Available online at www.sciencedirect.com Procedia Technology 3 (2012 ) 1 8 The 2012 Iberoamerican Conference on Electronics Engineering and Computer Science Pre-equalization for MIMO Wireless Systems Using Spatial Modulation M. G. González-Pérez, J. M. Luna-Rivera and D. U. Campos-Delgado Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, Av. Salvador Nava s/n, Zona Universitaria, 78290, San Luis Potosí, México. +52(444) 8262491 Abstract Spatial modulation (SM) has emerged as a very promising method to exploit the potential multiple-input multiple-output system gains using low-complexity transceivers. SM uses the antenna index at the transmitter as an additional source of information. Although SM avoids inter-channel interference, its performance tends to decrease when the number of constellation points per antenna grows up. The main contribution of this paper is to introduce a novel pre-equalization stage for SM which allows to mitigate the fading effect of the wireless channel. Its principle is to pre-distort the points generated by SM, so that the receiver has an approximation of a M-QAM scheme, where the minimum distance between constellation s points is increased and consequently, the system s performance is improved. In order to perform the preequalization, a complete channel state information is assuming at the transmitter. It is shown through Monte Carlo simulations that the proposed scheme significantly enhances the performance of SM. c 2012 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Keywords: Spatial Modulation, MIMO, pre-equalization, M-QAM, digital modulation, Rayleigh fading channel 1. Introduction Multiple-input multiple-output (MIMO) systems have raised as a key technology in the development of modern wireless communications. A MIMO system uses multiple antennas at both the transmitter and receiver to exploit the spatial domain for spatial multiplexing and/or spatial diversity. Whereas spatial multiplexing has been generally used to increase the spectral efficiency of a MIMO link [1, 2], the purpose of spatial diversity is to improve the system s performance [3]. However, MIMO scheme has some well known drawbacks, such as [4 6]: (a) it suffers from high inter-channel interference (ICI) due to simultaneous transmissions on the same frequency from multiple antennas, (b) it increases the complexity and cost associated with the use of several antenna elements, (c) inter-antenna synchronization (IAS) is necessary to avoid performance degradation. Spatial Modulation is a recently transmission scheme for MIMO systems proposed in [4]. One of the key advantages of SM, unlike other MIMO schemes, is that it exploits the antenna index to convey information Corresponding author Email address: miryam_gonzalez@alumnos.uaslp.edu.mx (M. G. González-Pérez) 2212-0173 2012 Published by Elsevier Ltd. doi:10.1016/j.protcy.2012.03.001 Open access under CC BY-NC-ND license.
2 M. G. González-Pérez et al. / Procedia Technology 3 ( 2012 ) 1 8 without any bandwidth expansion. In addition to this, SM enjoys of relatively low-complexity structures both at the transmitter and receiver. The basic concept of SM is to map a group of bits to a symbol, which it is then transmitted by one of the transmit antennas. Since SM activates only one transmit antenna at each time then it reduces the hardware at the transmitter by avoiding the use of multiple radio frequency (RF) chains. Moreover, SM abstains from the use of advanced detection mechanisms due to the absence of interference in the system. The simplest form of SM, referred as Space Shift Keying (SSK) modulation, is introduced in [7] where the information is only represented by the presence or absence of energy at the antenna activated for transmission. Because SSK uses only the antenna indices to relay information, the complexity of detection is lowered for the SSK compared to SM. In [7], the optimum maximum-likelihood (ML) detector is derived for both SM and SSK. An extension of SSK, referred as Generalized SSK (GSSK), is presented in [8]. The GSSK scheme basically allows the activation of a subset of the transmit antennas rather than just one as performed by SM or SSK. Indeed, the advantages of spatial modulation can be summarized as [4, 7 11]: (a) the spectral efficiency is increased without a bandwidth expansion, (b) only oneantennaremains active, then just onerf chain is needed, (c) since ICI is avoided then a low-complexity detector is required, and (d) spatial diversity can be exploited at the receiver. Despite having the above mentioned benefits, SM has some limitations which include [7, 8, 11 13]: (a) the spectral efficiency of SM grows logarithmically with the number of transmitter antennas, consequently, when a high spectral efficiency is required, the number of antennas is very large, making the technology impractical, (b) given the nature of the SM mapping, it requires that the number of transmit antennas is a power of two and (c) unlike other MIMO schemes, SM does not provide transmit diversity gain. Even though, SM has some advantages, like any traditional digital modulation system, its performance decreases when the number of constellation points that are transmitted per antenna grows up. The main contribution of this paper is to introduce a novel pre-equalization stage for SM which allows to mitigate the fading effect of the wireless channel. The proposed pre-equalization stage is based on the idea to pre-distort the points generated by SM, so that the receiver has an approximation of a M-QAM scheme. In this way, the constellation s points are scattered among themselves increasing the minimum distance, as a result the system s performance is improved. At the receiver, a maximum likelihood (ML) detection algorithm is considered to estimate the bit-error-rate (BER) performance. This stage will prove to enhance significantly the SM s performance through Monte Carlo simulations results. The rest of the paper is organized as follows. In Section 2, we introduce the generic system model for SM. The pre-equalization stage is presented in Section 3, and next the proposal is validated through numeric simulations in Section 4. Finally, conclusions are drawn in Section 5. Throughout this paper, the following notation is used. Bold and capital letters denote matrices, whereas bold and lowercase indicate vectors. Italicized symbols denote scalar values. We use ( ) T to denote transpose of a vector or matrix. The notation F is used to indicate the Frobenius norm of a matrix. On the other hand, CN(m,σ 2 ) represents samples of a complex random variable whose real and imaginary parts are Gaussian distributed with mean m and variance σ2 2.Alsoweusep( ) for the probability of a given event, p Y ( ) for the probability density function (PDF) of a random variable Y. 2. SM System Model Consider a generic SM system with N T transmitter antennas and N R receive antennas as shown in Figure 1. The input bit stream given d 1, d 2, d 3, is sent through the SM modulator, where a group of N T bits is mapped to the transmitted vector x = [x 1, x 2,..., x NT ] T C N T. In this way, at each time the symbol transmitted is given by x = [0 0, x t, 0 0] T C N T with 1 t N T,wherex t represents the symbol transmitted by the t-antenna [4]. Recall that, at each time, only one antenna convey information, while the remaining antennas still idle. At the receiver, the signal can be expressed as follows: y = ρhx + n (1) where H represents the N R N T channel matrix with complex gains h t,r CN(0, 1) and n = [n 1, n 2, n NR ] T the noise vector with n r CN(0, 1), where 1 r N R and 1 t N T. The signal to noise ratio (SNR) at
M. G. González-Pérez et al. / Procedia Technology 3 ( 2012 ) 1 8 3 Fig. 1. Block diagram of SM each receive antenna is denoted by ρ. In order to illustrate this idea easily but without loss of generality, let us consider the example given in Table 1 for a SM system using N T = 4andM = 2, where M represents the number of constellation points per antenna. Table 1. SMmapping with N T = 4andM = 2 [d 1 d 2 d 3 ] T [x 1 x 2 x 3 x 4 ] T 0 0 0 +1 0 0 0 0 0 1 0 +1 0 0 0 1 0 0 0 +1 0 0 1 1 0 0 0 +1 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 0 0 0 1 In general, we observe that the transmission efficiency for SM grows logarithmically with N T and M [1], i.e. η SM = log 2 (M) + log 2 (N T ) bits/symbol (2) The detector s main function is to determine the index of the antenna activated for transmission, and the symbol sent through it. Since the channel inputs are assumed equally likely, the optimal detector is performed using the maximum likelihood (ML) principle, which can be expressed by [7]: ˆd = arg max i p Y (y ˆx i, H), ˆx i C N T, 1 i L (3) where ˆd represents the estimated word of N T bits over the set of L possible transmit sequences, while p Y (y ˆx i, H)isgivenby[14] p Y (y ˆx i, H) = exp( y ρh ˆx i 2 F ). (4) Equivalently, the maximization problem in (4) can be rewritten, as follows, 3. Pre-equalization for Spatial Modulation π N R ˆd = arg min i y ρh ˆx i 2 F. (5) Even though SM exhibits a good performance with respect to the bit error rate (BER), when M grows, the performance tends to decrease [4]. To mitigate this effect, it is proposed a pre-equalization stage; which,
4 M. G. González-Pérez et al. / Procedia Technology 3 ( 2012 ) 1 8 Fig. 2. Block diagram of SM with pre-equalization besides to improve the system performance, also reduces the receiver s complexity. The block diagram in Figure 2 shows the SM system with the proposed pre-equalization stage. To carry out the pre-equalization, it is assumed the complete channel state information (CSI) at the transmitter. Then we seek to locate the received symbols in a strategic and convenient way by pre-distorting the transmitted signal. For this purpose, it is the aim of the pre-equalizer to maximize the minimum distance between the points of the constellation at the receiver, since the system performance depends on it [14]. For this end, the pre-equalization scheme is designed to map the distorted transmitted symbols into a rectangular M-QAM [15] at the receiver. It is important to highlight that this first approach seeks to evaluate the concept of the pre-equalization scheme, therefore, the fact of using a rectangular M-QAM represents a suboptimal scheme in the sense that it does do not maximally space the constellation points for a given energy. Future work will address the design of pre-equalization with an optimal space constellation points at the receiver. (a) (b) Fig. 3. Mapping of the constellation points of SM at channel s output with N R = 1: (a) without pre-equalization and (b) including pre-equalization. To illustrate the process carried-out for the pre-equalization stage, Figure 3(a) shows the noiseless received signals corresponding to the transmission of all constellation symbols for a given channel matrix in a SM without pre-equalization. Equivalently, Figure 3(b) presents the same scenario but now applying the pre-equalization scheme. In order to show the procedure performed in the pre-equalization it is necessary to define a set of vector {a 1, a 2,...,a M } for the pre-equalizer, where a m = [ a 1, a 2,...,a NT ] T C N T represents a pre-equalization factor which depends on the wireless fading channel. The transformation rule can be described as follows: according to the SM s modulator output, the non-zero value of x is mapped to a M-QAM constellation s point. For the sake of accomplish the transformation is assumed that {s 1, s 2,...,s M } corresponds to the
M. G. González-Pérez et al. / Procedia Technology 3 ( 2012 ) 1 8 5 well-known M-QAM constellation. Hence, the constellation s points are given by s 1 = ρha 1, s 2 = ρha 2,. s M = ρha M, (6) In (6) as a m is a complex variable, we have two unknowns per equation, corresponding to the real and imaginary parts, therefore, to solve the system of equations generated from the constellation s points, it is necessary to use the symmetry of the rectangular M-QAM modulation scheme. In this way, the system of equations has a unique solutions, that correspond to the pre-ecualization factor. At each time the symbol transmitted is given by x EQ = [0 0, a m, 0 0] T C N T. While the received signal for SM EQ through the Rayleigh flat fading channel is given by where n, as the original SM scheme, is the noise vector with n r CN(0, 1) y = ρhx EQ + n, (7) 4. Simulations Results In simulations, only one receive antenna is assumed in the system to give clarity of the advantage provided by the pre-equalizer. For a set-up N T = 4, M = 2 and a target of 3bits/s/Hz (8-QAM) it is necessary to compute the pre-equalization vector as a = [a 1, a 2, a 3, a 4 ] T C N T, which will be responsible for mapping the conventional SM system to SM EQ (SM with pre-equalization). Once solved the system of equations in (6) the constellation for SM EQ is given by the set {a 1, a 2, a 3, a 4, a 1, a 2, a 3, a 4 },valuesofa??are not shown, because as it is assumed that the wireless channel changes at each instant of time, then a is recalculated for each Monte Carlo realization. Histogram 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Constellation s minimum distance (a) Histogram 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Constellation s minimum distance (b) Fig. 4. Minimum distance for (a)smand (b) SM EQ system To verify the method s feasibility, we first obtained and compared the histograms of the minimum distances obtained by SM and SM EQ, through Monte Carlo simulations, with 10 5 channel realizations. In addition, the distance was established by the Euclidean norm. Figure 4 shows the gain obtained by the procedure in terms of distance. As can be seen, while the average of SM is 0.18, when the pre-equalization is implemented it is possible to reach an average of 0.51. Figure 5 shows the points received when the pre-equalization is implemented, as can be seen, these are aligned alike 8-QAM as expected. In Table 2, it is shown the mapping rule of SM without and with the pre-equalization stage implemented.
6 M. G. González-Pérez et al. / Procedia Technology 3 ( 2012 ) 1 8 3 3 2 2 1 1 Im(y) 0 Im(y ec ) 0 1 1 2 2 3 3 2 1 0 1 2 3 Re(y) (a) 3 3 2 1 0 1 2 3 Re(y ) ec (b) Fig. 5. Noiseless received signals at receiver with (a)smand (b) SM EQ Table 2. Mapping table for SMand SM EQ with N T = 4,M = 2. [d 1 d 2 d 3 ] T [x 1 x 2 x 3 x 4 ] T [x EQ1 x EQ2 x EQ3 x EQ4 ] T 0 0 0 +1 0 0 0 +a 1 0 0 0 0 0 1 0 +1 0 0 0 +a 2 0 0 0 1 0 0 0 +1 0 0 0 +a 3 0 0 1 1 0 0 0 +1 0 0 0 +a 4 1 0 0 1 0 0 0 a 1 0 0 0 1 0 1 0 1 0 0 0 a 2 0 0 1 1 0 0 0 1 0 0 0 a 3 0 1 1 1 0 0 0 1 0 0 0 a 4 Through Monte Carlo simulations with 10 5 channel realizations we evaluate the BER performance of the SM EQ system assuming the ML detector at the receiver. In Figure 6 we plot BER as a function of the signal to noise ratio for a system target of 3 bits/s/hz using E = 1, N T = 4andM = 2, i.e. L = 8 constellation points. We compare the performance of the proposed scheme to that of SM. As can be seen, SM EQ outperforms SM approximately 4 db for a BER = 10 2. We also compare the performance of the proposed SM EQ when the system s target is 4 bits/s/hz with the following set-ups: M = 4, N T = 4andM = 2, N T = 8 with E = 1andN R = 1 in both cases. Figure 7 shows that although both systems have the same spectral efficiency, when the diversity order at the transmitter is increased, the system s performance is improved, further than, SM EQ exhibits a similar performance for both systems, when SNRis less than 8dB, useful for systems where power is a critical factor. Based on the results presented here, we remark that the proposed pre-equalization scheme offers a significant performance gain to the SM scheme as a function of the diversity order at the receiver. It is observed that a higher diversity order gain can be achieved when the number of antennas is decreased at the receiver, a desirable condition in practical communications systems where placing a high number of transmitter antennas is a critical constraint. 5. Conclusions The benefits offered with spatial modulation can be improved by techniques such as the pre-equalization scheme presented in this paper. It was shown that SM EQ outperforms about 4 db in BER the original SM scheme, when a ML detector is used. However, the trade-off remains to have a complete channel information at transmitter. As future work, we seek to exploit spatial diversity at the receiver not only to
M. G. González-Pérez et al. / Procedia Technology 3 ( 2012 ) 1 8 7 10 0 SM, N T =4, M=2 SM EQ, N T =4, M=2 10 1 BER 10 2 10 3 10 4 0 2 4 6 8 10 12 14 16 18 20 SNR(dB) Fig. 6. Performance of SM EQ with N T = 8, M = 2andN T = 4, M = 4 improve the performance of SM EQ but also of the general SM system. This is possible due to the extra degrees of freedom introduced by the multiple antenna elements available at the receiver but always keeping the complexity to a tractable level, allowing the system to be implemented. 10 0 SM EQ, N T =8, M=2 SM EQ, N T =4, M=4 SM, N T =8, M=2 BER 10 1 10 2 0 2 4 6 8 10 12 14 16 18 20 SNR(dB) Fig. 7. Performance of SM EQ with M = 4andN T =2,8 References [1] E. C. H.Haas, E.Schulz, Increasing spectral efficiency by data multiplexing using antenna arrays, The 13th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (2002) 610 613. [2] N. J. A Goldsmith, S.A. Jafar, S. Vishwanath, V-blast: an architecture for realizing very high data rates over the rich-scattering wireless channel, URSI International Symposium on Signals, Systems, and Electronics, 1998 (1998) 295 300.
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