2012 IEEE 23rd International Symposium on Personal, Indoor and Mobile Radio Communications - (PIMRC)

Spatial and Space Shift Keying in Single Carrier Communication Pritam Som and A. Chockalingam Department of ECE, Indian Institute of Science, Bangalore 560012, India

Abstract—Spatial modulation (SM) and space shift keying (SSK) linear convolution of channel with data to circular convolu- are relatively new modulation techniques which are attractive in tion, which is equivalent to multiplication in frequency do- multi-antenna communications. Single carrier (SC) systems can main. This enables low complexity frequency domain pro- avoid the peak-to-average power ratio (PAPR) problem encoun- tered in multicarrier systems. In this paper, we study SM and cessing at the receiver. Comparison between OFDM and sin- SSK signaling in cyclic-prefixed SC (CPSC) systems on MIMO- gle carrier systems has been done extensively in the literature ISI channels. We present a diversity analysis of MIMO-CPSC both for single antenna scenario [14]-[19], as well as MIMO systems under SSK and SM signaling. Our analysis shows that scenario [20], [21]. Wireless standards like 3GPP LTE has (𝑛 ,𝑛 ) (𝑛 ,𝑛 , the diversity order achieved by 𝑡 𝑟 SSK scheme and 𝑡 𝑟 adopted single carrier system in the uplink [22]. Θ𝑀 ) SM scheme in MIMO-CPSC systems under maximum- likelihood (ML) detection is 𝑛𝑟,where𝑛𝑡,𝑛𝑟 denote the number In the context of SM and SSK in frequency selective chan- of transmit and receive antennas and Θ𝑀 denotes the modula- nels, the focus so far has been on the MIMO-OFDM [5],[23]- 𝑀 tion alphabet of size . Bit error rate (BER) simulation results [24]. [5] presented a matched filter (MF) based detection al- validate this predicted diversity order. Simulation results also show that MIMO-CPSC with SM and SSK achieves much bet- gorithm for SM on each subchannel. [6] presented the opti- ter performance than MIMO-OFDM with SM and SSK. mal detection scheme for SM in flat fading MIMO channels. Keywords: Spatial modulation, space shift keying, cyclic-prefixed single This scheme can be easily extended to MIMO-OFDM sce- carrier systems, MIMO-CPSC, diversity order. nario on individual subcarriers. However, to the best of our knowledge, neither SM nor SSK has been studied for single I. INTRODUCTION carrier systems. The use of multiple antennas at the transmitter and receiver In this paper, we study SM and SSK signaling in cyclic-prefix sides can significantly enhance the capacity and reliability of single carrier (CPSC) systems on MIMO-ISI channels. We wireless links [1]. However, multi-antenna operation faces present a diversity analysis of the MIMO-CPSC system under significant challenges due to complexity and cost of the hard- SSK and SM signaling. Our analysis shows that the diversity (𝑛 ,𝑛 ) (𝑛 ,𝑛 , Θ ) ware owing to the requirement of inter-antenna synchroniza- order achieved by 𝑡 𝑟 SSK scheme and 𝑡 𝑟 𝑀 tion and maintenance of multiple radio-frequency (RF) chains SM scheme in MIMO-CPSC systems under maximum-likeli- 𝑛 𝑛 ,𝑛 [2]. Spatial modulation (SM) is a relatively new modulation hood (ML) detection is 𝑟, where 𝑡 𝑟 denote the number Θ technique for multiple antenna systems which addresses these of transmit and receive antennas and 𝑀 denotes the mod- 𝑀 issues [3]. This modulation technique was first proposed in ulation alphabet of size . Bit error rate (BER) simulation [4], and later improved further in [5]-[7]. Space shift keying results for SSK and SM are presented to validate this pre- (SSK) is another signaling technique which can be thought of dicted diversity order. In addition, we present a comparison as the special case of SM [8]-[11]. between MIMO-CPSC and MIMO-OFDM performance un- der SSK and SM signaling. Simulation results show that the High data rate communication system designs necessitate lar- optimal BER performance of SM and SSK in MIMO CPSC ger bandwidths, which, in turn, result in frequency selectiv- are significantly better compared to those in MIMO-OFDM. ity of the wireless channel. In frequency selective channels, the presence of multipath components cause inter-symbol in- II. SYSTEM MODEL terference (ISI). To mitigate ISI, use of multi-carrier tech- Consider a MIMO channel with 𝑛𝑡 transmit and 𝑛𝑟 receive niques such as orthogonal frequency division antennas. The channel between each pair of transmit and re- (OFDM) is popular due to simple equalization/receiver com- ceive antennas is assumed to be frequency selective with 𝐿 plexity. However, high peak to average power ratio (PAPR) ℎ(𝑙) at the OFDM transmitter caused by the IFFT operation makes multipath components. Let 𝑗,𝑖 denote the channel gain from 𝑖th transmit antenna to the 𝑗th receive antenna on the 𝑙th mul- the design of amplifier a challenging task. Single carrier com- 𝒞𝒩(0, Ω ) munication [12], [13] in frequency selective channels is of tipath component (MPC), which is modeled as 𝑙 . interest because it does not suffer from PAPR limitations. The following power-delay profile of the channel is consid- ered: In single carrier (SC) systems, each data block is transmit- 2 (𝑙) −𝛽𝑙 ted along with either zero padding (ZP) or cyclic prefix (CP), Ω𝑙 = 𝔼[∣ℎ𝑗,𝑖∣ ]=Ω010 ,𝑙=0, ⋅⋅⋅ ,𝐿− 1, (1) which avoids inter-datablock interference. CP converts the where 𝛽 denotes the rate of decay of the average power in * This work was supported in part by a gift from the Cisco University the MPCs in dB. The power in the channel is assumed to be Research Program, a corporate advised fund of Silicon Valley Community ∑𝐿−1 Foundation. normalized to unity, i.e., 𝑙=0 Ω𝑙 =1.

978-1-4673-2569-1/12/$31.00 ©2012 IEEE 1962 Consider a CPSC scheme, where transmission is carried out The index of each transmit antenna represents a certain com- in frames. Each frame consists of a cyclic prefix (CP) part bination of information bits. For example, with 𝑛𝑡 =4, followed by a data part. The length of the CP part must be at transmit antennas with indices 1, 2, 3, 4 can be mapped to bit least 𝐿 − 1 channel uses in order to avoid inter-frame inter- sequences 00, 01, 10, 11, respectively. Therefore, at the re- ference. Let 𝐾 denote the length of the data part in number ceiver, detection of a certain transmit antenna index leads to of channel uses. The total frame length is 𝐾 + 𝐿 − 1 chan- conveying the group of information bits associated with that nel uses. In each channel use in the data part, a data vector index. The spectral efficiency in SSK grows logarithmically 𝑛 x is transmitted using 𝑡 transmit antennas. Let 𝑞 denote the with number of transmit antennas, i.e., log2 𝑛𝑡 bits per chan- transmitted symbol vector at time 𝑞, 0 ≤ 𝑞 ≤ 𝐾 − 1. The nel use (bpcu). A consequence of this type of spatial mapping entries of the x𝑞 vector depends on the type of modulation is that number of transmit antennas has to be a power of 2. used. Here, we consider spatial modulation (SM) and space Let us take the known signal transmitted by the active antenna shift keying (SSK), which are described in the following sub- to be 1. Then, for a SSK system with two transmit antennas, sections. We assume that the channel gains remain constant 𝑇 𝑇 the possible signal set is given by 𝕊2 ≡{[1, 0] , [0, 1] }.A over one frame duration. The received signal vector at time 𝑞 zero in the signal vectors above is silence in the correspond- can be written as ing transmit antenna. In general, a SSK signal set for 𝑛𝑡 trans- 𝐿−1 ∑ mit antenna system is given by y𝑞 = H𝑙 x𝑞−𝑙 + n𝑞,𝑞=0, ⋅⋅⋅ ,𝐾− 1, (2) 𝑙=0 𝕊 ≡{s : 𝑗 =1, ⋅⋅⋅ ,𝑛 }, 𝑛𝑡 𝑗 𝑡 (5) y ∈ ℂ𝑛𝑟 ×1 H ∈ ℂ𝑛𝑟 ×𝑛𝑡 𝑇 where 𝑞 , 𝑙 is the channel gain matrix s.t. s𝑗 =[0, ⋅⋅⋅ , 0,  1 , 0, ⋅⋅⋅ , 0] . 𝑙 𝑗 𝑖 for the th MPC such that its on th row and th column of is 𝑗 (𝑙) th coordinate ℎ𝑗,𝑖. After removing the CP at the receiver, the vector channel x ∈ 𝕊 ∀𝑞, 𝑞 = equation can be written in an equivalent form as In the CPSC system under consideration, 𝑞 𝑛𝑡 , 0, ⋅⋅⋅ ,𝐾− 1, i.e., x𝑞 vector in (2) is drawn from 𝕊𝑛 in (5). y = Hx+ n, (3) 𝑡 where H is the 𝐾𝑛𝑟 × 𝐾𝑛𝑡 equivalent channel matrix given B. Spatial Modulation (SM) x at the bottom of this page, is the combined vector of data Spatial modulation is a generalization of SSK signal design. symbols transmitted in one data frame, given by In SM, the stream of bits to be transmitted in one channel [ ] 𝑇 𝑇 𝑇 𝑇 use are divided into two groups. One group determines the x = x0 x1 ⋅⋅⋅ x𝐾−1 . transmit antenna index on which transmission will take place, Likewise, the vectors y and n of size 𝐾𝑛𝑟 ×1 are constructed and the second group determines the symbol to be transmit- as ted from the chosen antenna. The symbol transmitted on the [ ] [ ] 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 𝑇 chosen antenna can be from a regular modulation alphabet y = y0 y1 ⋅⋅⋅ y𝐾−1 , n = n0 n1 ⋅⋅⋅ n𝐾−1 , like 𝑀-QAM, denoted by Θ𝑀 ≡{𝜃𝑚 : 𝑚 =1, ⋅⋅⋅ ,𝑀}. 2 where each entry in the additive noise vector n is 𝒞𝒩(0,𝜎 ), Therefore, the number of bits that belong to the first and sec- 𝛾 = 𝐸𝑠 𝐸 so that the received SNR is given by 𝜎2 , where 𝑠 is the ond group, respectively, are log2 𝑛𝑡 and log2 𝑀. This gives average symbol energy. a spectral efficiency of (log2 𝑛𝑡 +log2 𝑀) bpcu. The SM 𝑛 𝕋 A. Space Shift Keying (SSK) signal set for 𝑡 antenna system, 𝑛𝑡,𝑀 , is given by In SSK, a group of information bits are used to choose one 𝕋 ≡{t : 𝑗 =1, ⋅⋅⋅ ,𝑛 ,𝑚=1, ⋅⋅⋅ ,𝑀}, 𝑛𝑡,𝑀 𝑗,𝑚 𝑡 (6) transmit antenna, i.e., an 𝑚-bit sequence chooses one antenna 𝑇 𝑚 s.t. t𝑗,𝑚 =[0, ⋅⋅⋅ , 0,𝜃𝑚 , 0, ⋅⋅⋅ , 0] ,𝜃𝑚 ∈ Θ𝑀 . from a total of 𝑛𝑡 =2 antennas. A known signal (which is  known to the receiver) is transmitted on this chosen antenna. 𝑗th coordinate The remaining 𝑛𝑡 −1 antennas remain silent. By doing so, the problem of detection at the receiver becomes one of merely In the considered CPSC scheme, x𝑞 vector is drawn from the 𝕋 finding out which antenna is transmitting. This leads to a SM signal set 𝑛𝑡,𝑀 in (6). significantly reduced complexity at the receiver.

⎡ ⎤ H0 000. 0H𝐿−1 H𝐿−2 H𝐿−3 . H2 H1 ⎢ H H 00. 00H H . H H ⎥ ⎢ 1 0 𝐿−1 𝐿−2 3 2 ⎥ ⎢ H2 H1 H0 0 . 00 0H𝐿−1 . H4 H3 ⎥ ⎢ ⎥ ⎢ . . . ⎥ ⎢ ...... ⎥ ⎢ . . . ⎥ ⎢ H𝐿−2 H𝐿−3 H𝐿−4 . H0 0 ..000H𝐿−1 ⎥ H = ⎢ ⎥ . (4) ⎢ H𝐿−1 H𝐿−2 H𝐿−3 . H1 H0 0 ..00 0⎥ ⎢ ⎥ ⎢ 0H𝐿−1 H𝐿−2 . H2 H1 H0 0 ..00⎥ ⎢ ⎥ ⎢ 00H𝐿−1 ..H2 H1 H0 0 .. 0 ⎥ ⎢ ⎥ ⎣ . . . ⎦ ...... 000...... H2 H1 H0

1963 C. Detection The pairwise error probability (PEP) of detecting vector x as Frequency selectivity in the channel causes inter-frame inter- x′, given the channel H, is given by ference (IFI) and inter-symbol interference (ISI). Though IFI √ 𝑃 (x → x′∣H)=𝑄( 𝜇∥H(x − x′)∥2), is avoided using CP in this system, the receiver processing (9) still has to deal with ISI within the frame. Joint detection of where 𝜇 is a scalar multiple of the average SNR. The block the entire frame of data optimally is of interest. In particu- circulant channel matrix H in (4) can be written in the form lar, the diversity order achieved by optimal detection in SSK H = D ⊗ I GD ⊗ I , and SM with CPSC signaling needs to be established. These 𝐾 𝑛𝑟 𝐾 𝑛𝑡 (10) are addressed in the next section. This, to our knowledge, where D𝐾 is the 𝐾-point DFT matrix, given by has not been reported before for SSK and SM with CPSC ⎡ ⎤ signaling in MIMO-ISI channels. Also, the performance of 𝜌0,0,𝜌0,1 ⋅⋅⋅ 𝜌0,𝐾−1 ⎢ ⎥ optimal detection is compared with the performance of low 1 ⎢ 𝜌1,0,𝜌1,1 ⋅⋅⋅ 𝜌1,𝐾−1 ⎥ complexity linear receivers like ZF and MMSE receivers. A D𝐾 = √ ⎢ ⎥ , (11) 𝐾⎣ . . . . ⎦ performance comparison between MIMO-CPSC and MIMO- . . . . 𝜌 ,𝜌 ⋅⋅⋅ 𝜌 OFDM is also carried out that illustrates the performance ad- 𝐾−1,0 𝐾−1,1 𝐾−1,𝐾−1 vantage in MIMO-CPSC. 2𝜋𝑢𝑣 where 𝜌𝑢,𝑣 = exp(−𝑗 𝐾 ). G is a block diagonal matrix, III. OPTIMAL DETECTION AND DIVERSITY ANALYSIS given by ⎡ ⎤ G0 ⎢ 0 ⎥ In this section, we present the optimum signal detection sche- ⎢ G1 ⎥ me for SSK and SM with CPSC signaling. We also present G = ⎢ . ⎥ , (12) ⎣ .. ⎦ an analysis of the diversity achieved by ML detection. 0 G𝐾−1 G A. Optimum Detection where 𝑞 is given by Assuming that all the signal vectors in the signal set are equally 𝐿∑−1 likely, the maximum likelihood (ML) decision rule for SSK G𝑞 = 𝜌𝑞,𝑙H𝑙 in CPSC system is given by 𝑙=0 ⎡ ⎤ [x𝑞,𝑀𝐿 : 𝑞 =1, ⋅⋅⋅ ,𝐾] H0 ⎢ ⎥ arg max [ ] ⎢ H1 ⎥ = 𝑃 (y∣H, x𝑞 : 𝑞 =0, ⋅⋅⋅ ,𝐾− 1) = 𝜌 I 𝜌 I ⋅⋅⋅ 𝜌 I ⎢ ⎥ x𝑞 ∈ 𝕊𝑛 : 𝑞 =0, ⋅⋅⋅ ,𝐾− 1 𝑞,0 𝑛𝑟 𝑞,1 𝑛𝑟 𝑞,𝐿−1 𝑛𝑟 . 𝑡 ⎣ . ⎦ arg min 2 = ∥y − Hx∥ . (7) H𝐿−1 x𝑞 ∈ 𝕊𝑛𝑡 : 𝑞 =0, ⋅⋅⋅ ,𝐾− 1   △ Likewise, the optimum decision rule for SM is given by √ = H = 𝐾D𝑞,𝐿 ⊗ I H, [x𝑞,𝑀𝐿 : 𝑞 =1, ⋅⋅⋅ ,𝐾] 𝐾 𝑛𝑟 (13) arg max = 𝑃 (y∣H, x𝑞 : 𝑞 =0, ⋅⋅⋅ ,𝐾− 1) D𝑞,𝐿 𝐿 𝑞 x𝑞 ∈ 𝕋𝑛𝑡,𝑀 : 𝑞 =0, ⋅⋅⋅ ,𝐾− 1 where 𝐾 denotes the first elements in the th row of D𝐾 . From (10), we can write = arg min ∥y − Hx∥2. x ∈ Γ : 𝑞 =0, ⋅⋅⋅ ,𝐾− 1 (8) 𝑞 𝑛𝑡,𝑀 H(x − x′)=D ⊗ I GD ⊗ I (x − x′) 𝐾 𝑛𝑟 ⎡ 𝐾 𝑛𝑡 ⎤ G0w0 ⎢ ⎥ B. Diversity Analysis = D ⊗ I ⎣ . ⎦, 𝐾 𝑛𝑟 . (14) A popular approach to determine the diversity order in a com- G𝐾−1w𝐾−1 munication system is to find the minimum value of the expo-   nent of SNR in the denominator of the pairwise error prob- =△ Gw ability expression. This approach comes from the consider- w =[w𝑇 w𝑇 ⋅⋅⋅ w𝑇 ]𝑇 w = D𝑞 ⊗ ation of union bound on the error performance, when an ex- where 0 1 𝐾−1 such that 𝑞 𝐾 I (x−x′) D𝑞 𝑞 D G w act expression for the error rate of the system becomes dif- 𝑛𝑡 and 𝐾 denotes the th row of 𝐾 .Now, 𝑞 𝑞 can be written as ficult to obtain. Establishing the diversity order of CPSC (√ ) signaling in single-input single-output (SISO) ISI channels G w = 𝑣𝑒𝑐(G w )=𝑣𝑒𝑐 𝐾(D𝑞,𝐿 ⊗ I )Hw 𝑞 𝑞 𝑞 𝑞 𝐾 𝑛𝑟 𝑞 has been the topic of investigation in [25]-[28], where it has √ ( ) been shown that, though in the asymptotic regime where SNR = 𝐾 w𝑇 ⊗ (D𝑞,𝐿 ⊗ I ) 𝑣𝑒𝑐(H), 𝑞 𝐾 𝑛𝑟   (15) →∞, CPSC does not extract diversity, in the moderate SNRs △ regime (corresponding to typical BERs used in practice), CPSC = h extracts some diversity [28]. Here, we carry out a diversity where h is the vector of all channel gains obtained by vector- analysis for CPSC on MIMO-ISI channels and obtain the di- ization of H. From (15), we can write Gw as versity orders for SSK and SM under ML detection. Gw = Vh, (16)

1964 where Considering the union bound of the probability of error, at ⎡ ( ) ⎤ very high SNRs, the PEP term corresponding to minimum 𝜌 w𝑇 ⊗ (D0,𝐿 ⊗ I ) ⎢ 0 𝐾 𝑛𝑟 ⎥ will dominate other constituent terms of the upper bound ex- √ ⎢ ⎥ V = 𝐾 ⎢ . ⎥ . pression [1]. Hence, the minimum value of 𝜌 obtained among . (17) ′ ⎣ ( ) ⎦ all possible choice of {x, x } pairs gives the diversity of the w𝑇 ⊗ (D𝐾−1,𝐿 ⊗ I ) 𝐾−1 𝐾 𝑛𝑟 system. We computed this minimum 𝜌, and, hence, the diver- sity order for SSK and SM in MIMO-CPSC systems under From (14) to (17), we have ML detection for different 𝑛𝑡,𝑛𝑟,𝐿,𝑀. The results are pre- ′ 2 2 sented in Tables I and II for SSK and SM, respectively. The ∥H(x − x )∥ = ∥(D𝐾 ⊗ I𝑛 )Vh∥ 𝑟 tables show one of the possible combinations of {x, x′} for = h𝐻 V𝐻 (D ⊗ I )𝐻 (D ⊗ I )V h. 𝐾 𝑛𝑟 𝐾 𝑛𝑟 (18) which the minimum number of non-zero eigenvalues of B △ B = B is obtained, and the corresponding set of eigenvalues of . From the results in Tables I and II, we conclude that the di- B in (18) is a 𝐿𝑛𝑟𝑛𝑡 ×𝐿𝑛𝑟𝑛𝑡 Hermitian matrix, whose eigen versity order achieved in a (𝑛𝑡,𝑛𝑟,𝐿) MIMO-CPSC system decomposition is under SSK and SM signaling is 𝑛𝑟. In the next section, we B = U𝐻 ΛU, (19) present BER simulation results which validate this predicted diversity order. Λ = 𝑑𝑖𝑎𝑔{𝜆 , ⋅⋅⋅ ,𝜆 } U where 1 𝐿𝑛𝑟 𝑛𝑡 and is a unitary matrix. Combining (18) and (19), we write IV. SIMULATION RESULTS AND DISCUSSIONS

𝐿𝑛∑𝑟 𝑛𝑡 In this section, we present the BER performance of SM and ′ 2 ˜𝐻 ˜ ˜ 2 ∥H(x − x )∥ = h Λh = 𝜆𝑖∥h𝑖∥ , (20) SSK signaling schemes as a function of the average received 𝑖=1 SNR obtained through simulation. Figures 1 and 2 depict the diversity order achieved by MIMO-CPSC under ML detec- h˜ = Uh h where is obtained from , the vector of all chan- tion for SSK and SM, respectively. Figures 3 and 4 show the h˜ 𝑖 nel gains, through unitary transformation, and 𝑖 is the th comparative performance in MIMO-OFDM versus MIMO- h˜ h˜ h element of . Therefore, is identically distributed as . CPSC for SSK and SM, respectively. In all simulations, we Entries of h are zero-mean independent complex normal ran- have assumed 𝛽 =3dB, 𝐿 =3, 𝑛𝑡 =2. We consider perfect dom variables. The corresponding covariance matrix is there- CSIR, i.e. channel state information is available at the re- fore a diagonal matrix. From (9) and (20), the expression for ceiver. We also assume quasi-static channel, i.e., the channel conditional PEP can be written as  remains constant during the transmission of one data block. ⎛ ⎞  𝐿𝑛∑𝑟 𝑛𝑡 Diversity order: Figure 1 shows the uncoded BER perfor- ′ ⎝⎷ ˜ 2⎠ 𝑃 (x → x ∣H)=𝑄 𝜇 𝜆𝑖∥h𝑖∥ , (21) mance of SSK for 𝑛𝑡 =2, 𝐾 =3, and 𝑛𝑟 =1, 2.We −1 −2 𝑖=1 have also plotted the 𝑐1𝑆𝑁𝑅 (diversity 1) and 𝑐2𝑆𝑁𝑅 (diversity 2) lines. We have run the BER simulations up to Applying Chernoff bound, sufficiently lower BERs to observe error rate curves to run ( ) 𝑛 =1 𝐿𝑛∑𝑟 𝑛𝑡 parallel to some diversity line. It is seen that for 𝑟 and ′ 1 1 ˜ 2 −1 𝑃 (x → x ∣H) ≤ exp − 𝜇 𝜆𝑖∥h𝑖∥ , (22) 𝑛𝑟 =2, the simulated BER curves run parallel to 𝑐1𝑆𝑁𝑅 2 2 −2 𝑖=1 line and 𝑐2𝑆𝑁𝑅 line, respectively, at high SNRs. This forms a simulation validation of the prediction in the previ- The unconditional PEP 𝑃 (x → x′) can be obtained by taking ous section (Table-I) that the achievable diversity order is 𝑛𝑟. expectation of this upper bound with respect to h˜. The num- ′ Figure 2 shows a similar set of plots for SM with BPSK mod- ber of non-zero 𝜆𝑖’s in (22) will depend on the {x, x } pair ulation. Here again, we see that, as predicted in the previous under consideration since B is a function of {x, x′}.Let𝜌 be section (Table-II), the diversity order observed through simu- the number of these non-zero eigenvalues, and, without loss lation is also 𝑛𝑟. of generality, let 𝜆𝑖, 𝑖 =1, ⋅⋅⋅ ,𝜌be these non-zero eigenval- ˜ 2 ues. We also observe that each ∥h𝑖∥ is a chi-square random Comparison with MIMO-OFDM: In Figs. 3 and 4, we present 2 variable with 2 degrees of freedom. Let 𝜎𝑖 be the variance of a ML performance comparison between MIMO-OFDM and ˜ 𝑛 =2 𝐾 =6 𝑛 =2 h𝑖. Then we can write the expectation as [1] MIMO-CPSC with 𝑡 , , and 𝑟 . Figure 3 is [ ] for SSK. Figure 4 is for SM with BPSK modulation. Perfor- 𝜌 𝜌 1 ∑ ∏ 1 mance of linear receivers (zero forcing and minimum mean 𝔼 exp(− 𝜇 𝜆 ∥h˜ ∥2) = . h˜ 𝑖 𝑖 𝜎2𝜇𝜆 (23) 2 1+ 𝑖 𝑖 square error receivers) are also shown for comparison. It is 𝑖=1 𝑖=1 2 seen that MIMO-CPSC scheme achieves significantly better Therefore, at very high SNRs, we can write the upper bound BER performance compared to MIMO-OFDM, particularly −3 of unconditional PEP as in the low-to-moderate SNR regime. For example, at 10 BER, MIMO-CPSC performs better by about 4 to 5 dB. This 2𝜌−1 ∏𝜌 1 𝑃 (x → x′) ≤ . performance advantage, coupled with the ‘no PAPR problem’ 𝜇𝜌 𝜎2𝜆 (24) 𝑖=1 𝑖 𝑖 advantage, makes MIMO-CPSC to be a credible alternative to

1965 Space Shift Keying (𝑛𝑡,𝑛𝑟) 𝐿 =3,𝐾 =3 𝐿 =2,𝐾 =3 Vector pair Eigenvalues Diversity Vector pair Eigenvalues Diversity (2,1) x: [(1,0),(1,0),(1,0)] 6, 1 x : [(1,0),(1,0),(1,0)] 4, 1 x′: [(0,1),(0,1),(0,1)] 5 zeros x′: [(0,1),(0,1),(0,1)] 3 zeros (2,2) x: [(1,0),(1,0),(1,0)] 6,6, 2 x: [(1,0),(1,0),(1,0)] 4,4, 2 x′: [(0,1),(0,1),(0,1)] 10 zeros x′: [(0,1),(0,1),(0,1)] 6 zeros (4,3) x: [(1,0,0,0),(1,0,0,0),(1,0,0,0)] 6,6,6, 3 x: [(1,0,0,0),(1,0,0,0),(1,0,0,0)] 4,4,4, 3 x′: [(0,1,0,0),(0,1,0,0),(0,1,0,0)] 33 zeros x′: [(0,1,0,0),(0,1,0,0),(0,1,0,0)] 21 zeros (4,4) x: [(1,0,0,0),(1,0,0,0),(1,0,0,0)] 6,6,6,6, 4 x: [(1,0,0,0),(1,0,0,0),(1,0,0,0)] 4,4,4,4, 4 x′: [(0,1,0,0),(0,1,0,0),(0,1,0,0)] 44 zeros x′: [(0,1,0,0),(0,1,0,0),(0,1,0,0)] 28 zeros TABLE I ′ DIVERSITY ORDER AND CORRESPONDING (x, x ) PAIR ALONG WITH CORRESPONDING EIGEN VALUES OF B FOR DIFFERENT CONFIGURATION OF (𝑛𝑡,𝑛𝑟,𝐿) FOR SPACE SHIFT KEYING IN MIMO-CPSC SYSTEMS.

Spatial Modulation (𝑛𝑡,𝑛𝑟, Θ𝑀 ) 𝐿 =3,𝐾 =3 𝐿 =2,𝐾 =3 Vector pair Eigenvalues Diversity Vector pair Eigenvalues Diversity (2,1,BPSK) x: [(1,0),(1,0),(1,0)], 12, 1 x: [(1,0),(1,0),(1,0)] 8, 1 x′: [(-1,0),(-1,0),(-1,0)] 5 zeros x′: [(-1,0),(-1,0),(-1,0)] 3 zeros (2,2,BPSK) x: [(1,0),(1,0),(1,0)], 12,12, 2 x: [(1,0),(1,0),(1,0)] 8,8, 2 x′: [(-1,0),(-1,0),(-1,0)] 10 zeros x′: [(-1,0),(-1,0),(-1,0)] 6 zeros (2,3,4-QAM) x: [(1+j,0),(1+j,0),(1+j,0)] 12,12,12, 3 x: [(1+j,0),(1+j,0),(1+j,0)] 8,8,8, 3 x′: [(1-j,0),(1-j,0),(1-j,0)] 15 zeros x′: [(1-j,0),(1-j,0),(1-j,0)] 9 zeros (4,4,4-QAM) x: [(1+j,0,0,0),(1+j,0,0,0), 12,12,12,12, 4 x′: [(1+j,0,0,0),(1+j,0,0,0), 8,8,8,8, 4 (1+j,0,0,0)] 44 zeros (1+j,0,0,0)], 28 zeros x′: [(1-j,0,0,0),(1-j,0,0,0), x′: [(1-j,0,0,0),(1-j,0,0,0), (1-j,0,0,0)] (1-j,0,0,0)] TABLE II ′ DIVERSITY ORDER AND CORRESPONDING (x, x ) PAIR ALONG WITH CORRESPONDING EIGEN VALUES OF B FOR DIFFERENT CONFIGURATION OF (𝑛𝑡,𝑛𝑟,𝐿,Θ𝑀 ) FOR SPATIAL MODULATION IN MIMO-CPSC SYSTEMS.

0 0 10 10 SSK, n =1, ML SM, BPSK, n =1, ML r r −1 10 c1/SNR −1 10 c3/SNR SSK, n =2, ML SM, BPSK, n =2, ML r r −2 10 2 2 c2/SNR −2 10 c4/SNR

−3 10 −3 10 −4 10

Bit Error Rate Bit Error Rate −4 10 −5 10 SSK, CPSC, SM, CPSC K=3, n =2, t K=3, L=3, n =2, −5 t −6 L=3, β=3 dB 10 10 β =3 dB, BPSK

−7 −6 10 10 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 Average SNR (dB) Average SNR (dB)

Fig. 1. Diversity order of MIMO CPSC system with SSK. 𝐾 =3, 𝑛𝑡 =2, Fig. 2. Diversity order of MIMO CPSC system with SM. 𝐾 =3, 𝑛𝑡 =2, 𝐿 =3, 𝛽 =3dB, 𝑛𝑟 =1, 2. 𝐿 =3, 𝛽 =3dB, BPSK, 𝑛𝑟 =1, 2.

MIMO-OFDM in MIMO-ISI channels. We note that the car- future extension to this work. dinality of the search space in the ML detection is of the or- 𝑀 𝐾 𝑛 𝐾 der of 𝑡 in both SSK and SM, resulting in exponential V. C ONCLUSIONS computation complexity in 𝐾. So for the MIMO-CPSC ap- We studied spatial modulation and space shift keying in the proach to be feasible in SSK and SM for large 𝐾 (which will context of cyclic-prefixed single carrier systems on MIMO- be needed in MIMO-ISI channels with large delay spreads), ISI channels. Our diversity analysis revealed that the diver- there is a need to devise low-complexity near-optimal algo- sity order achieved by SM and SSK in MIMO-CPSC systems rithms that scale well for large 𝐾, which is being pursued as

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1967