Energy-Efficient M-QAM Precoder Design with Spatial Peak Power Minimization for MIMO Directional Modulation Transceivers

Transcription

1 1 Energy-Efficient M-QAM Precoder Design ith Spatial Peak Poer Minimization for MIMO Directional Modulation Transceivers Ashkan Kalantari, Christos Tsinos, Mojtaba Soltanalian, Symeon Chatzinotas, Wing-Kin Ma, and Björn Ottersten arxiv: v1 [cs.it] Feb 17 Abstract Spectrally efficient multi-antenna ireless communications is a key challenge as service demands continue to increase. At the same time, poering up radio access netorks increases CO footprint. Hence, for an efficient radio access design, e design a directional modulation precoder for M- QAM modulation ith M = 4, 8, 16,. First, extended detection regions are defined in these constellations using analytical geometry. Then, constellation points are placed in the optimal positions of these regions hile the imum Euclidean distance to neighbor constellation points and detection region boundary is kept as in the conventional M-QAM modulation. For further energy-efficiency, relaxed detection regions are modeled for inner points of M = 16, constellations. The modeled extended and relaxed detection regions as ell as the modulation characteristics are utilized to formulate convex symbol-level precoder design problems for directional modulation to imize the transmission poer hile preserving the imum required SNR at the destination. In addition, the extended and relaxed detection regions are used for precoder design to imize the output of each poer amplifier. Results sho that compared to the benchmark schemes, the proposed methods perform better in terms of poer and peak poer reduction as ell as symbol error rate reduction for a long range of SNRs. Keyords Directional modulation, energy efficiency, extended detection region, spatial peak poer, M-QAM modulation, symbollevel precoding. I. INTRODUCTION According to Cisco s prediction [1], mobile data traffic, primarily driven by video demand, ill increase eleven-fold from 1 to. The extreme groth of video content on the Internet, the advent of mobile devices, e.g., smart phones and tablets, and the market appetite for them are the most important elements hich have contributed to the tremendous surge in mobile traffic. Conventional approaches such as orthogonal frequency division multiplexing access and time division multiplexing access [], [] are used to utilize the frequency and time resources in order to improve the data communication This ork as supported by the National Research Fund (FNR of Luxembourg under AFR grant for the project SeMIGod. Ashkan Kalantari, Christos Tsinos, Symeon Chatzinotas, and Björn Ottersten are ith the Interdisciplinary Centre for Security, Reliability and Trust (SnT, The University of Luxembourg, 9, avenue JF Kennedy, L-18 Luxembourg-Kirchberg, Luxembourg, ( s: {ashkan.kalantari,christos.tsinos,symeon.chatzinotas,bjorn.ottersten}@uni.lu. Mojtaba Soltanalian is ith the Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 667, (msol@uic.edu. Wing-Kin Ma is ith the Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong, China ( kma@ieee.org. rate. To use the spatial dimension, Multiple-input Multipleoutput (MIMO communication systems emerged, e.g., Long Term Evolution (LTE and WiFi technologies. MIMO systems provide spatial degree of freedom in the design at the expense of interference among the transmitted data streams. In order to better utilize the available degrees of freedom, pre/post-processing at the transmitter and/or receiver ends are employed [4], [] to reduce the interference among the data streams. Recently, there has been a groing interest in directional modulation [6] [11] and symbol-level precoding for constructive interference [1] [1] techniques to mitigate interference in MIMO communication systems. In directional modulation, the channel realization and the symbols are used to design the the antenna eights. These eights are designed such that the Radio Frequency (RF signals get modulated after passing through the channel and result in communicating the desired symbol at the desired direction (antenna. Depending on the design, this can result in no interference [9], [11] or limited interference [7], [1] among the communicated data streams. Both digital symbol-level precoding and directional modulation focus on multiplexing gain in MIMO communication systems. Directional modulation and digital symbol-level precoding for constructive interference differ in the folloing ay. The former focuses on applying array eights in the analog domain to have the desired amplitude and phase for the received signals, hile the latter uses symbol-level precoding for digital signal design at the transmitter to create constructive interference at the receiver. Apart from an increasing data demand, ireless communications consume a large amount of energy and have a considerable share in environmental pollution [16]. Not only reducing the energy consumed in the radio access netorks is environmental friendly, but it also decreases the communication costs for both the operators and users [17]. The research orks in [9], [1], [1], [18], [19] study the relaxed design in constructive interference and directional modulation ith the goal of reducing the energy consumption at the transmitter. Although directional modulation offers transmission ith no or limited interference as ell as energy-efficiency, the hardare limitations at the transmitter need to be considered in the precoder design process. Among the hardare limitations, e focus on keeping the poer amplifier outside its saturation region. In the architecture of a directional modulation transmitter, the RF oscillator signal is equally divided among the RF chains [6] [9]. Each poer amplifier needs to operate ithin

2 a specific range to avoid nonlinear distortion of the amplified signal []. To avoid distortion in the amplified signal, e need to design the antenna eights such that the output of each poer amplifier does not go into the saturation region. In this direction, the references [1], [] consider constant envelope precoding for a single-user massive MIMO system. Hoever, the design condition of a constant envelop design is restrictive and results in a conservative design. The authors in [] consider a lo peak poer to average ratio design based on constructive interference for M-PSK modulation to limit the amplifier output poer belo a specific value here there is a strict constraint on the phase of the received symbols. A relaxed lo peak poer design for M-PSK modulation in order to include both energy efficiency and hardare limitation is proposed in [1]. A. Contributions and Main Results Based on the above descriptions, e tackle the design of a system hich jointly takes into account the user demand, poer amplifier saturation as a hardare limitation, and energy efficiency hen finite-alphabet input is considered. The designed precoders for Gaussian input signals can be used to precode finite-alphabet inputs, hoever, this may result in considerable system performance reduction [4]. The orks of [9], [1], [1], [1], [18], [19] focus on M- PSK precoder design. Hoever, there is no ork on designing the M-QAM directional modulation precoder for M = 4, 8, 16,, hile jointly utilizing the extended and relaxed detection regions of the constellation as ell as controlling the poer of the amplifier output signal to avoid saturation. To pursuit this design, e use the concept of directional modulation to jointly address the data rate and energy-efficiency issues of ireless communications hile considering the limit on poer amplifier outputs. In the directional modulation, instead of producing the symbols at the transmitter and sending them, the channel state information and the symbols to be communicated are used to produce the phase and amplitude of each transmitter RF chain. Then, the transmitted RF signals are modulated after passing through the channel and create the required symbols on the intended antennas. Depending on the design, directional modulation can induce exact symbols, hich can be translated into the interference-free communication, or induce a close value (amplitude and phase to the symbols, hich can be translated into communication ith interference, on the receiving antennas. In this paper, e bring the folloing contributions: 1 We define the extended detection regions of M-QAM modulation for M = 4, 8, 16, and model these regions using analytical geometry. By extended detection region, e mean a region in hich a constellation point can be placed, given that it preserves the standard distance ith the neighbor constellation points and the detection boundaries. Here, the Euclidean distance among the constellation points in conventional M- QAM is considered as the standard distance. The ork of [1] considers symbol-level precoder design for M = 16. In addition to the extended detection regions, e characterize relaxed detection regions here the inner points of M = 16, constellations can be placed in a region rather than being fixed. Using the extended detection region, e can setup a trade-off beteen further energy-efficiency at the transmitter and symbol error rate (SER at the receiver. The research [11] performs directional modulation precoder design for M = 4, 8, 16, constellations by only using extended detection region ithout considering peak poer imization. The ork of [1] does not design the precoder hen relaxed detection region is considered for inner point of M = 16 constellations. We design the optimal M-QAM directional modulation precoder using the characterized extended detection regions for M = 4, 8, 16, to imize the transmission poer hile satisfying the required SNR at the antennas of the receiver. Furthermore, e re-design the optimal precoder for M = 16, constellations hile considering the relaxed detection region for inner constellation points in addition to the extended detection regions for outer constellation points and investigate the energyefficiency and SER. 4 We design the optimal precoder to imize the instantaneous peak poer of the amplifier output signal for each RF chain of the directional modulation transmitter. We refer to this as spatial peak poer imization defined as P spatial max = max k=1,...,n t H E k. (1 We carry out this design using the characterized extended detection regions for M = 4, 8, 16, hile preserving the required SNR at the antennas of the receiver. Also, e repeat the precoder design hile considering both relaxed and extended detection regions for M = 16,. In [], the spatial peak poer imization is carried out for M-PSK modulation ithout considering extended and relaxed detection regions. Through extensive simulations, e reveal the benefits of using extended detection regions for M-QAM directional modulation transmission in terms of poer reduction at the transmitter and SER improvement at the receiver. As a hardare limitation, e evaluate the performance in terms of the mentioned metrics hen imizing the instantaneous spatial peak poer. In addition, e quantify the trade-off beteen poer consumption and SER hen relaxed detection regions are used hile this is not investigated in [1]. The design for M = 4, 8 translates into interference-free communication since the symbols keep the standard Euclidean distance. In addition, the precoder design hen considering fixed inner constellation points for M = 16, translates into the interference-free communication. On the other hand, the precoder design hen considering relaxed inner constellation points for M = 16, translates into the communication ith interference.

3 It is orth mentioning that the precoder design for 18- QAM is similar to -QAM. Furthermore, the precoder design for 64-QAM and 6-QAM are similar to 16-QAM. Hence, e consider precoding design for M = 4, 8, 16, constellation points in this ork. From practical point of vie, M-QAM modulations up to 64-QAM modulation are used in the long term evolution (LTE standard []. B. Paper Organization The remainder of this paper is organized as follos. In Section II, e introduce the signal and system model. The extended and relaxed detection regions are defined and modeled in Section III. In Section IV, the optimal M-QAM precoder design problems for total and spatial peak poer imization are formulated and transformed into standard convex forms. In Section V, e evaluate the proposed methods and compare them versus the benchmark scheme through simulations. At the end, e mention the conclusions in Section VI. C. Notation Upper-case and loer-case bold-faced letters are used to denote matrices and column vectors, respectively. The superscripts ( T, (, ( H, and ( represent transpose, conjugate, Hermitian, and Moore-Penrose pseudo inverse operators, respectively. I N N denotes an N by N identity matrix, E k has one unit-valued element on the k-th diagonal entry ith the rest of the elements being zero, Ẽk has to unit-valued elements on the k-th and (N t +k-th diagonal entries ith the rest of the elements being zero, diag(a denotes a diagonal matrix here the elements of the vector a are its diagonal entries, a b is the element-ise Hadamard product, is the all zero vector, is the Frobenius norm, and represents the absolute value of a scalar. Re (, Im (, and arg ( represent the real valued part, imaginary valued part, and angle of a complex number, respectively. Card( shos the number of set members. II. SIGNAL AND SYSTEM MODEL Let us consider a directional modulation transmitter, denoted by T, having N t antennas that communicates ith a receiver, denoted by R, equipped by N r antennas using M-QAM modulation here M = 4, 8, 16,. The received signal, y, at R is y = H + n, ( here y is an N r 1 vector denoting the received signals by R, H = [h 1,..., h n,..., h Nr ] T is an N r N t matrix denoting the channel from T to R, h n is an N t 1 vector containing the channel coefficients from all the transmitter antennas to the n- th antenna of R, and is the vector containing the eights of radio frequency (RF chains hich is the design variable in this ork. The random variable n CN (, σ I Nr N r denotes the additive hite Gaussian noise at R, here CN denotes a complex and circularly symmetric random variable. The vector s = [s 1,..., s n,..., s Nr ] contains the M -QAM symbols to be communicated beteen T and R using directional modulation [ ] T technology, the elements of H = s 1,..., s n,..., s N r are the induced M -QAM symbols on the antennas of R here s n is the induced M -QAM symbol on the n-th antenna of R. To detect the symbols, R can apply conventional detectors on each receiving antenna. In this ork, a single carrier is used to communicate symbols over a narro band channel. Since the transmit precoder is designed such that the received signals have the desired amplitude and phase, a simple multiple-antenna receiver is considered hich does not perform processing on the received signals. In the next section, e characterize the extended and relaxed regions of the mentioned M-QAM constellations. III. CHARACTERIZATION OF EXTENDED AND RELAXED DETECTION REGIONS In this part, e geometrically characterize the extended detection regions for M = 4, 8, 16, constellations and relaxed detection regions for M = 16, constellations. To do so, e derive analytical expressions hich concisely describe these extended and relaxed detection regions. The extended detection regions are shon by solid areas and dashed lines in Fig. 1. As one can see, the extended detection region is an area in hich the constellation point can be placed hile keeping the distance to other constellations points more than or equal to the standard distance. Here, the Euclidean distance among the constellation points of conventional M- QAM modulation is considered as the standard distance. We divide each constellation into multiple sets, as illustrated in Figures 1(a to 1(e, and continue to analytically model the extended and relaxed detection region of each set. A. The Case of M = 4 Consider s n as a symbol to be communicated ith the n-th antenna of the receiver receiver. Considering that the received signal on the n-th antenna of the receiver is h T n, the extended detection region for s n s 14 in the first quadrant of Fig. 1(a can be modeled as Re ( h T n γre (s n, Im ( h T n γim (s n, ( here γ is the imum required amplification for the induced symbol at the receiving antennas. The value of γ is derived from the required SNR value in db at the receiver as γ = 1 SNR/1. The extended detection region defined in ( cannot be used in other quadrants due to change in the sign of real and imaginary parts of s n. Hoever, e can multiply both sides of the equations by Re (s n and Im (s n to make both sides positive. Note that the direction of inequalities does not change since Re (s n and Im (s n have the same sign as Re ( h T n and Im ( h T n, respectively, at the optimal point of precoder design. Hence, the extended detection region for any 4-QAM symbol, s n s 14, in Fig. 1(a can be geometrically characterized using the folloing general expression: Re (s n Re ( h T n γre (s n, Im (s n Im ( h T n γim (s n. (4

4 4 Quadrature Amplitude s 14 s 14 s 14 s 14 s Quadrature Amplitude s 18 s 18 s s 4 s s 18 s Quadrature Amplitude s s 4 s 14 1 s In-phase Amplitude (a Extended detection regions for 4-QAM constellation. Quadrature Amplitude 7 1 d In-phase Amplitude (d Extended and relaxed detection regions for 16- QAM constellation. s s 4 s 14 1 s In-phase Amplitude (b Extended detection regions for 8-QAM constellation. Quadrature-phase amplitude s 6 7 6s 1 s s s s s s 8 s 9 1 s 19 s 18 1 s In-phase amplitude (e Extended detection regions for -QAM constellation. s - - In-phase Amplitude (c Extended detection regions for 16-QAM constellation. Extended detection regions Quadrature-phase amplitude s 6 7 6s 1 s s s s s s 8 s 9 1 s 19 s 18 1 s In-phase amplitude (f Extended and relaxed detection regions for -QAM constellation. Fig. 1: Characterization of extended and relaxed detection regions for M-QAM constellations. The numbers sho the decimal equivalent of the gray code for each symbol. The extended detection regions are shon by dashed lines and solid regions for M = 4, 8, 16,. s B. The Case of M = 8 The extended detection region of s n s 18 for 8-QAM constellation in Fig. 1(b can be characterized in the same ay as (4. Next, the upper and loer sides of s n s can be characterized, respectively, as Re ( h T n = γre (s n, Im ( h T n γim (s n, ( Re ( h T n = γre (s n, Im ( h T n γim (s n. (6 The characterizations of s in ( and (6 can be fused to get a unified expression hich describes the points in the set s as C. The Case of M = 16 Re ( h T n = γre (s n, Im (s n Im (h n γim (s n. (7 For s n of 16-QAM constellation in Fig. 1(c, the extended detection region is characterized using (4. In addition, for s n s in Fig. 1(c, the characterization is the same as (7. The right-hand side and left-hand side extended detection regions of s n can be characterized, respectively, as Re ( h T n γre (s n, Im ( h T n = γim (s n, (8 Re ( h T n γre (s n, Im ( h T n = γim (s n, (9 hich can be compressed into Re (s n Re ( h T n γre (s n, Im ( h T n = γim (s n. (1 In the case s n s 4, the points can be characterized in the folloing to ays. 1 Fixed detection region characterization for s n s 4 : In this approach, the points of s n s 4 are considered in their on place. This satisfies the imum standard distance beteen the constellation points and does not increase the SER compared to conventional 16-QAM. Hoever, this modeling

5 does not improve the energy-efficiency. Accordingly, the constellation points s n s 4 can be modeled as Re ( h T n = γre (s n, Im ( h T n = γim (s n. (11 Relaxed detection region characterization for s n s 4 : In another approach, the points of s n s 4 can be considered to be placed ithin a square-shaped detection region. This approach is illustrated in Fig. 1(d ith gray squares. The relaxed detection region design results in a further energyefficient transmitter, hoever, it increases the SER at the receiver since the imum Euclidean distance beteen the received constellation points as in conventional M-QAM does not hold anymore. The related expressions to describe the relaxed detection region are γre (sn d Re ( h T n γre (s n + d, γre (sn d Im ( h T n γim (s n + d, (1 here d is the Euclidean distance beteen the edge of the relaxed detection region and the constellation point as shon in Fig. 1(d. Choosing the proper value of d depends on the target metric such as total consumed poer and SER. As an approach to derive the optimal value of d, e can perform a 1D search over d to maximize the goodput [6] over the total consumed poer, hich is defined as follos: η = R (1 SER, (1 here R is the bit per symbol, and is the total consumed poer at the transmitter. We ill quantify the metric η in Section V and find its optimal value. D. The Case of M = The extended detection regions for s n s and s n s 6 in the first quadrant of Fig. 1(e are characterized as γ Im ( h T n Re ( h T n γ, (14 Re ( h T n + γ Im ( h T n, Re ( h T n γ. (1 To model the extended detection region for s n s and s n s 6 in the other quadrants of Fig. 1(e, e can rotate them so that they fall ithin s n s and s n s 6 in the first quadrant. Then, e can use the the developed expressions for the first quadrant points in (14 and (1 to characterize the extended detection region for s n s and s n s 6 hich are are not in the first quadrant. The extended detection regions of s n s and s n are modeled similarly as in (7 and (1, respectively. For s n s 4, the fixed points are modeled as (11 and the relaxed points are modeled as (1. The relaxed detection regions for s n s 4 of -QAM modulation are shon in Fig. 1(f using gray squares. In the next section, e design the optimal symbol-level precoders for M-QAM directional modulation transmitter. IV. SYMBOL-LEVEL PRECODER DESIGN FOR DIRECTIONAL MODULATION In this part, e design optimal M-QAM directional modulation precoders using the developed characterized extended and relaxed detection regions of Section III. To this end, first, e formulate optimal precoder design problems in Section IV-A aig to imize the transmitter poer hile satisfying the SNR constraints at the receiver antennas. Second, e consider imizing the output poer of the amplifier signal in each transmitter RF chain in Section IV-B hile satisfying the required SNR at the receiver antennas. We refer to this as spatial peak poer imization. A. Optimal Precoder Design: Transmit Poer Minimization In this part, e formulate and design the optimal M-QAM MIMO directional modulation precoder hen the objective is to imize the total transmission poer hile satisfying the required SNR at the receiving antennas. The design problem for 4-PSK case can be ritten as s.t. Re s n1 Re (h T n 1 γre s n1, sn1 s 14 Im s n1 Im (h T n 1 γim s n1. (16 We can cast the optimal precoder design for 8-QAM modulation as s.t. Re s n1 Re (h T n 1 γre s n1, sn1 s 18 (17a Im s n1 Im (h T n 1 γim s n1, (17b Re h T n = γre s n, sn s (17c Im ( s n Im ( hn γim ( s n. (17d Next, e can formulate the optimal precoder design problem for 16-QAM as follos s.t. Re s n1 Re (h T n 1 γre s n1, sn1 (18a Im s n1 Im (h T n 1 γim s n1, (18b Re h T n = γre s n, sn s (18c Im ( s n Im hn γim s n, (18d Re s n Re (h T n γre s n, sn (18e Im h T n = γim s n. (18f Re h T n 4 = γre s n4, sn4 s 4 (18g Im h T n 4 = γim s n4. (18h

6 6 In the case of relaxed detection region design for inner points, the constraints (18g and (18h are replaced by the constraints in (1. Finally, the optimal precoder design problem for - QAM constellation is defined as s.t. Re h T n = γre s n, sn s (19a Im ( s n Im hn γim s n, (19b Re s n Re (h T n γre s n, sn (19c Im h T n = γim s n, (19d Re h T n 4 = γre s n4, sn4 s 4 (19e Im h T n 4 = γim s n4, (19f γ Im ( ht n Re ( ht n γ, s n s (19g Re ( ht n6 + γ Im ( ht n6, s n6 s 6 (19h Re ( ht n6 γ, (19i here h T n = h T n e iϕn, h T n 6 = h T n 6 e iϕn 6, ϕ n is the phase difference beteen s, shon in Fig. 1(e, and s n s, ϕ n6 is the phase difference beteen s, shon in Fig. 1(e, and s n6 s 6. In the case of relaxed design for inner constellation points, e can replace the constraints (19e and (19f by the constraints in (1. Here, e proceed ith transforg the -QAM precoder design in (19 into a standard convex form hen considering fixed and relaxed detection regions for the points s n4 s 4. A similar approach can be applied to the design problems in (16 to (18 in order to transform them into standard convex forms. After applying a series of algebraic operations on (19, hich are shon in Appendix A, the simplified -QAM design problem can be cast as s.t. A a, B = b, ( here A, a, B, and b are defined in (1. In the case of relaxed detection region design for the points s n4 s 4, e can reach a problem ith the same format as in ( here A, a, B and b are defied as in (. As e see, ( is a convex linearly constrained quadratic programg problem and can be solved efficiently using convex optimization techniques. B. Optimal Precoder Design: Spatial Peak Poer Minimization In this part, e design the optimal M-QAM directional modulation precoders for M = 4, 8, 16, constellations aig at keeping the poer output of each poer amplifier as lo as possible. We pursuit this design hile satisfying the required SNR at the receiving antennas. Note that in another approach e can imize the transmission poer hile keeping the output poer of each RF chain belo a specific value and satisfying the required SNR at the receiving antennas. The optimal spatial peak poer imization precoder design problem for 4-QAM modulation is ritten as max H E k k=1,...,n t s.t. Ω M QAM, (1 here Ω M QAM is the constellation-specific constraint set hich can be found for M = 4, 8, 16, in (16, (17, (18, and (19, respectively. In the case of relaxed detection region design for s n4 s 4, the related constraints, defined in (18 and (19, are replaced by the constraints in (1. Here, e proceed ith transforg (1 for M = into a standard form. A similar approach can be applied to the design problems for M = 4, 8, 16. First, e move the objective of (1 to the constraint by introducing the auxiliary variable t as t s.t. H E k t, k = 1,..., N t Ω M QAM. ( To simplify ( for M =, e can apply a similar process used in Appendix A for (19 to get s.t. t T Ẽ k t, k = 1,..., N t A a, B = b, ( here A, a, B and b are as in (1. As e see, ( is a convex problem hich can be solved efficiently. If e consider the relaxed detection region design for s n4 s 4, e get a similar problem as in ( here A, a, B, and b are the same as (. V. SIMULATION RESULTS In this section, e demonstrate the performance of the proposed methods and compare them ith the benchmark schemes. We use average over various designed precoders to measure the performance metrics: total poer consumption, maximum spatial peak poer among RF chains, SER, and bit error rate (BER. Each designed precoder in the proposed method is used to communicate N r symbols. In all simulations, channels are considered to be quasi static block Rayleigh fading generated as i.i.d. complex Gaussian random variables ith distribution CN (, 1 and remain fixed during the communication of a group of N r M-QAM symbols. Also, the noise is generated using i.i.d. complex Gaussian random variables ith distribution CN (, σ. We assume adaptive coding and modulation in the simulation scenarios and consider specific SNR range in hich each modulation order operates. To save space, e use the acronyms DM, Opt LP, and PPM in the legend of the figures instead of the terms directional modulation, Optimal linear precoding method of [7], and spatial peak poer imization, respectively. In the folloing, e first mention the benchmark schemes and then proceed ith the simulation scenarios.

7 (a SNR = 1 db (b SNR = db. Fig. : Average total consumed poer ith respect to N t for the proposed M-QAM directional modulation precoding and the benchmark schemes hen total and spatial peak poer imization designs are considered ith N r = (a SNR = 1 db (b SNR = db. Fig. : Average maximum peak poer among the RF transmit chains ith respect to N t for the proposed M-QAM directional modulation precoding and the benchmark schemes hen total and spatial peak poer imization designs are considered ith N r = 1. A. Benchmark Schemes In this part, e mention the zero-forcing [8] at the transmitter, and optimal linear precoding [7] as the comparison benchmark schemes. 1 ZF: We consider zero-forcing (ZF at the transmitter [9] as one of the benchmark schemes since both directional modulation and ZF use the CSI knoledge at the transmitter to design the precoder and ZF results in interference-free MIMO communication. In the benchmark scheme, e apply the ZF precoder at the transmitter to remove the interference among the transmitted symbol streams. After applying ZF, the received signal at R is y = HWs + n, (4 here W = H H( HH H 1 is the precoding vector and vector s contains the symbols to be transmitted. Optimal linear precoding: The optimal linear precoding design problem using channel state information can be ritten as [7] 1,..., Nr N r i=1 i

8 (a (b Fig. 4: Average total consumed and spatial peak poers ith respect to the required SNR for the proposed M-QAM directional modulation precoding and the benchmark schemes hen N t = N r = Fig. : Average consumed poer ith respect to N t for the proposed M-QAM directional modulation precoding scheme ith total poer imization hen fixed and relaxed, d = 1, detection region designs for s n4 s 4 are considered ith N r = 1 and SNR = db. 1 1 Fig. 6: Average consumed poer ith respect to the required SNR at the receiver for the proposed M-QAM directional modulation precoding and the benchmark schemes hen N t = 11, N r = 1, and d =.4. s.t. N r j k h T k k h T j j + σ γ, k = 1,..., N r. ( The precoder design problem in ( can be solved using semidefinite programg and rank-one relaxation. If the solution to ( happens not to be rank-one, randomization can be used to derive a rank-one solution []. The instantaneous spatial peak poer imization version of (, can be cast as 1,..., Nr,t t s.t. 1 E i Nr E i Nr t i, i = 1,..., N t, h T k k N r j k h T j j + σ γ, k = 1,..., N r, (6 here t = [t 1,..., t Nr ]. A similar approach as in ( can

9 Fig. 7: Average SER ith respect to the required SNR at the receiver for the proposed M-QAM directional modulation precoding and the benchmark schemes hen N t = N r = Fig. 9: Average SER ith respect to the required SNR at the receiver for the proposed M-QAM directional modulation precoding hen fixed and relaxed, d =.4, detection region designs for s n4 s 4 are considered ith N t = N r = Fig. 8: Average bit error rate ith respect to the required SNR at the receiver for the proposed M-QAM directional modulation precoding ith and ithout LDPC forard error correction code hen code rate is /6 and N t = N r = Fig. 1: Average goodput over consumed poer ith respect to d for the proposed 16 and -QAM directional modulation precoding schemes hen N t = N r = 1 and SNR = 16 db. be used to solve (6. The spatial peak poer is imized in (6 over the precoding vectors i, hoever, in optimal linear precoding, the precoding vectors i ill be multiplied by the M-QAM symbols, summed as y t = N r i=1 is i, and transmitted. As a result, this may change the transmission poer of the antennas. To effectively imize the maximum element of the transmit signal y t, e need to imize each element of = N r i=1 i since the elements of each i are multiplied by the symbols and then summed up. B. Simulation Scenarios For the first scenario, e measure the transmitter s average consumed poer and spatial peak poer for the proposed M-QAM directional modulation as ell as the benchmark schemes ith respect to transmitter s number of antennas, N t. The average total consumed poer of the proposed and benchmark schemes versus N t are shon in Fig. for specific system parameters. As e see, the proposed 4, 8, 16, and -QAM directional modulation precoder designs ith poer imization consume considerably less poer than the ZF

10 1 1 1 Quadrature-phase amplitude Quadrature-phase amplitude In-phase amplitude (a In-phase amplitude (b Fig. 11: The constellation of the induced symbols at the receiver for the proposed -QAM directional modulation precoding ith N t = N r = and γ = hen fixed detection, Fig. 11(b and relaxed, Fig. 11(a, ith d =.1 region design of s n4 s 4 is considered. and optimal linear precoding schemes for specific range of N t, especially for close values of N t and N r. As the modulation order decreases, the difference beteen the consumed poer by the proposed and the benchmark schemes increases. For instance hen N t = N r = 1 and SNR = 1 db, 4-QAM and 8-QAM are 6.7 db,.7 belo the optimal linear precoding benchmark, also 16-QAM and -QAM are 6.98 db and 4.4 db belo the benchmark scheme for SNR = db. The reason is that in contrast to the conventional ZF and optimal linear precoding, directional modulation takes advantage of the available detection regions of the M-QAM constellation by symbol-level precoding. This lets the symbols be placed in the optimal location of the defined regions hile satisfying the required SNR at the receiving antennas. The MIMO communication systems usually operate in square mode, i.e., equal number of transmit and receive antenna, since the multiplexing gain of the system is the imum of transmit and receive antennas. Hence, the proposed scheme is a good candidate for MIMO systems since it provides the highest gain compared to optimal linear precoding for N t = N r, as Fig. shos. We investigate the spatial peak poer in the second scenario. The average spatial peak poer ith respect to N t is shon in Fig.. Average spatial peak poer ith respect to N t are presented in Fig.. The first observation is that as the number of transmit antennas, N t, increases, the transmitter ability to reduce the maximum output poer among the poer amplifiers increases. The second observation is that for loer modulation orders, the transmitter ith spatial peak poer imization is more capable of reducing the level of the poer amplifier signal hile satisfying the SNR requirements at the receiving antennas. It is seen that the benchmark schemes result in a higher spatial peak poer compared to the proposed schemes. To analyze the effect of relaxed detection region for s n4 s 4 of 16-QAM and -QAM constellations introduced in Section III, e have shon the average consumed poer of the relaxed design ith respect to N t in Fig.. As it is illustrated, the relaxed detection region design for s n4 s 4 results in a loer poer consumption. Interestingly, as N t increases, the average consumed poer of -QAM gets close to that of 16- QAM in a loer N t in the relaxed detection region design case compared to the fixed design. In the next scenario, e measure the average consumed poer, average maximum spatial peak poer, and the average symbol error rate ith respect to the required SNR at the receiver. The average consumed poer ith respect to the required SNR is shon in Fig. 4(a for N t = N r = 1. As it is observed, the consumed poer increases consistently ith respect to the required SNR. In relatively middle and high SNR regimes, the proposed scheme results in a loer poer consumption compared to the optimal linear precoding. The average maximum spatial peak poer ith respect to SNR is shon in Fig. 4(b. As it is seen, directional modulation results in a loer spatial peak poer compared to the optimal linear precoding in relatively middle and high SNR regimes. To study the effect of relaxed detection region design for s n4 s 4 ith respect to SNR, the average total consumed poer ith respect to SNR is presented in Fig. 6 for N t = 11 and N r = 1. The results shos that the relaxed design results in loer poer consumption in lo SNR regime. As SNR increases, the consumed poers by the fixed and relaxed designs converge. This is due to the fact that as the required SNR at the destination increases, the relaxed detection region gets relatively smaller compared to the required SNR and consequently the gain of relaxed design fades out. It is seen that -QAM ith relaxed design consumes loer poer than 16-QAM in lo SNR since -QAM has four times more constellation points ith relaxed design compared to 16-QAM.

11 11 The average SER ith respect to the required SNR is shon in Fig. 7. It is observed that the SER of ZF is close to our scheme since it totally neutralize the interference in the extended detection region design. On the other hand, the optimal linear precoding results in a higher SER compared to the proposed schemes, especially for M = 8, 16,, since it does not fully mitigate the interference. In addition, the difference in the SER of the proposed method and optimal linear precoding goes higher as the SNR increases. The effect of lo density parity check code (LDPC on BER of the proposed scheme is shon in Fig. 8. Next, e investigate the effect of the relaxed detection region on SER. The average SER ith respect to SNR is shon in Fig. 9. The relaxed detection region design increases the SER in relatively lo SNR regime, hoever, the SER gets close to the fixed design as SNR increases. As as shon, applying the relaxed detection region results in both poer reduction and SER increment. To figure out the optimal value for relaxation, d, e need to consider a metric hich captures both poer consumption and SER. To do so, let us consider the goodput over the total consumed poer, mentioned in (1, as the performance metric. The value of η ith respect to d is presented in Fig. 1. It is observed that there exists a value of d for both 16 and -QAM here it is possible to setup an optimal trade-off beteen poer consumption and SER. At the end, e present an example of the communicated -QAM constellation points at the receiver for both relaxed and fixed detection region designs of s n s 4 in Figures 11(a and 11(b. As e see, many symbols go above the required signal level at the destination, hich can results in loer SER. Particularly, e see that the optimally precoded points in the sets s and s 6 are concentrated at the boundary of the extended detection regions. VI. CONCLUSIONS In this paper, using the concepts of extended and relaxed detection regions, e formulated the optimal directional modulation precoder design problems for 4,8,16,-QAM constellation points ith total and spatial peak poer imization, to reduce the poer amplifier output in each RF chain, objectives and transformed these problems into standard convex forms. Through directional modulation precoding, symbols are placed in the optimal location of extended and relaxed detection region. In the simulation results, e shoed that the suggested M-QAM directional modulation precoding consumes less poer than the conventional ZF in all range of SNR, and less poer than the optimal linear precoding for relatively middle and high SNR regimes. We demonstrated that this difference in poer consumption increases as the modulation order decreases. Also, the results shoed that directional modulation results in loer spatial peak poer in all SNR ranges compared to ZF as ell as loer spatial peak poer compared to optimal linear precoding in middle and high SNR regimes. The results shoed that the difference beteen the average spatial peak poer of the proposed and the benchmark schemes increases as the modulation order decreases. In addition, e sa that the interference-free communication capability of directional modulation, considering fixed design for inner constellation points of M = 16,, results in loer SER at the receiver compared to optimal linear precoding. This difference in SER increases for M = 8, 16, as SNR increases. Considering that the proposed scheme can provide both loer poer consumption and SER depending on SNR and modulation order, it can provide communication at a specific rate using a loer amount of poer. In case of adaptive coding and modulation, it is possible to sitch to a higher order modulation, due to loer SER, using a loer poer consumption at the transmitter. We demonstrated that precoder design using the relaxed detection region for inner constellation points of 16-QAM and -QAM results in loer poer consumption and SER increment in a specific range of SNR, hich depends on the relaxation value. It as shon than the goodput over the total poer consumption can be optimized for a specific relaxation value. APPENDIX A TRANSFORMATION OF -QAM OPTIMAL PRECODER DESIGN PROBLEM To simplify (19, first, e stack the constraints of (19 as s.t. Re (H = s re, (7a Im (S Im (H s im, Re (S Re (H s re, (7b (7c Im (H = s im, (7d Re (H 4 = s re4, (7e Im (H 4 = s im4, (7f 1 Im H Re H 1, (7g Re H Im H6, (7h Re H6 1 6, (7i here s rej and s imj are the vectors that respectively stack the real and imaginary parts of the symbols s nj s j multiplied by γ, 1 j is a Card(s j 1 vector ith elements multiplied by γ, and S j is a diagonal matrix ith diagonal entries as s nj s j. Next, e proceed to remove the real and imaginary operators from (7. Similar as in [9], e have Re (H j = H ja, Im (H j = H jb, (8 [ ( here = Re T, Im ( ] T T, Hja = [Re (H j, Im (H j ], H jb = [Im (H j, Re (H j ], and =. Using the results in (8, e can

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