal to harvest the benefits of the MIMO system.
Space shift keying (SSK), as a type of spatial modulation (SM), is a MIMO technique that implicitly uses the antenna index as the means of conveying information. During each instant, a single antenna is activated where each information symbol is encoded into a unique antenna index. SSK overcomes the main complexities and challenges of MIMO systems by requiring only a single RF chain. Furthermore, it is also more robust to channel estimation errors since the detection depends on the differences between the transmit links rather than the actual link realization [3]. However, an acceptable data rate in an SSK system requires a large number of transmitting antennas, which increases the cost and the difficulty of implementation on small mobile devices. Therefore, SSK can be an efficient choice only in the downlink of cellular networks. By combining SSK with a collaborative MIMO system, this problem has been solved in [4], and a new scheme is presented and named distributed SSK (DSSK).
Several authors have investigated the potential benefits of combining SSK transmissions and relay-aided cooperation. The error performance of a cooperative SSK network is studied in [5][6][7][8], where the source applies SSK modulation and employs spatially distributed relays as a distributed antenna array to forward the signal to the destination. The relay(s) involved in the forwarding process is/are either equipped with a single antenna, as in [5,9], or with multiple antennas, as in [7,8]. Based on relaying principles and the number of relaying antennas, data in the signal space or antenna space is transmitted to the destination [8,10]. In [8,[11][12][13], the error performance of the decode and forward (DF) cooperative SSK network is studied, where there is a direct link between the source and destination and all nodes are equipped with multiple antennas. Selection combining is used at the destination to select either the direct link or any of the relay-destination links when more than one relay correctly decodes the source SSK symbol, based on the Euclidean distance between the SSK constellation points and the channel state of the transmission link(s).
To increase the diversity gain of a cooperative SSK network, transmit diversity can be incorporated at the source. In [14], transmit diversity is achieved by using space-time SSK, where a symbol is transmitted over many time slots by activating a sequence of transmit antennas. In [15], the diversity is achieved by dividing the transmit antennas into K antenna groups of the same size; then, according to channel state information (CSI), the receiver selects one group in each transmission block. In [4,16,17], transmit diversity is achieved by shifting the SSK modulation from the source to cooperating relays. SSK is applied to their spatial location and is called distributed SSK (DSSK) or cooperative SSK (CSSK). The broadcast symbol transmitted by the source is used to activate a single relay, similar to the activation of a single transmitting antenna in SSK, and the relay can transmit its own data in the conventional constellation domain. In [4,17], the cooperating relays are unaware of the decoding errors and simply map the received symbol into their spatial domain. Therefore, a direct link between the source and destination and a robust detector at the destination node is required to compensate for the forwarded errors. Moreover, the modulation order, the number of the cooperating relays, and the number of the receiving antennas at the destination node are restricted to "2" in [4]. In [16,17], the source and relays are equipped with a single antenna, while multiple antennas are configured at the destination. In contrast to the system in [17], the relays in [16] apply a cyclic redundancy check (CRC) to detect errors during the first broadcast phase. If a relay successfully decodes the source signal that coincided with its index, it transmits an unmodulated carrier; otherwise, it remains silent during the cooperating phase.
Most studies have focused on analyzing the error performance of SSK, while others have analyzed its channel capacity. The closed form of the channel capacity of SM and/or SSK is studied in [18][19][20]. In [21], the capacity of SM techniques in AF cooperative systems is derived, and a comparison between MIMO and SM in AF cooperative systems is presented. In this study, we analyze the capacity of the DSSK in the decode and forward (DF) cooperative system and present a closed form of its lower bound. To our knowledge, this is the first study to analyze the capacity of such a system.
To make SSK suitable for single-antenna systems, cooperative diversity is considered in this study. The space diversity of nearby distributed users is exploited to act as DSSK relays, and their antennas form a virtual antenna array. In this study, two detection methods are applied at the destination: maximum likelihood detection (MLD), which is common and optimal detection in SSK systems, and a new method called maximum signal detector (MSD). MSD does not require channel state information (CSI), thus leveraging the simplicity of the system with little degradation in error performance. The main objective of this study is to maintain a simple, low-cost, and efficient cooperative system. The remainder of this paper is organized into five sections. Sections 2, 3, and 4 present the system model, computational complexity, and theoretical analysis of the channel capacity, respectively. A simulation study is presented in Section 5, and the study is concluded in Section 6.

System Model
The study considers half-duplexed terminals of one source, one destination, and a set of terminals that are able and willing to cooperate if required. There is no direct link between the source and destination such that two orthogonal channels are required for communication purposes. The space diversity of either distributed cooperating users or dedicated relays is exploited to act as DSSK relays. The source broadcasts a symbol of n bits in the first transmission phase, and the surrounding terminals overhear and decode the symbol. 2 n of the terminals who successfully decode the transmission are selected to employ their distributed antennas in SSK modulation in the second transmission phase and forward the source signal through the index of the activated relay. The activated cooperating user is able to retain its transmission simultaneously with the forwarded symbol, unlike the conventional relay-aided network, where the cooperating user freezes its own information transmission. This cooperative framework retains the advantages of SSK, and furthermore, it can remove interrelay interference (IRI), and the need for interrelay synchronization (IRS) in multirelayaided cooperative networks. A time-varying Rayleigh fading channel is considered.
The transmission of data takes place in two different time slots to avoid collision between the data of two links. During broadcasting phase, the received signal in each relay node is where rϵf1, 2, ⋯, N r g, χ s is an (n bits) signal broadcasts from the source using one of two conventional modulation schemes MQAM/MPSK with a transmit power P s . The number of cooperating relays N r equals the constellation size M = 2 n . h sr is the Rayleigh fading channel between the source and rth cooperating relay, and η sr represents a zeromean independent complex Gaussian random variable of the link in the first broadcast phase.
In the second phase of transmission, N r relays that correctly decoded the signal from the source will be included in the SSK modulation process in a distributed manner to forward the received symbol to the destination. To portray the DSSK modulation, an example of 2 bits/s/HZ transmission is given in Table 1. Each of the four relays has a unique identification index. If the broadcast symbol matches a relay's index, it becomes active and transmits either its own signal or only an unmodulated carrier signal, whereas the other relays remain silent.

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The received signal at the destination node is given by P r is the transmit power of the active relay, and h rd = ½h 1d h 2d ⋯ h N r d , where h rd represents the fading channel between the rth relay and the destination. η rd = ½η 1d η 2d ⋯ η N r d and η rd denotes a zero-mean independent complex Gaussian random variable of the link from the rth relay to the destination. The destination considers the MLD algorithm to estimate b Φ, an estimate of the transmitted vector Φ r by the relays in the second transmission phase. The MLD is the optimal detector that minimizes the probability of error based on the channel state information (CSI) by solving the following problem: where S is SSK symbol. Minimization is performed over all possible transmitted vectors of Φ. Solving this problem also involves the calculation of the objective function for all the possible values of b Φ. The complexity of the MLD detector increases exponentially with the number of cooperating relays and requires the CSI of all relay-destination links. However, a blind version of the detection can be applied by solving the following problem: This is called the maximum-signal detector (MSD). Finally, the output is decoded to determine the index of the active relay and retrieve the source data.

Computational Complexity
Relying on the on/off state for embedding information, SSK imposes low complexity, either if it is applied in a conventional system or a cooperative relaying system [22]. The ability to have no amplitude/phase elements to generate the data constellation symbols (e.g., no RF chains required) is recommended as a more energy-efficient technique [23]. Further, in DSSK, one relay is active at the second transmission phase which relaxes the requirement of interrelay synchronization in contrast to conventional relay-aided networks.
MSD is easier to implement since it is blind detection and exhibits the lower computational complexity denoted by O MSD . To compare the computational complexity of the detection algorithm MLD, O MLD is found to represent the additional computational as a function of the number of transmitted bits in a symbol (n) in terms of the real arithmetic operations over O MSD .
It is important to note that each complex addition imposes two real additions as ðA + jBÞ ± ðC + jDÞ = ðA + CÞ ± jðB + DÞ, and each complex multiplication imposes four real multiplications as ðA + jBÞðC + jDÞ = ðA × C − B × DÞ + jðB × C + A × DÞ. As it is obvious that the MLD detector imposes higher computational complexity in addition to the cost of estimating the channel state of 2 n links to the destination.
On the other hand, to compare the complexity with other cooperative relay-aided cooperative scheme, distributed spatial modulation (DSM) in [24] is considered. It uses the habitual amplitude/phase constellation symbols and the index of transmit antenna/relay. Relying on the MLD detector to recover the embedded information, the detection computational complexity over O MSD in DSSK system is given by All the schemes consider one destination antenna. However, if more antennas at the destination are involved in the detection process, the computational complexity will face a multiplicative increase in the number of multiplication operations.

System Capacity
Hassan and Mustafa defined channel capacity as the channel's mutual information maximized over all possible input distributions [25]. In relay-aided cooperating networks, the transmission of a symbol takes place in two orthogonal time  3 Wireless Communications and Mobile Computing slots and channels. In the first transmission phase, the source transmits a conventional MPSK or MQAM constellation symbol. The symbol is received by N r spatially distributed antennas, and the capacity of the first phase can be calculated similarly to the SIMO system as given by ϱ sr = ½ϱ s1 ϱ s2 ⋯ ϱ sN r where ϱ 2 sr is the variance of the white Gaussian noise over the link from the source to the r th relay. In the second transmission phase, the 2 n distributed relays that have successfully decoded the transmitted data from the source cooperate to implement DSSK and act as a virtual antenna array that uses the antenna space to forward the data to the destination, i.e., the data stream is mapped to the index of a relay. The capacity of the second phase through the distributed antenna space of the cooperating relays can be defined as the mutual information between the channel input and output Hðγ rd Þ denotes the differential entropy of a continuously distributed random variable, which is interpreted as the expected value of log 2 1/ðpðγ rd ÞÞ . where Hðγ rd jΦ r Þ is the entropy of γ rd knowing Φ r and is formulated as the expected value of log 2 1/ðpðγ rd jΦ r ÞÞ. Assuming transmission over a flat fading channel, the PDF of the received signal vector γ rd given Φ r with a complex Gaussian distribution, is given as follows: σ rd = ½σ 1d σ 2d ⋯ ⋯σ N r d where σ 2 rd = kh rd k 2 P r + kϱ rd k 2 is the variance of the signal received at the destination from the rth relay. The transmit power of the active relay is only P r ≠ 0, and ϱ 2 rd is the variance of the white Gaussian noise over the rth-destination link. Φ r is a random vector signal because the index of the active relay is random, and the relays are equally likely to radiate the unmodulated signal with a probability of 1/N r : Then, pðγ rd Þ can be formulated as follows: where Φ denotes the space of all relayed signals. Then, the channel capacity in Equation (10) can be formulated as an integration over all possible γ rd Substituting Equation (11) and Equation (12) into Equation (13) To solve the last equation and find a closed-form expression for capacity, it can be defined as a sum of three terms and solve each term separately as follows: Since Ef f ðxÞg ≥ f ðEfxgÞ then Eflog 2 ð kσ rd k 2 Þg ≥ log 2 Ef kσ rd k 2 g, Based on the commonly known integration that Ð e −x 2 dx = ffiffiffi π p , Now, to solve the second term in Equation (14), Since is a convex function, then

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By converting the bound equation to the polar coordinates through the assumption of γ id j j= r then dγ id = rdrdθ, By taking a new variable as r 2 /jσ id j 2 = x and rdr = jσ id j 2 /2dx, The last third term is given by Since Ð 1/ðπ kσ rd k 2 Þe −kγ rd k 2 /kσ rd k 2 dγ rd = 1, as the integration of a probability density function over the entire space is equal to one, and −1/N r Eff ðγ rd Þg ≥ −1/N r f ðEfγ rd gÞ, This can also be written as A closed form for the lower bound of capacity in the second phase is the summation of the three terms, as given by Since the transmission takes place in two time slots, the overall system capacity is as follows:

Simulation Study
In this section, a computer simulation is performed to study the performance of the DSSK system model described in Figure 1, considering a time-varying Rayleigh fading channel. The study exposes the effect of changing the number of cooperating relays in transmission, considering two, four, and eight cooperating relays. The BPSK, QPSK, and 8-QAM modulations are applied to modulate the source symbols depending on the number of cooperating relays. For example, 8-QAM modulation is applied at the source when eight distributed relays are involved in the process. The study is conducted in terms of error performance and channel capacity. Figure 2 shows the error performance of the DSSK scheme for the purpose of comparing the performance of the two detection techniques described earlier in Section 2, MLD and MSD. The error performance as a function of signal-to-noise ratio (SNR) over time-varying Rayleigh fad-ing channels has been investigated for N r = 2, 4, and 8 cooperating relays. Obviously, the performance deteriorates when the number of spatially distributed relays in the DSSK system increases. However, the results show the potential of MLD over MSD when the number of cooperating relays is low (N r = 2 and N r = 4) at the expense of higher cost and complexity, where the channel state is required in the detection process. But the performance of both detection methods converges when more relays are involved in the DSSK system, although MSD is a blind detection method, as explained earlier. For example, at an error rate of 10 -4 , the results show the progress of MLD over MSD by 3.2 dB and 0.7 dB for two and eight cooperating relays, respectively.
In addition to the error performance, the channel capacity of the DSSK system is investigated as a function of SNR and illustrated in Figure 3, using different numbers of spatially distributed relays. The results reveal the lower bound of the transmission rate over the bandwidth. As the results indicate, the achievable channel capacity clearly depends on the number of cooperating relays. For example, over a wide range of SNR, a gain of nearly 0.4 bit/sec/Hz and 0.37 bit/s/Hz are recoded by the switch from 2 to 4 and 4 to 8 cooperating relays, respectively.
A comparison study in terms of the error performance and the channel capacity between DSSK and the two cooperative SM (CSM) schemes in [26], CSM1 and CSM2, where the SM is applied, either at the source or at a single relay. In CSM1, SM is applied at a source with four multiple antennas N s to communicate with the destination through a single antenna relay. While in CSM2, SM is applied at a single relay node with four antennas to retransmit the received constellation symbol from a single antenna source to the destination. The schemes in this comparison study consider a single antenna at the destination, N d = 1.    Wireless Communications and Mobile Computing In Figure 4, the performance of DSSK with N r = 4, CSM1 with N s = 4 and CSM2 with N r = 4 over the Rayleigh fading channel is presented for comparison purposes. In CSM1 and CSM2, the source applies BPSK in addition to the index modulation. The results illustrate the outperformance of DSSK where a gain of 9 dB and 5 dB is found over CSM1 and CSM2, respectively. Figure 5 illustrates a comparison of the channel capacity achievable by the DSSK scheme with four distributed relays N r . The results clearly present a gain of more than 2 bit/sec/ Hz in the channel capacity using the DSSK scheme over the CSM1 where SM is applied at the source node. However, when the SM is moved to the relay, less progress is recorded by DSSK where the achieved gain is more than 1 bit/sec/Hz.

Conclusion
In this study, DSSK has been proposed and studied as an energy-and spectral-efficient cooperative framework. The computational complexity and cost of the DSSK are lower than those of other types of distributed spatial modulation techniques because, at most, one cooperating relay is active at each signaling instant; therefore, no IRS is required. The error performance of the system has been investigated using MLD and MSD detection algorithms with different costs and complexities. It has been inferred that the performance of MSD closely approximates that of MLD as the size of the DSSK system increases with more relays involved, although MSD is a blind detection with lower complexity. In addition, the lower bound of the channel capacity has been derived and studied. The results show that the number of cooperating relays influences the attainable channel capacity.
Further, the comparison study with the CSM relay-aided system clearly illustrates the outperformance of the DSSK system in terms of the error performance and channel capacity, especially over the system where SM is applied at the source.

Data Availability
The simulation results of this study has been run by the software program (MATLAB 2016a).