Constellation Design for Spatial Modulation

Transcription

1 Constellation Design for Spatial odulation ehdi aleki Department of Electrical Akron, Ohio Hamid Reza Bahrami Department of Electrical Akron, Ohio Ardalan Alizadeh Department of Electrical Akron, Ohio Abstract Conventional amplitude-phase modulations (APs) are designed such that the minimum Euclidean distance between constellation points is maximized. Unlike these modulations, the error performance of spatial modulation (S) is not only a function of the Euclidean distances but also the energy of each symbol, i.e. constellation point. Therefore, conventional AP schemes that are designed solely based on the notion of Euclidean distance are not necessarily suitable for S transmission. In this paper, the constellation design for S is investigated it is shown that, by a proper constellation design, a significant performance gain is obtained compared to the well-known phase sift keying (PSK) or quadrature amplitude modulation (QA). Index Terms multi-input multi-output; spatial modulation; amplitude-phase modulation. I. INTRODUCTION The advent of multi-input multi-output (IO) communication systems has enabled new modulation domains (e.g. space code) in addition to widely used conventional domains such as amplitude-phase (i.e. inphase-quadrature) or frequency domains [], []. Furthermore, these new domains have been used in conjunction with conventional domains to design more advanced modulation schemes. In this context, joint constellation design optimal constellation breakdown between different domains have emerged as interesting yet deming problems that call for efficient solutions [3]. Spatial modulation (S) is one of these new modulation schemes that utilizes the amplitude-phase space domains for data transmission. In S, the index of the active transmitting antenna along with the inphase-quadrature components of transmit signal are used to map the information [4] [6]. One interesting important question is the problem of optimal signal design for S. In [3], optimal constellation breakdown for S is studied, it is shown that at any fixed transmission rate, there exists an optimal amplitudephase modulation (AP) size as well as the number of transmit antennas that minimize the symbol error rate (SER). The optimal S breakdown for the two common types of modulations, i.e. phase shift keying (PSK) quadrature amplitude modulation (QA), has been derived. Nevertheless, as far as the optimal design of S is concerned, applying PSK QA schemes to S does not necessarily result in the minimum SER. In fact, it is shown that the union bound of the SER of S is the function of the Euclidean distance between the AP constellation points as well as the norm of the symbols [3], [5], [6]. This is in sharp contrast to the notion that the performance of a modulation scheme is mainly a function of the minimum Euclidean distance. Star-QA is a quadrature amplitude modulation scheme which allows differential encoding/decoding increases bwidth efficiency in Rayleigh fading channels [7], [8]. In [9], it is shown that space-time shift keying (STSK) using star- QA outperforms STSK with conventional AP schemes. In [], star-qa is considered as a suitable cidate for S, a low-complexity algorithm is introduced to design twin-ring star-qa to minimize the SER. In this paper, we consider the design of AP schemes that are suitable for S in SER sense. We consider the general problem of AP design for S without imposing a predefined structure on the constellation diagram. We set up an optimization problem,, by solving it numerically, show that a generalized star-qa scheme is indeed a good cidate for S. While restricting structures, such as two-ring 6 star-qa with equal number of constellation points in each ring, are considered for star- QA in the literature, we consider the more general problem without imposing any limitation on the number of rings or the composition of the constellation points in different rings. Although the AP design, in the most general case, is obtained numerically, the impact of different system parameters such as the number of transmit receive antennas on the structure of optimal AP scheme for S are mathematically analyzed. ore precisely, by analysis we show that in special cases of large number of transmit or receive antennas, the optimal constellation converges to the well known PSK square-qa (SQA), respectively. The remainder of the paper is organized as follows. In Section II, we introduce the system model for S transmission. In Section III, we present the optimization problem for AP design by mathematical analysis, we evaluate the effect of different system parameters on the proposed constellation design. In Section IV, we present some numerical results, finally, Section V concludes the letter. II. SPATIA ODUATION The received signal for an S system with N transmit receive antennas can be written as y = p Hs l x n + v, n {,...,N}, l {,...,} ()

2 is the signal-to-noise ratio (SNR), s l is an -ary AP symbol, v is the zero-mean complex additive white Gaussian noise vector with unitary covariance matrix, x n is the n-th SSK constellation vector with one nonzero element equal to, H is the N uncorrelated Gaussian IO channel matrix with unit variance entries. The union bound of the average SER of S at high SNRs can generally be expressed as [3] P (, N) P () = P () = l=, l = l=,l = l6=l h (N ) P ()+ P i () s l + s l s l s l () (3). (4) Note that (4) is related to the probability of error corresponding to the AP. In conventional modulations (i.e. when N =), (3) is absent in the union bound equation, therefore, only Euclidean distances between AP symbols affect the error performance of such modulation schemes. The impact of the P () term on the performance of S increases by increasing the number of transmit antennas [3]. As a result, in addition to the Euclidean distances between AP symbols, especially at large number of transmit antennas, the amplitudes of the AP symbols play an important role in the performance. III. AP CONSTEATION DESIGN FOR S In this section, we introduce the optimization problem to design APs for S based on minimizing the union bound of the probability of error in (). Assuming s l = r l exp(j l ), r l is the amplitude l is the phase of the symbol, (3) (4) can, respectively, be rewritten as P () = rl + rl (5) P () = l=,l = l 6=l l=, l = r l + r l ( "l,l cos( l l )) (6) " l,l = r lr l. Note that P rl () does not depend on the +r l phases of the symbols, while P () depends on the phases. For a fixed, the optimization problem can be expressed as n ˆd = arg min (N ) P ()+ P o () d=[r,...,r,,..., ] T : <r l < p, apple l < rl =. (7) l= The optimization problem in (7) indicates that the optimal AP scheme can vary for different number of transmit antennas. Intuitively, increasing the distances between the AP symbols as much as possible minimizes the second term in the right side of (7), while equalizing the amplitude of different symbols will result in the minimization of the first term. Finding a solution to (7) can be interpreted as making a balance between these two terms for different number of transmit antennas N. To set up a tractable optimization problem, we make some valid assumptions. Based on the symmetry of P () P () with respect to real imaginary parts of symbols, we conjecture that the optimal constellation diagram is symmetric along both inphase quadrature axes. By this assumption, we can modify the optimization problem as follows n ˆd = arg min (N ) P ()+ P o () d=[r,...,r,,..., ] T : <r l < p, apple l < / 4 rl =. (8) l= Also, P () P () can be rewritten as P () = 6 P () = l=,l = l=, l = r l + r l r l + r l (9) l,l () () x +x R (x) = +, x < () 8 < l,l () = R l,l + R + l,l, l 6= l : R +. () l,l +, l = l ± l,l = " l,l cos ( l ± l ). Reformulating (7) as (8) expedites the execution of the numerical optimization algorithm by reducing the number of variables by a factor of 4. The rest of the symbols can be simply obtained by s l+ = s l, s l+/ = s l, s l+3 = s l, l =,..., (3) (.) sts for the conjugate operation. While obtaining closed-form solutions for the optimization problem in (8) is very difficult, we solve it numerically for several cases in Section IV to find out suitable APs we show that depending on different parameters, such as the number of transmit receive antennas, the optimal AP constellation approximately resembles a star-qa constellation with different number of rings. In the sequel, we investigate the effect of different parameters on the structure of the optimal constellation diagram.

3 N = N = 6 N = 64 N = 4 = = = 3 Fig.. The proposed 64-ary constellation diagram of S for different number of transmit receive antennas. A. Effect of the number of transmit antennas As the number of transmit antennas (N) increases, the impact of the portion of the error related to spatial domain increases the optimal AP scheme for S diverges from conventional APs. For very large values of N, the first term in the right side of (7) is dominant the second term can be neglected to find the optimal amplitudes of different symbols. The second term can in (7), however, be considered to find the optimal phases. This can be concluded from the following Proposition. Proposition : Asymptotically for very large number of transmit antennas, PSK is the optimal AP scheme for S. Proof: For very large number of transmit antennas, the optimization problem in (7) can be approximated as [ˆr,...,ˆr ] T = arg min [r,...,r ] T : <r l < p P () rl = l= (4) hˆ,...,ˆ i T = arg min [,..., ] T : apple l < P ( r l =ˆr l ). (5) Solving (4) is much easier than (7), in contrast to (7), its closed-form solution can be obtained using agrange multiplier method. The agrangian function can be written as! = rl + rl + (6) l=, l = is the agrange multiplier. By taking the derivative of with respect to r l,l=,...,, setting it to zero, we have ˆr l +ˆr l =, l =,...,. (7) l = l= Therefore, for any l l (l 6= l ), we have ˆr l +ˆr l = ˆr l +ˆr l l = l = r l (8) which can be simplified as ( ) ˆr l ˆr Q ˆr l +ˆr l, ˆr l +ˆr l l = (9) ˆr l +ˆr + l ˆr l +ˆr + l l =

4 Q (x, y) = P i= x i y i. Since for any x, y > Q (x, y) is positive, ˆr l =,l=,...,, satisfies (9). Based on this fact to minimize P (), PSK is the optimal AP scheme for S with very large number of transmit antennas. B. Effect of the number of receive antennas Studying the impact of the number of receive antennas () is not as straight forward as that of the transmit antennas. This is because appears in as the exponent factor in P () P () in (5) (6), respectively. To analyze the effect of, we express the following Proposition. Proposition : By increasing the number of receive antennas, the ratio between P () P () increases. Proof: A direct rigorous proof of this Proposition tends to be very difficult. However, several numerical trials verify the statement. Here, we introduce lower upper bounds for the ratio of P () P (), prove that both are increasing functions of. Considering (9) (), for this ratio we can write P () P () = P l=,l = P h l=, l = r l + r l l,l () i (r l + r l ) () It is easy to show that for any positive series of {a i } {b i } (a i,b i ), we have! a i. b i apple b i () i= l=,l = i= l,l () A i= a i.b i apple i= a i. i= Considering () (), one can conclude + apple P () P () apple l=,l = l,l () + () As a property of R (x), we have ( ) x +x R +(x) R (x) =x > (3) Therefore, l,l ( + ) > l,l (), as the result both upper bound lower bound in () are increasing functions of. Based on Proposition, we can conclude that increasing the number of receive antennas () increases the dominance of P () over P (). This means that for large number of receive antennas, the optimal AP for S is very close to conventional APs such as QA. Therefore in this case, the design of new AP schemes is unnecessary. Note that as a conventional AP scheme such SQA has a more favourable minimum distance property compared to PSK, according to Proposition, at a small N, SQA is Symbol Error Rate 3 N = SNR (db) PSK QA Two ring star QA Proposed AP N = 56 Fig.. SER performance comparison of S with PSK, QA proposed AP schemes ( =6, =3). Symbol Error Rate 3 N = PSK QA Two ring star QA Proposed AP SNR (db) N=64 Fig. 3. SER performance comparison of S with PSK, QA proposed AP schemes ( =64, =). generally a better choice for S than PSK. While at a very large N, based on Proposition, PSK is a better choice. Therefore, we expect that by increasing the number of transmit antennas, the optimal AP constellation diagram gradually changes from a constellation close to SQA to PSK. IV. SIUATION RESUTS In this section, we provide some numerical results to investigate the AP design for S. We also simulate the error performance of the obtained AP schemes compare them with conventional modulation schemes. We first use ATAB Optimization Toolbox to find the solution to the optimization problem in (8). In Fig., the constellation diagrams of proposed APs with size = 64 are illustrated for different numbers of transmit receive antennas. As shown in Fig., for N =, the numerically derived AP is very close to SQA. However, as the number of transmit antennas increases, the proposed AP gradually diverges away from SQA toward PSK which is in

5 agreement with Proposition. Also, Fig. demonstrates the effect of the number of receive antennas on the AP constellation diagram which verifies the validity of Proposition. As seen from the figure, the proposed numerically derived AP does not generally resemble well-known modulation schemes. However, upon further reexamining the figure, it turns out that the proposed AP constellation for different cases is close to star-qa with varying number of rings number of constellation points in each ring. Therefore, one can conclude that star-qa can be considered as a suitable AP modulation scheme for S. Adopting a structured modulation such as star- QA has an added advantage of reduced detection complexity by using a constellation quantization (slicing) function [] to avoid exhaustive search over different constellation symbols. Note that unlike the star-qa proposed in [], the number of the constellation points in different rings of the proposed numerically derived star-qa is not constant. In fact, the outer rings include more constellation points. Also in [], a predetermined number of rings is considered for all the cases. However, as it is shown in Fig., as the number of transmit antennas increases, the number of rings in the proposed constellation decreases, asymptotically (N >>) converges to one, i.e. PSK constellation. We next compare the SER performance of S with proposed AP with that of S with PSK QA in Fig. for = 6, N = 56 = 3. As a benchmark, the performance of a single-antenna transmitter exploiting 64- QA 64-PSK is also shown. As shown in the figure, although conventional QA outperforms PSK in single-antenna systems, PSK is a better choice for S. In fact, as seen, the performance of S with proposed AP is every close to that of S with PSK in this case. This means that in this case, the number of transmit antennas is large enough (N = 56), S with PSK matches the performance of optimal S (see Proposition ). Fig. 3 shows the SER performance of S for = 64, N = 64 =. As illustrated in this figure, S with proposed AP outperforms S with QA PSK with db 5dB, respectively. On the other h, unlike the previous case, QA turns out to be a better constellation for S compared to PSK. As a benchmark, the performance of single-antenna 6- QA 6-PSK modulations is also shown. Interestingly, the performance of single-antenna PSK is almost the same as that of S with PSK. This is because, in this case, the error related to AP is dominant to the error related to space. the numerically derived constellation closely resembles a star- QA with different number of rings. With simulations, we showed that the SER performance of S with the proposed AP is superior to that of S with QA or PSK. REFERENCES [] J. Jeganathan, A. Ghrayeb,. Szczecinski, A. Ceron, Space shift keying modulation for IO channels, IEEE Trans. Wireless Commun., vol. 8, no. 7, pp , July 9. [] S. Sugiura, S. Chen,. 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Webb, Quadrature Amplitude odulation: From Basics to Adaptive Trellis-Coded, Turbo-Equalised Space-Time Coded OFD, CDA C-CDA Systems. Wiley- IEEE Press, 4. [9] S. Sugiura, S.. N. C. u,,. Hanzo, Reduced-complexity iterative-detection aided generalized space-time shift keying, IEEE Trans. Veh. Technol., vol. 6, no. 8, pp , Oct.. [] P. Yang, Y. iao, B. Zhang, S. i,. El-Hajjar,. Hanzo, Star- QA signaling constellations for spatial modulation, IEEE Trans. Veh. Technol., to appear, 4. [] R. Y. esleh, H. Haas, S. Sinanovic, C. W. Ahn, S. Yun, Spatial modulation, IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 8 4, July 8. V. CONCUSION We investigated the design of AP schemes for S. An optimization problem was first formulated, the effect of the number of transmit receive antennas on the structure of the optimal constellation was studied analytically. We showed that in two extreme cases of large number of transmit receive antennas, the optimal constellation converges to SQA PSK, respectively. The optimization problem was then solved numerically, the AP constellations were derived for different cases. It was argued that, in many cases,