Hybrid Bit-to-Symbol Mapping for Spatial Modulation

Transcription

1 IEEE Hybrid Bit-to-Symbol Mapping for Spatial Modulation Yue Xiao, Ping Yang, Lu Yin, Qian Tang, Shaoqian Li, Senior Member IEEE, Lajos Hanzo, Fellow IEEE Abstract In spatial modulation (SM), the information bit stream is divided into two different sets: the transmit antenna inde-bits (TA-bits) as well as the amplitude phase modulation-bits (APMbits). However, the conventional bit-to-symbol mapping (BTS-MAP) scheme maps the APM-bits the TA-bits independently. For eploiting their joint benefits, we propose a new BTS-MAP rule based on the traditional two-dimensional (-D) Gray mapping rule, which increases the Hamg distance (HD) between the symbol-pairs detected from the same transmit antenna (TA) simultaneously reduces the average HD between the symbol-pairs gleaned from different TAs. Based on the analysis of the distribution of imum Euclidean distance (MED) of SM constellations, we propose a criterion for the construction of a meritorious BTS-MAP for a specific SM setup, without the need for any additional feedback-link or etra computational compleity. Finally, Monte Carlo simulations are conducted for confirg the accuracy of our analysis. Inde Terms Gray-mapping, Hamg distance, spatial modulation (SM), multiple-input multipleoutput (MIMO). I. Introduction Spatial modulation (SM) is a new three-dimensional (3- D) hybrid modulation scheme conceived for multiple-input multiple-output (MIMO) transmission, which eploits the indices of the transmit antennas (TAs) as an additional dimension invoked for transmitting information, apart from the classic two-dimensional (-D) amplitude phase modulation (APM) [] [5]. In SM, the information bit stream is divided into two different sets: the bits transmitted through the TA indices the APM constellations Copyright (c) 5 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubspermissions@ieee.org. P. Yang, Y. Xiao, L. Yin, Q. Tang S. Li are with the National Key Laboratory of Science Technology on Communications, University of Electronic Science Technology of China, 673, Sichuan, China. Y. Xiao is also with the National Mobile Communications Research Laboratory, Southeast University. ( yplw@63.com, iaoyue@uestc.edu.cn, @qq.com, qian.tang@63.com, lsq@uestc.edu.cn). L. Hanzo is with the School of Electronics Computer Science, University of Southampton, Southampton SO7 BJ, U.K. ( lh@ecs.soton.ac.uk). The financial support of the European Research Council s Advanced Fellow Grant, the National Science Foundation of China under Grant number 6479, the National High-Tech R&D Program of China ( 863 Project under Grant number 4AAA77) the open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 3D5). are gratefully acknowledged. [6]. For simplicity, we refer to these two sets of bits as TA-bits APM-bits. In conventional -D APM constellations, the choice of the bit-to-symbol mapping (BTS-MAP) rule plays an important role in detering the achievable bit-error ratio (BER) performance [7]. For eample, an optimized BTS- MAP is capable of providing a low error floor in both bit-interleaved coded-modulation as well as in its iterative decoding demodulation aided counterpart (BICM- ID) [8]. It is widely known that for equally likely statistically independent -D APM constellations, Gray mapping is optimal in terms of imizing the BER [7]. In general, the optimal BTS-MAP depends on the specific geometry of the signal constellation, especially on the location of the phasors separated by the imum Euclidean distance (MED) [9]. Compared to APM schemes, SM has a higher constellation dimension a different distribution of MED due to its hybrid modulation principle. Hence, the classic Gray mapping proposed for -D constellations will no longer achieve the optimal BER in fading MIMO channels. Recently, the wide-ranging studies disseated in [], [4], [6], [] [4] have characterized some of the fundamental properties of SM, such as its energy efficiency [], [3] the effects of power imbalance [4]. In these contributions, a general framework was established for the BTS-MAP rule of SM [], where the APM-bits are mapped according the classic Gray mapping rule, while the TA-bits are mapped to the active TA inde. Due to its intuitive nature low-compleity, this rule has been considered in diverse SM-based systems [5] [8]. For eample, in [5], [6], this general framework was developed for an arbitrary number of TAs hence strikes a fleible trade-off in terms of the attainable BER performance capacity. In [7], [8], this philosophy has been etended to trellis coded modulation (TCM) aided SM systems for the sake of achieving reliable digital transmission. However, this classic BTS-MAP scheme maps the APM-bits TAbits to symbols independently hence may sacrifice the classic Gray-coded benefits. Moreover, the MED distribution of SM was not considered in the design process of the conventional BTS-MAP scheme, which is a measure of separation between two constellation points as a result, it has a doant influence on the BER. Against this background, the novel contributions of this treatise are as follows. We propose a new BTS-MAP rule based on the classic Gray coded principles for eploiting the interaction

2 IEEE of the TA-bits the APM-bits in fading channels, which differs from the conventional BTS-MAP, since it reduces the average Hamg distance (HD) between the SM symbol-pairs of the different TAs simultaneously it increases the HD between the symbol-pairs of the same TA. Based on the analysis of the distribution of the MED, we propose a criterion for the construction of a beneficial BTS-MAP rule for a specific SM-MIMO setup, which is achieved without the need for an additional feedback-link without any etra computational compleity. Finally, performance comparisons with the conventional BTS-MAP scheme of [] are provided for different constellation sizes signal-to-noise ratios (SNRs). Notation: ( ), ( ) T ( ) H denote conjugate, transpose, Hermitian transpose, respectively. The probability of an event is represented by P ( ). Furthermore, denote the Euclidean norm magnitude operators, respectively, while ε(, y) is the Hamg distance between the binary strings y. N t is the number of TAs M is the size of the APM constellation adopted. Let b m i be the transmit bit vector mapped to the SM symbol i the APM symbol s m i, which corresponds to TA i, while b k j is the transmit bit vector mapped to the SM symbol j related to TA j the APM symbol s k j. The average HD between the symbol pair gleaned from different TAs, termed as HDD, is defined as HDD = N t(n t )M ε(b m i, bk j ). Moreover, i j,i,j {,...,N t} the average HD between the symbol pair detected from the same TA, termed as HDS, is defined as HDS = ε(b m i, bk i ). For a fied MIMO channel N tm(m ) i {,...,N t} H, d Same (H) is the MED between the symbol pair of the same TA, (H) is the MED between the symbol pair of different TAs, the overall MED of a specific SM is d = {d Same (H), (H)}. II. Conventional BTS-MAP Rule for SM Consider a MIMO system having N t transmit N r receive antennas. The (N r N t )-element channel matri H is used for modelling a flat-fading channel with elements having comple Gaussian distributions with unit variance. We focus our attention on the independent identically distributed (i.i.d) Rayleigh case, but also discuss Nakagami-m channels. Let b = [b,..., b L ] be the transmit bit vector in each time slot, which contains L= log (N t M) bits. As shown in Fig., the input vector b is divided into two sub-vectors of log (N t ) log (M) bits, denoted as b b, respectively. The bits in the sub-vector b are used for selecting a unique TA inde q for activation, while the bits in the sub-vector b are mapped to a Gray coded APM symbol s q. Hence, the resultant SM symbol l C Nt is formulated as [6] = s q l e q, () where e q ( q N t ) is selected from the N t -dimensional stard basis vectors. b b bl bl =[,...,,..., ] Source bits b SM h () -h () h () -h () TA-bits b = [ b,..., b ] log N t APM-bits b = [ b,..., b ] log Nt + Conventional BTS-MAP Input bits TA inde APM symbol h L TA Inde APM symbol Im () h () -h () h () ( ) { h, h } < { h h, h + h } ( b) { h, h} { h h, h+ h} a Fig.. The conventional BTS-MAP of SM: an eample for the BPSK-modulated ( )-element SM. (a) The imum Euclidean distance encountered on the same TA, where the conventional BTS- MAP is optimal; (b) The imum Euclidean distance encountered on different TAs, where the conventional BTS-MAP is suboptimal. To epound a little further, we eemplify the BPSKmodulated ( )-element SM in Fig.. As seen in Fig., L = input bits are divided into two single-bit streams, then the first bit deteres the activated TA ( or ), while the second single bit generates the classic BPSK symbol (+ or -). The above-mentioned BTS- MAP method considers the APM-bits TA-bits independently hence facilitates a simple implementation of SM. However, this BTS-MAP method may result in a Graycoding penalty [9], which degrades the BER. Specifically, for the eample of BPSK-modulated ( )-element SM in Fig., assug that the channel matri is H = [h, h ], there are four received constellation points denoted as h, h, h h we investigate two scenarios: (a) the MED is d = h ; (b) the MED is d = { h h, h + h }. Note that scenario (a) corresponds to the case when the MED is encountered on the same TA (TA ), while scenario (b) corresponds to the case when the MED is encountered on different TAs (between TAs ). For scenario (a), the most likely erroneously-detected pattern is given by the nearest constellation points ( h, h ). If the conventional BTS- MAP is adopted, these points differ only in a single bit, namely by the difference between the bits in Fig. (a), while other adjacent constellation points may differ in more than one bits. This mapping rule obeys the concept of Gray mapping, where the probability of having a bit error is imized, hence it has an optimal performance in this specific scenario. However, for scenario (b), it is suboptimal, because the nearest constellation points become to ( h, h ), which differ in two bits. This implies that more than one bit errors are associated with the most likely error pattern of ( h, h ), hence a Re

3 IEEE 3 performance penalty will occur. This indicates that the MED distribution should be considered in the design of BTS-MAP. III. Proposed BTS-MAP for SM A. The Principle of the Proposed BTS-MAP We found in Section II that the design of the BTS-MAP scheme depends on the MED, which may be achieved between the symbol pair gleaned from the same TA or between the symbol pair from different TAs. The conventional BTS-MAP rule does not consider this distribution of the MED hence it becomes sub-optimal for some channel scenarios, as illustrated in Fig. (b). For eploiting the mapping gain of SM, we propose a new BTS-MAP scheme based on traditional Gray coded modulation. To be specific, similar to the conventional BTS-MAP of SM, the L-bit input vector b = [b,..., b L ] is divided into a pair of subvectors d = [d,..., d log (M)] = [b,..., b log (M)] s = [s,..., s log (N t)] = [b log (M)+,..., b L ]. As a result, the transmit bit vector can be represented as b = [d, s]. Then, the first sub-vector d is mapped to an APM symbol s q, rather than the TA inde in conventional BTS-MAP. l Then, the sub-vector s is transformed to a new input vector by using the bit-by-bit XOR operation with jointly considering the last APM-bit, which is represented as s = [s, s,..., s log (N ] t) = [d log (M) s, s s,..., s log (N t) s log (N t)]. = [b log (M) b log (M)+,..., b log (L) b log (L)] () where denotes the XOR operation. Then, the bit vector ( s is mapped) to a specific TA inde of q = log (N t) log (Nt) k s k +, which is used for transmitting k= the APM symbol. The rationale of introducing the XOR operation in () is to increase the HDS, while decreasing the HDD. To be specific, the last APM-bit b log (M) of d the sub-vector s = [b log (M)+,..., b L ] form a new vector s = [b log (M), s] for generating the TA-bits in (), where we have b log (M) = or b log (M) =. Assug that there are two sub-vectors s a s b, then provided that two different sub-vectors s a = [, s a ] s b = [, s b ] map to the same TA inde, the HD ε( s a, s b ) is maimized to log (N t ) +. For eample, the vector s a = [,, ] s b = [,, ] are mapped to the same TA inde for SM using N t = 4 TAs. This result implies that the XOR operation maps the sub-vector pair s a s b having the highest HD to the same TA hence it is capable of increasing the HDS, while decreasing the HDD. B. Eample Selection Criterion Compared to Fig., we present the new BTS-MAP table in Fig. for the simple BPSK-modulated ( )- element SM eample mentioned in Section II, where the HDD is reduced to the HDS is increased to. This b b bl bl =[,...,,..., ] Source bits Fig.. b SM TA-bits = [ b,..., b] b log ( M) + L APM-bits b =[ b,..., b ] log ( M ) XOR s = [ s, s,..., s ] log ( N t ) APM symbol Proposed BTS-MAP Input bits TA inde APM symbol The proposed BTS-MAP for SM. TA inde BTS-MAP may be more suitable for the channel scenario (b) of Fig., because it performs a Gray mapping, when considering the adjacent constellation points h h associated with the MED. As indicated in Section II, the design of BTS-MAP depends on the distribution of MED, which can be encountered either on the same TA or on different TAs. To be specific, for a given channel matri H, the MED between the SM symbol pair ( i, j ) of different TAs is given by (H) = H( i j ) F i, j X, i j,i j = (h i s m s m i,sk j Q,i j i h js k j ), (3) where X is the set of all legitimate transmit symbols, h i is the ith column of H s m i s k j represent the classic APM constellation points from the set Q. Moreover, the MED between the symbol pair of the same TA is defined as d Same (H) = h i (s m,k i Q, i s k i ). (4) s m i s m i sk i Based on (3) (4), the probability of the MED d encountered on different TAs is defined as P Diff = P ( (H) < d Same (H)), while the probability of the MED d on the same TA is defined as P Same = P ( (H) > d Same (H)). To imize the probability of having a bit error, the HD of the more likely adjacent constellation points having the MED should be lower than that of other points located at a higher distance than the MED. This is also the basic concept of Gray mapping for a -D APM constellation. In the proposed BTS-MAP, the HDD is lower than the HDS hence it is more suitable for the specific scenario, when the MED occurs more often in the contet of different TAs, which can be epressed as P Diff > P Same. In other words, when we have P Diff > P Same for a SM setup, most of the error events are typically imposed by the SM symbols of different TAs, hence the imization of the HD between these nearest points (i.e. the HDD) leads to directly imizing the probability of bit errors. Based on this observation, the BTS-MAP selection criterion conceived for a specific SM scheme is formulated as: Selection criterion: if P Diff > P Same (or P Diff ) is satisfied by a specific SM, then the proposed BTS-MAP is

4 IEEE 4 superior to the conventional one in terms of reducing the BER. Otherwise, the conventional BTS-MAP is preferred. IV. Theoretical Analysis Mapping Optimization In this section, the probabilities of P Diff P Same are derived, which are used as an evaluation criterion for selecting a meritorious BTS-MAP for a specific SM setup. As shown in [4], [], the PSK modulation schemes are preferred in SM, hence PSK is adopted for our theoretical analysis. A. M-PSK Modulated ( )-Element SM For the M-PSK modulated ( )-element SM, the associated channel matri can be epressed as H = [h, h ], where h h are the fading coefficients of the first second TAs, respectively, which have zero mean unit variance. The corresponding MED (H) of (3) the MED d Same (H) of (4) are given by { ( ) ( ) d Same (H) = { sin π π h, sin h } M M. (5) (H) = { h e j kπ M h, k =,, M } Now, we can derive the distribution functions of (H) d Same (H). Since the amplitudes of h i (i =, ) obey the Rayleigh distribution having probability density functions (PDFs) of f hi () = e, i =,, the cumulative distribution functions (CDFs) of η i = sin ( ) π M hi, i =, are formulated as F ηi () = sin( M π ) e d = e. (6) 8 sin ( M π ),, i =,. Based on the distribution function of η i in (6), the CDF PDF of the rom variable d Same (H) = {η i, i =, } in (5) are given by F d Same (H) () = e 4 sin ( M π ), (7) [ d f () = d Same (H) ] F d Same (H)() d. (8) = sin ( π M ) e 4 sin ( M π ),. Let us now drive the PDF of the variable (H). Since the amplitudes of β k = h e j kπ M h, k =,, M obey the Rayleigh distribution having PDFs of f βk () = e 4, k =,, M, the associated CDF functions are F βk () = f β k () d = e 4,, k =,, M.. (9) Based on the theory of order statistics, the CDF PDF of (H) = {β k, k =,, M } are ) M F (H) () = ( ( e 4 ) = e M 4,. () d[fddiff f (H) () = (H)()] d M e 4, = M. () As illustrated in Section III, for a fied channel matri H provided that the proposed BTS-MAP performs better than the conventional BTS-MAP the inequality P Diff = P {d Same (H) > (H)} > should be satisfied, which is equivalent to P {z } >, where z = (H) d Same (H). Based on (8) (), the probability P Same = P (z > ) for the M-PSK-modulated ( )-element SM is given by P Same [ = P (z > ) ] + = f (H) (z + y) f d Same (H) (y) dy dz ( ) ( + M(z+y) = e M(z+y) ) y y 4 sin ( M π ) e 4 sin ( M π ). dydz = (M sin ( π M )+) () Due to the constraint of P Diff + P Same =, the probability P Diff = P (z ) is calculated as P Diff = P (z ) = P (z > ) = M sin ( π M )+. (3) According to (3), the values of P Diff for BPSK, QPSK, 8-PSK 6-PSK are.67,.67,.54.39, respectively. The result in (3) indicates that P Diff > P Same is satisfied for M 8. In this case, the proposed BTS-MAP performs better than the conventional scheme. B. BPSK-Modulated (4 )-Element SM Net, consider the case of N t >. In this subsection, we investigate the (4 )-element SM using BPSK. Let us denote the channel coefficients by H = [h, h, h 3, h 4 ]. In this system, the MEDs of (3) (4) can be represented as d Same (H) = { h, h, h 3, h 4 }, (4) (H)={ h ± h, h ± h 3, h ± h 4,... h ± h 3, h ± h 4, h 3 ± h 4 }. (5) Similar to Section IV-A, if the proposed BTS-MAP outperforms the conventional BTS-MAP for the fading channel, the inequality P {d Same (H) > (H)} should be satisfied. To simplify the analysis, we shall assume that the EDs in the receive constellation are statistically independent. Strictly speaking, this is not true, since the constellation points created by each channel are indeed inter-dependent through the transmit symbols. But nonetheless, based on regression analysis [], we can state that the correlation between the d Same (H) (H) is low. Moreover, we will demonstrate using Monte Carlo simulations in Table I that this assumption does not impose a high inaccuracy. Firstly, for a normalized transmit constellation the received vectors h i (i =,, 3, 4) obey the Rayleigh distribution of f hi () = e 8 (i =,, 3, 4). (6) 4

5 IEEE 5 TABLE I The metrics of HDS HDD of the conventional BTS-MAP the proposed BTS-MAP of SM for different MIMO setups. Moreover, the corresponding metrics P Same P Diff are also provided. APM HDS/HDD HDS/HDD P Same P Diff N t/n r scheme (Conventional) (Proposed) (%) (%) / BPSK./.5./ / QPSK.33/../ / BPSK./.83 3./ / QPSK.33/.33.67/ / BPSK./.5./ / QPSK.33/../ / BPSK./.83 3./ / QPSK.33/.33.67/ Based on the theory of order statistics [], on the four distances h i (i =,, 3, 4) in the receive SM constellation, the CDF the PDF of the rom variable d Same (H) is F d Same (H) () = [ F hi () ] 4 = e 8 4 = e. f d Same d[fdsame (H) () = d = e, >. (H)()] (7) (8) Let us now derive the PDF of the MED (H). Since h h are Gaussian rom variables, the PDF of h i ± h j (i j) formulated in (5) obeys the Rayleigh distribution, which can be epressed as f hi h j () = e 4,. (9) Then, the CDF the PDF of the rom variable (H) are given by F (H) () =, > () e 3 f (H) () = d(f ddiff (H)()) d = 6e 3, >. () Similar to the M PSK-modulated ( )-element SM, we have the following probability: P ( z = (H) d Same (H) > ) = = = f (H) (y + z) f d Same (H) (y)dydz 6 (y + z) e 3(y+z) ye y dydz () ye 7 y dy = 7. From (), we have P Same = P (z > ) = 7 P Diff = P (z ) = P (z > ) = 6 7, which satisfies the condition P Diff > P Same. Hence, for the BPSK-modulated (4 )-element SM, the proposed BTS-MAP is preferred. C. Other MIMO Setups In case of a high modulation order M a large number of TAs N t, there eist too many received distances associated with different values. In this case, it may be a challenge to theoretically evaluate the probability P { d Same (H) > (H) }, because the eact distribution of the rom variable (H) depends on both the channel matri on the symbol alphabet. To deal with these challenging scenarios, the statistical P Diff P Same results based on Monte Carlo simulations can be invoked for selecting the appropriate 3-D mapping schemes. To be specific, we can create a parameter-lookup table for the SM schemes associated with the MIMO setups considered, similar to Table II. For a specific SM transmission, we assume that the relevant statistical information, concerning the fading type, the MIMO antenna setup the PSK scheme adopted, is available for the transmitter. Then, we can use this information to select the appropriate BTS-MAP scheme according the lookup table off-line designed. Moreover, if we consider the adaptive SM schemes of [], [3], we can use a feedback link for appropriately selecting the BTS-MAP directly by using the information (H) d Same (H). If the constraint of P Diff > P Same ( (H) < d Same (H) for adaptive SM) is satisfied for a specific MIMO setup, the proposed BTS- MAP is adopted. Otherwise, the conventional BTS-MAP scheme is utilized. V. Performance Results A. HDD HDS Metrics for Different BTS-MAP Schemes In this subsection, the Hamg distances HDD HDS of the proposed BTS-MAP of the conventional BTS-MAP are compared under different MIMO setups. The simulation setup is based on -4 bits/symbol transmissions over independent flat Rayleigh block fading channels. Furthermore, the probabilities P Diff P Same of the occurrence of the MED d are also investigated. As shown in Table I, the XOR operation of () allows the proposed BTS-MAP scheme to achieve higher HDD lower HDS values compared to that of the conventional BTS-MAP. Moreover, the inequality P Diff > P Same is satisfied in diverse MIMO setups in Table I. It means that the MED d is encountered between different TAs with a high probability hence the proposed BTS-MAP, which has a lower HDD, is preferred. For eample, P Diff of the SM system associated with N t = 4, N r = BPSK modulation is higher than 86.6%, while the HDD is reduced from.83 to.5 by using the proposed scheme. The imization of this HD between these nearest points leads to a BER performance gain. Moreover, Table I shows that the simulation results of P Diff match the theoretical results for the BPSK-

6 IEEE 6 P Diff Probability of Modulation order (M) Proposed BTS-MAP Conventional BTS-MAP BER - (N t =,N r =) (M= 6) BPSK Pro. BPSK Con. QPSK Pro. - QPSK Con. 8PSK Pro. 8PSK Con. 6PSK Pro. 6PSK Con SNR(dB) Fig. 3. The probability P Diff for SM under various modulation orders different antenna configurations N t N r. modulated ( )- (4 )-element SM systems of Section IV. Note that the modest difference observed between the theoretical simulation results is due to the approimation process invoked for the evaluation of P Diff in Section IV. Furthermore, observe in Table I that as the modulation order increases, the corresponding P Diff is reduced. To epound a litter further, we investigate the effect of the modulation order the number of TAs on the probability P Diff in Fig. 3. Eplicitly, observe in Fig. 3 that a higher modulation order may achieve a lower P Diff value for a fied (N t N r )-element MIMO. This is due to the fact that if M is significantly higher than N t, the APM symbol errors doate the performance of SM. By contrast, if the number of TAs N t is increased while maintaining a fied value of M, we have an increased value of P Diff due to the fact that the TA decision errors doate the performance of SM. Moreover, since the increase of N r can reduce both the TA APM decision errors in SM, hence the specific effect of this parameter depends on the particular SM setup considered. As shown in Fig. 3, our BTS-MAP rule is that if we have P Diff >.5, then the proposed BTS-MAP may achieve a better BER performance. Otherwise, the conventional BTS-MAP can be utilized. Note that even if the statistics of P Diff are available for an SM-based MIMO system (such as the adaptive SM of [] [3]), our BTS- MAP selection rule still remains appropriate. Moreover, the proposed scheme can also be readily etended to other types of fading channel distributions, such as Rician Nakagami fading [9]. B. BER Performance In this subsection, we characterize the BER performance of the proposed BTS-MAP compared to the conventional BTS-MAP in MIMO Rayleigh Nakagami-m fading channels. Moreover, the optimal ML detector is adopted. Fig. 4. BER performance of the proposed BTS-MAP the conventional BTS-MAP schemes having N t =, N r = employing M-PSK signal sets. BER - - (N t =4, N r =) BPSK Pro. BPSK Con. QPSK Pro. QPSK Con. 8PSK Pro. 8PSK Con. 6PSK Pro. 6PSK Con. (M= 6) SNR(dB) Fig. 5. BER performance of the proposed BTS-MAP the conventional BTS-MAP schemes associated with N t = 4, N r = M-PSK schemes. Here, the notation Pro. represents the proposed BTS- MAP scheme, while Con. denotes the conventional BTS- MAP. Fig. 4 shows the BER performance of the ( )-element SM systems associated with different PSK schemes. As epected, in Fig. 4 the proposed BTS-MAP provides SNR gains of about.9 db for M =.6 db for M = 4 at BER= over the conventional BTS-MAP scheme. More importantly, similar to the result achieved by conventional Gray mapping for classic -D constellations, the specific SNR value only has a modest effect on the mapping gain of the proposed scheme [9]. Observe in Fig. 4 that for the case of M > 8, the conventional BTS- MAP outperforms the proposed BTS-MAP. This result is consistent with the findings of Fig. 3, where the constraint of P Diff > P Same is no longer met. Additionally, for the case

7 IEEE 7 BER (N t =, N r =) BPSK Con. BPSK Pro. QPSK Con. QPSK Pro. m=.5 m= SNR(dB) Fig. 6. BER performance of the proposed BTS-MAP the conventional BTS-MAP schemes for ( )-element Nakagami-m channels. of M = 8, it is found that the proposed BTS-MAP the conventional BTS-MAP achieve almost the same BER performance. This is due to the fact that for this scheme we have P Diff.5. The above-mentioned trends of these BTS-MAP schemes recorded for SM are also visible in Fig. 5, where (4 )-element SM systems are considered. Moreover, in Fig. 6, the performance of the proposed BTS- MAP is investigated in Nakagami-m fading channels. As shown in Fig. 6, the proposed scheme outperforms the conventional one in ( )-element MIMO channels having m =.5 m =.8. Since we have a higher P Diff for the case of m =.5, the corresponding BER gain is more attractive than that of m =.8. VI. Conclusions A novel BTS-MAP scheme was proposed for SM systems with the objective of increasing the HDS simultaneously reducing the average HDD. Based on the theoretical analysis of the MED distribution of SM constellations, a criterion was proposed for the construction of beneficial BTS-MAP scheme for a specific MIMO setup. The proposed mapping rule ehibits is attractive for employment in SM systems. For achieving a further improved BER performance, our further work will be focused on the integration of adaptive SM channel coding with the proposed scheme. References [] R. Mesleh, H. Haas, S. Sinanović, C. W. Ahn, S. Yun, Spatial modulation, IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 8-4, Jul. 8. In our simulations, the value of P Diff for m =.5 is approimately.7, while this value for m =.8 is about.55. Moreover, our proposed BTS-MAP can also be directly etended to the SM in conjunction with M-QAM modulation. 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