A Low Complexity HD Detector for Dual Polarized ... - IEEE Xplore

2017 IEEE/CIC International Conference on Communications in China (ICCC)

A Low Complexity HD Detector for Dual Polarized Spatial Modulation ∗ School

Jiliang Zhang∗ , Lei Yuan∗ , and Yang Wang† of Information Science and Technology, Lanzhou University, Lanzhou, China, 730000, Email: [email protected] † Harbin Institute of Technology (Shenzhen), Shenzhen, P. R. China. Email: [email protected]

Abstract—Dual Polarized Spatial Modulation (DP-SM) systems are recently developed to pursue high spectral efficiency while keeping a low cost and complexity transceiver design. In this paper, we propose a Hard Decision (HD) detector for the DP-SM system which has a lower computational complexity compared to the traditional optimal Maximum Likelihood (ML) detector. Considering multiplication and addition of a complex number as an operation, the computational complexity of the HD receiver and ML receiver are characterized and compared with that of the SM receiver. Moreover, the analytical Symbol Error Probability (SEP) of the HD receiver is given in closed-form and compared to that of the ML detector under independent identically distributed (i.i.d.) Rayleigh channels. Numerical results show that SEPs of the HD receiver and the ML receiver are close from each other when the signal constellation size is larger than 8, even through the HD receiver has a lower complexity.

I. I NTRODUCTION As a state of the art multi-antennas system, the concept of single RF Multiple-Input-Multiple-Output (MIMO) system, covering Spatial Modulation (SM) [1]–[6] and Space Shift Keying (SSK) [7]–[13], has been proposed to reduce the complexity and cost of multiple-antenna schemes. Recently, Single-RF MIMO in different channel conditions and system configurations is analyzed theoretically [14]–[17]. Single-RF MIMO systems are also evaluated under realistic measured channel transfer functions [18], [19]. Our previous works on the topic of Single-RF MIMO has been published in [20]–[23]. In multi-antenna systems, multi-polarized antennas can exploit polarization domain resources to achieve channel multiplexing/diversity gain and reduce the required antenna spacing [24]–[28]. Zain ul Abidin, et al. exploited circular polarization and developed a modulation system switching left-hand and right-hand circular polarizations to carry 1bit information per channel use [29]. However, circular polarization requires two activated transmit antennas at a same time, which loses advantages of single RF system. Exploiting polarization domain resources into single RF wireless system, G. Zafari et al. proposed a Dual Polarized SM (DP-SM) modulation scheme using the polarization as an information-bearing unit to increase the overall spectral efficiency [30]. It is found that DP-SM system performs better than the traditional SM system. In [30], the optimal Maximum Likelihood (ML) detector with Correspondence to: Lei Yuan, 423 Lab, Feiyun Building, Lanzhou University, No.222 South Tianshui Road, Chengguan District, Lanzhou 730000, Gansu Province, P.R.China

978-1-5386-4502-4/17/$31.00 ©2017 IEEE

high computational complexity, is employed to detect the polarized SM signal. In [31], the performance of the DP-SM system is evaluated over correlated fading channels. Cross Polarization Interference Cancellation (XPIC) technology consists of employing the same carrier frequency for the simultaneous transmission and reception of two independent streams of data, so that the links capacity is doubled without using new spectrum [33]. Recently, attention has been paid to XPIC investigation [34], [35]. In this paper, a low complexity Hard Desision (HD) detector is proposed and investigated inspired by recent progress of XPIC. Achieving the same rate using the same signal constellation, the complexity of the HD DP-SM detector is lower than that of ML DP-SM detector. Analytical and approximate Symbol Error Probabilities (SEPs) of DP-SM using the HD and the ML receivers are derived. Through analytical analysis and numerical simulation, we found that if the order of the signal constellation is larger than 8, then the proposed HD detector can achieve a SEP that nearly equal to optimal ML detector. The organization of this paper is as follows. Section II introduces the system scheme, including the mapping and detection process. Section III gives analytical results over Rayleigh fading channels. Section IV provides numerical results. Finally, we conclude the paper in Section IV. In this paper, ρ denotes the average signal-to-noise Es ratio N , M denotes the size of signal constellation, 0 NT and NR are respectively the number of transmit and receiver antennas, m, nT and p denote the index of transmitted signal, the transmitting antenna and the polarization state, ˆ◦ is the estimation of variable at the receiver end, ◦V and ◦H denote vertical and horizontal polarization states respectively, CN (μ, σ 2 ) is a complex Gaussian distributed random variable with an expectation of μ and a variance of σ 2 , χ2υ is a Chi-squared distributed random variable with υ degrees of freedom, Q(x) = ∞ u2 π x2 √1 e 2 du = π1 02 e− 2 sin2 θ dθ, and R(κ, NR ) 2π x NR q NR −1 q 1 κ 1 κ ×C . 1 − 1 + NR −1+q 2 q=0 2 κ+1 κ+1 II. T RANSMISSION SCHEME A. Polarized Channel with XPIC Assuming NT dual-polarized antennas are equipped at the transmitter (Tx) and NR dual-polarized antennas are equipped at the receiver (Rx). Considering a polarized wireless channel


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Fig. 1. An example of a 2 × 2 DP-SM system scheme with the BPSK constellation. The frequency efficiency is 3bpcu.

between the nT th Tx antenna and the nR th Rx antenna, ynR =

T hnR nT xnT + wnR where xnT = xnT ,V xnT ,H denote transmitted symbol through vertically polarized Tx antenna and horizontally-polarized Tx antenna respectively. ynR =

T ynR ,V ynR ,H denote the received signals through vertically polarized Rx antenna and Rx horizontally-polarized

T wnR ,V wnR ,H antenna respectively. wnR = are received additive white complex Gaussian noises with zero-mean and variance ofσn2 . In this paper, perfect XPIC is assumed 0 hnR nT ,V , where hnR nT ,V ∼ and hnR nT = 0 hnR nT ,H CN (0, 1) and where hnR nT ,H ∼ CN (0, 1). B. Transmitter As shown in [30], at the transmitter, the bit stream is divided into three data streams. The bits in the first stream are used to select the transmit antenna that is switched on for transmission, whereas all the other antennas are kept silent. Bits in the second stream are mapped onto the index of the polarization pattern selected for transmission (i.e., use vertically polarized antenna to transmit signals when the bit is 0 and use horizontally polarized antenna to transmit signals when the bit is 1). The bits in the third stream are used to choose a symbol in the signal constellation diagram. In the DP-SM system, only one polarization pattern of one antenna is activated at the transmitter. For example, in a 3 bit per channel use (bpcu) transmission with 2 polarized antennas, information bits are mapped onto a BPSK signal, and transmitted on one of the two available polarization patterns, thus the data rate is 3bpcu. The scheme of the 3bpcu DP-SM system is shown in Fig. 1. Under this transmission scheme, some special cases of DP-SM is given as follows. (1) Polarized SSK: When M = 1, the DP-SM system is named Polarized SSK (PSSK) modulation, which is a low complexity and low data rate version of DP-SM where only the polarization and spatial constellation diagrams are exploited for data modulation. From another perspective, PSSK modulation is a system that exploits polarization domain resources in SSK modulation systems. (2) Polarization Shift Keying: When M = 1 and NT = 1, the PSSK system is named Polarization Shift Keying (PolarSK) which use only the polarization constellation diagram

to transmit bits. The HD receiver of PolarSK is extremely simple and can be constructed by only two energy detectors and comparators. The PolarSK system was generalized in [36]. (3) Polarization Modulation: When NT = 1, the DPSM is named Polarization Modulation (PM), where only the polarization and signal constellation diagrams are exploited for data modulation. In PM system, the transmission can get one bit multiplexing gain when the transmitter equipped only one antenna. C. Hard Decision Receiver Traditionally, DP-SM systems are ML detected as [32] [ˆ p, nˆT , m] ˆ = arg min y − hpnT xm p,nT ,m

(1)

The idea of the HD receiver is based on the fact that if the signal is transmitted by a p polarized transmit antenna, then the received power of p polarized receiver antennas will probably be larger than that of 2 − p polarized receiver antennas. There are two estimation processes in the HD DPSM receiver: 1) The transmit polarization index is estimated by a simple comparison of the received power of vertically polarized receive antennas and horizontally polarized receive antennas, and 2) the transmitted symbol and the antenna index are estimated using SM estimators. It is needed to emphasize here that there are two methods for the second step named optimal ML estimation [32, eq. (4)] and Maximum Ratio Combining (MRC) estimation [37, eq. (3)] respectively. Since our topic is to investigate the role of polarization in the single RF system, we selected the ML SM receiver. Its application to the DP-SM system using MRC SM receiver is not trivial. Two estimation processes are described in (2) and (3) respectively.

pˆ =

1 ||yV ||2 > ||yH ||2 , 2 ||yV ||2 ≤ ||yH ||2 .

[m, ˆ n ˆ T ] = arg min {ypˆ − hpn ˆ T xm } . m,nT

(2)

(3)

III. P ERFORMANCE ANALYSIS A. Computational complexity In this section, we analyze computational complexity and SEP of the HD receiver. Considering only multiplication and addition of a complex number as an operation, the computational complexity of (2) is 4NR − 1. According to [32], the computational complexity of (3) is 2NR NT + NT M + M . Therefore the total computational complexity for the HD DPSM receiver is 4NR + 2NR NT + NT M + M − 1. Using MRC SM receiver, the total computational complexity for HD DP-SM receiver is 4NR + 2NR NT + NT M + f (M ) − 1, where f (M ) depends on the type of demodulation used.


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Fig. 2. Receiver complexity comparison of 4-8bpcu transmissions that uses SM and DP-SM modulation. It is found that the complexity of the HD DP-SM receiver under a same bit rate is lower than that of the ML DP-SM receiver and the SM receiver. Since in subsection V, we will prove that when the signal constellation size is larger than 8, the SEP performance of the HD DP-SM receiver is nearly equal to that of the ML DP-SM receiver, receiver complexity of DP-SM can be thought of lower than that of SM.

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Fig. 3. SEP of 2 × 2 and 4 × 4 HD DP-SM receiver with PSSK, BPSK, QPSK, 8PSK and 16PSK signal constellations under isolated polarized i.i.d. Rayleigh channel. The frequency efficient of 2 × 2 DP-SM is 2bpcu, 3bpcu, 4bpcu, 5bpcu and 6bpcu respectively and the frequency efficient of 4 × 4 DP-SM is 3bpcu, 4bpcu 5bpcu, 6bpcu, 7bpcu respectively. 0

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−1

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4NT4NR Simulation 8QAM PSM 16QAM PSM 32QAM PSM

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Following [32], the computational complexity of the ML DP-SM receiver is computed by

−3

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8NR NT + 2NT M + M.

−4

Achieving the same rate using the same MIMO setup, complexities of DP-SM and SM receivers are shown in Fig. 2. It is found that the complexity of the HD DP-SM receiver under the same bit rate is lower than that of the SM receiver. B. Symbol Error Probability In this subsection, analytical closed-form SEPs of HD DP-SM systems are given. The reasons for analytical SEP derivation is [14]: 1) It allows a deeper understanding of the system performance, 2) It enables a simpler comparison with other competing transmission technologies ,and 3) It provides opportunities for system optimization. The derivation of the analytical SEP has two folds: 1) The estimation of the polarization pattern, 2) The estimation of the transmit antenna number and the transmitted symbol. These two processes are assumed to be independent in the calculation. The bits are correctly recovered only if all estimates are correct. To compute the overall SEP , let SEPPolar denote the probability that the polarization pattern is wrongly detected and let SEPSM denote the probability that the transmit antenna index and/or transmitted symbol is wrongly detected. Then, the retrieved DP-SM bits are correct if and only if both the polarization pattern and the SM symbol are correctly estimated, thus we have SEP = 1 − (1 − SEPPolar )(1 − SEPSM ).

(4)

Firstly, closed-form SEPPolar , which denotes the symbol error caused by wrong transmitted polarization pattern estimation, is calculated as SEPPolar =

M 1 P r(p = pˆ|m), M m=1

(5)

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Fig. 4. SEP of 2 × 2 and 4 × 4 HD DP-SM receiver with 8QAM, 16QAM and 32QAM signal constellations under isolated polarized i.i.d. Rayleigh channel. The frequency efficient of 2 × 2 DP-SM is 5bpcu, 6bpcu and 7bpcu respectively and the frequency efficient of 4 × 4 DP-SM is 7bpcu, 8bpcu and 9bpcu respectively.

and P r(p = pˆ|m) denotes the probability that polarization is wrongly detected when xm is transmitted at the transmitter. Closed-form P r(p = pˆ|m) is represented as (6), which is proved in Appendix A. P r(ˆ p = p|m) =

N R −1 q=0

NR −1 CN (ρ|xm |2 + 1)q R +q−1

(ρ|xm |2 + 2)NR +q

.

(6)

Plugging (6) into (5), we obtain final version of SEPPolar : N −1 M NR −1 CNRR+q−1 (ρ|xm |2 + 1)q 1 . (7) SEPPolar = M m=1 q=0 (ρ|xm |2 + 2)NR +q If a constant-modulus signal constellation, then we have |xm | ≡ 1 (e.g. PSK DP-SM, PSK PM or PSSK) is employed, SEPPolar =

NR −1 (ρ + 1)q CN R +q−1

(ρ + 2)NR +q

.

(8)

Secondly, closed-form SEPSM , which denotes the symbol error caused by wrong transmitted signal point and/or antenna index estimation, is calculated using the improved upper bound introduced in [14], SEPSM = SEPSignal + SEPSpatial + SEPJoint

(9)


⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎧

2 3π 2 π M > 4, ⎪ ⎨ 2 R ρ sin M , NR + R ρ sin M , NR ∼ SEPSignal,PSK = 2R (ρ/2, NR ) M = 4, ⎪ ⎩ R (ρ, NR ) M = 2, 2 , SEPSignal,QAMS = 2Pk;QAMS − Pk;QAM S SEPSignal,QAMR = Pk;QAMR + Pl;QAMR − Pk;QAMR Pl;QAMR , M 2 R ρ|x2m | , NR , SEPSpatial ∼ = NTM−1 m=1 M M 2 2 NT −1 ∼ SEPJoint = M R ρ(|xm | 4+|xmˆ | ) , NR , ˆ =m=1 ⎧ m=1 m 27ρ ∼ ⎪ √ Pk;QAMS = 2 1 − 1M R 2M3ρ−2 , NR + R 2M , NR , ⎪ −2 ⎪ ⎪ ⎪ ⎪ 3ρ 27ρ 1 ⎪ R 5M/2−2 , NR + R 5M/2−2 , NR , = 2 1 − √2M ⎨ Pk;QAMR ∼ ⎧ ⎪ 3ρ 27ρ ⎪ ⎨ 2 1− √1 R 5M/2−2 , NR + R 5M/2−2 , NR ⎪ ⎪ M/2 ⎪ Pl;QAMR ∼ ⎪ = ⎪ ⎪ ⎪ 3ρ ⎩ ⎩ R 5M/2−2 , NR

(10)

M = 8, M = 8.

⎧ SEPML ≤ SEPPolar,ML + SEPSM , ⎪ ⎪ ⎪ M ⎪ 1 2 ⎪ ⎨ SEPPolar,ML = M p = p; m, maxxmˆ |xm ˆ | ), m=1 P r(ˆ ⎧ ρ maxxm |x ˆ |2 } ⎨ ρ|xm |2 ˆ { m ⎪ ⎪ QAM, ≤ πR , NR R , NR ⎪ P r(ˆ 8 8 ⎪ p = p, m) ⎪ ρ ⎩ ⎩ PSK/PSSK. = R 8 , 2NR

10 10 10

SEP

10 10 10

technologies, in this section, we provide a tight high-SNR approximation and analyse both the diversity order and the coding gain of DP-SM modulation. For arbitrary κ > 0, following [14, Proposition 4], we have NR 1 NR lim R(κ, NR ) = C2N , (12) R −1 κ→+∞ 4κ

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and the approximate PEP is computed by (13), from which we observe that the diversity gain OSM is

4NT4NR Approximation PSSK QPSK PSM 16QAM PSM 16PSK PSM 5

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OSM = OSignal = OSpatial = OJoint = NR . Fig. 5. Analytical SEPs and approximate SEPs of HD DP-SM receiver with PSSK, QPSK, 16QAM and 16PSK signal constellations under isolated polarized i.i.d. Rayleigh channel. The frequency efficient of 2 × 2 DP-SM is 2bpcu, 4bpcu, 6bpcu and 6bpcu respectively and the frequency efficient of 4 × 4 DP-SM is 3bpcu, 5bpcu, 5bpcu, 7bpcu and 7bpcu respectively.

where SEPSignal , SEPSpatial and SEPJoint are respectively computed by [14, eq.(16)], [38, eq. (18)] and [39, eqs. (12),(19)] as (10). The SEP of HD DP-SM receiver is obtained by combining (4), (7), (9) and (10). The SEP of ML DP-SM receiver SEPML is obtained following the idea of [14] as (11). Due to the page limit, the detail process of derivation is omitted in this paper. C. Diversity To enable a deeper understanding of the achievable SEP performance and a simpler comparison with other transmission

(14)

According to (7), we have OPolar = NR ,

(15)

and therefore the diversity order of the DP-SM system with a HD detector is NR . It is shown in Fig. 5 that the approximations calculated by the diversity order asymptotically converge to the analytical SEP in the high SNR regime. IV. S IMULATION R ESULTS In Fig. 4, we study the tightness of the obtained SEP of the HD DP-SM receiver in (4) against Monte Carlo simulations. It is observed that analytical SEPs are very tight and agree well with the simulation results. Since the Monte Carlo simulations are not accurate and time consuming, we will analyse the DP-SM performance using analytic SEP upper bounds in the following section only.


⎧ NR π −NR ⎪ C2N ρ−NR M ≥ 4, 2 4 sin2 M ρ1 ⎪ R −1 ⎪ SEP = ⎪ Signal,PSK ⎪ N ⎪ M = 2, 4−NR C2NRR −1 ρ−NR ⎪ ⎪ ⎪ M ⎪ −NR NR ρ1 ⎪ N −1 −N 2 ⎪ SEPSpatial = TM C2NR −1 ρ R , 2|xm | ⎪ ⎪ ⎪ m=1 ⎪ ⎪ ⎨ M M ρ1 NR 2 −NR SEPJoint = NTM−1 C2N ρ−NR , |xm |2 + |xm ˆ| R −1 ⎪ m=1 m ˆ =m=1 ⎪ ⎪ M −1 −NR NR ρ1 ⎪ ⎪ SEPSignal,QAMS = 4 1 − √1M C2NR −1 ρ−NR , ⎪ ⎪ 6 ⎪ ⎪ ⎪ 5M −4 −NR NR ρ1 ⎪ 1 ⎪ ⎪ SEPSignal,QAMR = 2 2 − √2M − √1 C2NR −1 ρ−NR , ⎪ 24 M/2 ⎪ ⎪ ⎪ ⎩ ρ1 NR SEPPolar = C2N ρ−NR . R −1

V. C ONCLUSIONS AND F UTURE W ORKS

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Fig. 6. Comparison of SEPs using HD receiver and ML receiver using PSK signal constellation and 4 × 4 polarization antenna arrays.

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In this paper, we proposed an HD DP-SM receiver under the assumption of perfect XPIC. It is shown that the complexities of the proposed HD DP-SM receiver is nearly a half of the traditional ML receiver that is optimal in term of SEP. We investigated the HD DP-SM receiver and derived analytical SEP performances in a closed-form under i.i.d. Rayleigh polarization channels. The derived analytical SEP and the simulation result were shown to closely agree with each other in the high SNR regime. Based on the analytical SEP, an approximate SEP is given and employed to analyse the diversity order of the HD DP-SM modulation. Finally, SEP performances of the proposed HD receiver and the ML receiver under typical system configurations are compared. Numerical results show that the SEP performance of the HD receiver is nearly equal to that of the optimal ML receiver when the signal constellation size is larger than 8. It indicates that the HD receiver achieves a near optimal SEP performance with a low receiver complexity. In this paper, a perfect XPIC is assumed. If the XPIC is not perfect, it is predictable that the gap between HD and ML is higher. In the future, analytical results of HD DP-SM receivers taking imperfect XPIC have to be proposed. ACKNOWLEDGEMENT

10-6 -7

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(13)

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Fig. 7. Comparison of SEPs using HD receiver and ML receiver using QAM signal constellation and 4 × 4 polarization antenna arrays.

In Figs. 6-7, we study the SEP performance of the HD receiver as compared with the optimal ML receiver for PSK and QAM PSM under a 4 × 4 MIMO setup. Interestingly, when the signal constellation size is larger than 8, the SEP performance of the HD receiver is nearly equal to that of the optimal ML receiver. It means that the proposed HD receiver can achieve an optimal SEP performance with a low receiver complexity when the signal constellation size is larger than 8.

The research is funded in part by National Natural Science Foundation of China (61501137, 61371101), in part by the European Union’s Horizon 2020 research and innovation programme-is3DMIMO (734798), and in part by the Fundamental Research Funds for the Central Universities (lzujbky2017-38, lzujbky-2017-188). A PPENDIX A: P ROBABILITY OF pˆ = p U SING C OMPARISON R ECEIVER The analytical closed form probability of polarization estimation error under an SNR of ρ is derived in this Appendix. If p is wrongly estimated, according to the procedure of comparison DP-SM receiver, √ P r(ˆ p = p, m) = P r(|| ρhnT p xm + np ||2 ≤ ||n2−p ||2 ). (16)


In a i.i.d. Rayleigh fading channel, hp ∼ CN NR (0, INR ), np ∼ CN NR (0, INR ) and n2−p ∼ CN NR (0, INR ). Therefore, 1 √ || ρhnT p xm + np ||2 ∼ (ρ|xm |2 + 1)χ22NR , 2 ||n2−p ||2 ∼ and

1 2 , χ 2 2NR

(17)

(18)

1 2 1 2 2 P r(ˆ p = p; m) = P r (ρ|xm | + 1)χ2NR ≤ χ2NR . (19) 2 2

Plugging PDF of chi-square distribution

N −1 − x x R e 2 x≤0 2NR Γ(NR ) Pχ22N (x) = R 0 otherwise

(20)

into −NR

P r(ˆ p = p; m) = 2NRA 2 R )] (21) +∞ +∞ N 2−1 N[Γ(N x+Ay −1 − × 0 x R y R e 2A dxdy, y ∞ where A = ρ|xm |2 + 1, and using 0 tn e−at ≡ n!a−n−1 , we obtain P r(ˆ p = p, m) as (6). R EFERENCES [1] M. Di Renzo et al., “Spatial modulation for multiple-antenna wireless systems: a survey,” IEEE Commun. Magazine, vol. 49 , no.12, pp, 182191, 2011. [2] P. Yang, et al., “Design guidelines for spatial modulation,” IEEE Commun. Surveys Tuts., vol. 17, no. 1, pp. 6-26, 2015. [3] P. Yang, et al., “Single-carrier SM-MIMO: a promising design for broadband large-scale antenna systems,” IEEE Commun. Surveys Tuts., to appear [4] M. Di Renzo, et al., “Spatial modulation for generalized MIMO: challenges, opportunities, and implementation,” IEEE Proc., vol. 102, no. 1, pp. 56-103, 2014. [5] S. Guo, et al., “Link adaptive mapper designs for space shift keying modulated MIMO systems,” IEEE Trans. Veh. Technol., vol. 65, no. 10, pp. 8087-8100, 2016. [6] S. Guo, et al., “Spatial modulation via 3-D mapping,” IEEE Wireless Commun. Lett., vol. 20, no. 6, pp. 1096-1099, 2016. [7] J. Jeganathan et al., “Space shift keying modulation for MIMO channels,” IEEE Trans. Wirel. Commun., vol. 8, no. 7, pp. 3692-3703, 2009. [8] J. Jeganathan et al., “Generalized Space Shift Keying Modulation for MIMO Channels,” IEEE PIMRC, pp. 1-5, 2008. [9] M. D. Renzo and H. Haas, “Improving the performance of space shift keying (SSK) modulation via opportunistic power allocation” IEEE Commun. Lett., vol. 14, no. 6, pp. 500-502, 2010. [10] M. Di Renzo et al., “Space shift keying (SSK-) MIMO with practical channel estimates,” IEEE Trans. Commun., vol. 60, no. 4, pp. 998-1012, 2012. [11] M. Di Renzo and H. Haas, “Space shift keying (SSK) modulation with partial channel state information: Optimal detector and performance analysis over fading channels,” IEEE Trans. Commun. , vol. 58, no. 11, pp. 3196-3210, 2010. [12] M. Di Renzo and H. Haas, “A general framework for performance analysis of space shift keying (SSK) modulation for MISO correlated Nakagami-m fading channels,” IEEE Trans. Commun., vol. 58, no. 9, pp. 2590-2603, 2010. [13] S. S. Ikki and R. Mesleh, “A general framework for performance analysis of space shift keying (SSK) modulation in the presence of Gaussian imperfect estimations,” IEEE Commun. Lett., vol. 16, no. 2, pp. 228230, 2012. [14] M. Di Renzo and H. Haas, “Bit error probability of SM-MIMO over generalized fading channels.” IEEE Trans. Vehicular Tech., vol. 61, no. 3, pp. 1124-1144, 2012.

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