Spatial Peak Power Minimization for Relaxed Phase M-PSK MIMO Directional Modulation Transmitter

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1 Spaial Peak Power Minimizaion for Relaxed Phase M-PSK MIMO Direcional Modulaion Transmier Ashkan Kalanari, Chrisos Tsinos, Mojaba Solanalian, Symeon Chazinoas, Wing-Kin Ma, and Björn Oersen Absrac The burs in media conen and access o smar phones has creaed an increasing demand for daa. A he same ime, powering up mobile base saions conribues noably o CO 2 fooprin. To address hese issues, we need o design energy efficien communicaion sysems wih higher daa raes while considering pracical limiaions. As a soluion, we design an opimal M-PSK direcional modulaion precoder wih spaial peak power imizaion where he communicaed symbol on each receiving anenna is placed in he opimal locaion of a predefined region. Such an approach allows less sringen design and resuls in furher energy efficiency. In his work, we characerize he relaxed region, formulae he opimal symbollevel precoder design problem, and ransform i ino a sandard form. The simulaion resuls show ha he relaxed design reduces he consumed power while he symbol error rae incremen a he receiver due o he relaxed phase design is negligible. Keywords Direcional modulaion, energy efficiency, M-PSK modulaion, spaial peak power, symbol-level precoding I. INTRODUCTION The adven of mobile smar devices, e.g., smar phones and ables, and he boom of Inerne conen, has led o a fas growh in users demand for daa communicaion. Based on Cisco s whie paper [1, i is prediced ha he mobile Inerne raffic will increase eigh folds from 2015 o Therefore, i is necessary o effecively use he available resources in order o address he increasing rae demand. Oher han ime and frequency resources [2, [3, precoding a he ransmier [4, [5 in order o use he spaial dimension has shown a significan poenial o improve he daa rae by reducing he inerference and saisfying he qualiy of service a he users. As anoher paradigm o comba he inerference, direcional modulaion [6 [9 and consrucive inerference [10 [12 have been developed o communicae muliple inerference-limied or inerference-free sreams wih he receiver. In direcional modulaion, he anenna weighs are designed such ha he phase and signal-o-noise raio (SNR) of he receiving signal on each anenna of he receiver are respecively equivalen o he phase and SNR of a specific desired symbol. In fac, in he direcional modulaion, he modulaion happens while he radio frequency (RF) signal passes hrough he fading channel. This work was suppored by he Naional Research Fund (FNR) of Luxembourg under AFR gran for he projec SeMIGod. Ashkan Kalanari, Chrisos Tsinos, Symeon Chazinoas, and Björn Oersen are wih he Inerdisciplinary Cenre for Securiy, Reliabiliy and Trus (SnT), The Universiy of Luxembourg, 29, avenue JF Kennedy, L-1855 Luxembourg-Kirchberg, Luxembourg, ( s: Mojaba Solanalian is wih he Deparmen of Elecrical and Compuer Engineering, Universiy of Illinois a Chicago, Chicago, IL 60607, Wing-Kin Ma is wih he Deparmen of Elecronic Engineering, The Chinese Universiy of Hong Kong, Shain, Hong Kong, China In he archiecure of he direcional, he power of he RF oscillaor is equally divided among he RF chains [6 [9 and he power amplifier in each RF chain needs o operae in a specific range o avoid nonlinear disorion of he amplified signal [13. To keep he amplifier in he linear region, we need o consider an upper limi on he oupu power of he amplifier in each RF chain. As a soluion, we can design a direcional modulaion ransmier wih spaial peak power imizaion. To his end, he references [14, [15 consider consan envelope precoding for a single-user massive MIMO sysem and he auhors in [16 consider a peranenna power imizaion based on consrucive inerference where here is a sric consrain on he phase of he received symbols. In addiion o he hardware consideraions, he mobile communicaions consume a large amoun of energy which is responsible for a considerable amoun of environmenal polluion [17. Reducing he energy consumed a he radio access neworks is no only environmenal friendly, bu also i reduces he mobile communicaions cos. The works of [9 [11, [18, [19 consider a relaxed design approach o improve he energy efficiency a he ransmier. Based on he above descripions, i is in he bes ineres of boh mobile operaors and users o have a sysem which joinly akes ino accoun he increasing rae demand, hardware limiaion, and energy efficiency. A consan envelop design such as [14, [15 resuls in non-convex problems and conservaive designs. Also, he consrucive inerference approach in [16 considers a fixed phase design for low peak power o average raio, which resuls in a sringen design. However, here is no work on addressing a join design considering boh amplifier disorion and energy efficiency. To address his, we design an M-PSK direcional modulaion ransmier wih spaial peak power imizaion while a relaxed region for furher energy efficiency of M-PSK symbol is considered. In his relaxed design, he M-PSK symbol can be placed in he opimal locaion of a defined region. This resuls in lower power consumpion while saisfying he SNR requiremen a he receiver. The remainder of his paper is organized as follows. In Secion II, he signal model is inroduced. The spaial peak power imizaion problems for he fixed and relaxed designs are formulaed ino sandard forms in Secion III. In Secion IV, we presen simulaion resuls by comparing he proposed mehod wih he benchmark schemes and draw conclusions in Secion V. Noaion: Upper-case and lower-case bold-faced leers are used o denoe marices and column vecors, respecively. The superscrips ( ) T, ( ), ( ) H, and ( ) represen ranspose, ISBN EURASIP

2 conjugae, Hermiian, and Moore-Penrose pseudo inverse operaors, respecively. I N N denoes an N by N ideniy marix, E k has one uni-valued elemen on he k-h diagonal enry wih he res of he elemens being zero, Ẽ k has wo univalued elemens on he k-h and (N +k)-h diagonal enries wih he res of he elemens being zero, diag(a) denoes a diagonal marix where he elemens of he vecor a are is diagonal enries, a b is he elemen-wise Hadamard produc, 0 is he all zero vecor, is he Frobenius norm, and represens he absolue value of a scalar. Re( ), Im( ), and arg( ) represen he real par, imaginary par, and angle of a complex number, respecively. II. SIGNAL AND SYSTEM MODEL Consider a single carrier ransmier wih N anennas, denoed by T, ha funcions using he direcional modulaion concep and communicaes wih a receiver wih N r anennas, denoed by R. The received signal, y, a he receiver can be wrien as y = Hw+n, (1) where y is an N r 1 vecor denoing he received signals by R, H = [ h 1,...,h n,...,h Nr is an Nr N marix denoing he channel from T o R, h n is an N 1 vecor conaining he channel coefficiens from he ransmier anennas o he n-h anenna of R, and w = [ w 1,...,w n,...,w N is he ransmi vecor. The random variable n CN(0,σ 2 I Nr N r ) denoes he addiive whie Gaussian noise a R where CN denoes a complex and circularly symmeric random variable. In he direcional modulaion, he elemens of Hw = [ s 1,...,s n,...,s N r are he induced M -PSK symbols on he anennas of R where s n is he induced M -PSK symbol on he n-h anenna of R, s = [ s 1,...,s n,...,s Nr are he M - PSK symbols o be communicaed beween T and R wih insananeous uni energy, i.e., s n 2 = 1. To deec he received symbols, R can apply convenional deecors on each receiving anenna. If we show he insananeous oupu signal in he n-h RF chain as w n, he maximum oupu power among he RF chains can be defined as P spaial max = max k=1,...,n w H E k w, (2) which we refer o i as he spaial peak power. In he nex secion, we aim o imize he spaial peak power. III. SPATIAL PEAK POWER MINIMIZATION FOR DIRECTIONAL MODULATION In his secion, we formulae he fixed benchmark and relaxed designs wih he goal of spaial peak power imizaion for a M-PSK direcional modulaion ransmier and design he opimal symbol-level precoder for each case. A. Peak Power Minimizaion: Fixed Design In his secion, we formulae and ransform he M-PSK precoder design problem ha saisfies he exac required phase and he imum required SNR for he signal received on each anenna of R, he induced desired symbol, while imizing he spaial peak power. Since a fixed phase is required a he desinaion, increasing he required signal level a he receiver resuls in a proporional incremen in real and imaginary pars of he received signal. Hence, we can break he SNR consrain ino wo consrains on real and imaginary pars of he induced M-PSK symbol, s n, as Re ( h T nw ) γre(s n ), Im ( h T nw ) γim(s n ), (3) where γ is he required signal power a he receiver defined as γ = 10 SNR/10. Puing down he SNR consrain as in (3) helps us avoid a non-convex design. Since he phase of he received signal is fixed, we only need o use eiher he real or imaginary par in (3). Using (3), he design problem is cas as w H E k w s.. arg ( h T nw ) = arg(s n ), n = 1,...,N r (4a) Re(s n )Re ( h T nw ) γre 2 (s n ). n = 1,...,N r (4b) The consrain (3), will no saisfy he signal level requiremen if Re ( h T nw ) < 0. To address his issue, boh sides of (4b) are muliplied by Re(s n ). This does no affec he inequaliy since Re ( h T nw ) and Re(s n ) have he same sign a he opimal poin. By inroducing as an auxiliary variable, we can ransform (4) ino a more familiar form as s.. w H E k w, k = 1,...,N (5a) arg ( h T nw ) = arg(s n ), n = 1,...,N r (5b) Re(s n )Re ( h T nw ) γre 2 (s n ). n = 1,...,N r (5c) We can wrie he phase consrain in (5b) in he linear from as α n Re ( h T nw ) Im ( h T nw ) = 0 and sack he consrains ses in (5b) and (5c) o ge s.. w H E k w, (6a) ARe(Hw) Im(Hw) = 0, (6b) Re(S)Re(Hw) γs r, (6c) where S = diag(s), s is an N r 1 vecor conaining he M -PSK symbols o be communicaed, s r = Re(s) Re(s), A = diag(α), α = [ α 1,...,α n,...,α Nr, and αn = an(arg(s n )) = Im(s n )/Re(s n ). To remove he real and imaginary pars from (6), we can use he resuls of [9 as Re ( Hw = H 1 w, Im ( Hw = H 2 w, (7) [ ( where w = ) Re w T,Im ( w T) [ [ ), H1 ) = Re( H, Im( H, H 2 = Im( H,Re( H, and w 2 = w 2. Using (7), (6) ransforms ino s.. w T Ẽ k w, (8a) (AH 1 H 2 ) w = 0, (8b) Re(S)H 1 w γs r. (8c) Similar as in [9, we can pu w in he null space of AH U1 H U2. If he SVD of AH 1 H 2 is given by UΣV H, ISBN EURASIP

3 a 1 a 2 y = b 1x +a 1 1 boundary M M Afer defining he boundaries of he relaxed region, we exend i o oher symbols. As he firs sep o define he relaxed region for he symbol s n, we need o muliply i by e iϕn so ha i roaes ino he defined relaxed region of Fig. 1 shown by he check board paern. The erms Re ( h T nw ) and Im ( h T nw ) are equivalen o x and y in he 2D plane of Fig. 1, herefore, he relaxed region of symbol s n can be defined using he following wo inequaliies: b 1 Re ( h T nwe iϕn) +a 1 Im ( h T nwe iϕn), Im ( h T nwe iϕn) b 2 Re ( h T nwe iϕn) +a 2. (11) y = b2x 2 +a2 boundary Relaxed region The parameers of lines y 1 and y 2 can be defined as ( π ( π b 1 = an, b 2 = an, M) M) a 1 = b 1 γ, a2 = b 2 γ. (12) Fig. 1. Characerizaion of he relaxed region for M -PSK modulaion. he orhonormal basis for he null space of AH 1 H 2 are he las 2N r columns of he marix V wih r being he rank of AH 1 H 2 [20. Using his, we can wrie w as w = Vλ where V = [ v r +1,...,v 2N, λ = [ λ1,...,λ 2N r. (9) By replacing w wih Vλ, (8) reduces ino λ, s.. λt B λ, Cλ a, (10) where B = V T Ẽ i V, C = Re(S)H 1 V, and a = γs r. The design in (10) is a linear program wih quadraic and linear consrains which can be solved via sandard approaches. B. Energy Efficien Peak Power Minimizaion: Relaxed Design In his par, we design he opimal spaial peak power imizaion precoder for direcional modulaion ransmier by leing he phase of he received signal on each anenna of R vary in a defined region, shown by check board paern in Fig. 1. This resuls in a less sringen design compared o [14 [16 and consequenly reduces he power consumpion a he ransmier. To do so, firs, we characerize he relaxed phase region in which he phase of he receiving signal on each anenna of R can vary. For an easy design, we characerize he relaxed region on he real axis, as shown in Fig. 1, and hen roae he oher symbols ino his region and apply he defined consrains on hem. In M-PSK, each symbol has a deecion region wihin π M degrees from each direcion. Also, considering ha we wan o saisfy a specific signal level, γ, for each induced symbol a he receiver, we need o consider he relaxed region ouside he circle of Fig. 1. One way o saisfy he required SNR and leaving enough disance o he boundaries of he deecion region is drawing lines y 1 and y 2 parallel wih he deecion region boundary so ha hese lines pass hrough he signal level hreshold, γ, as illusraed in Fig. 1. Using he defined relaxed region in (11), he relaxed opimal precoder design problem is defined as s.. w H E k w, k = 1,...,N (13a) Im ( h T nwe iϕn) b 1 Re ( h T nwe iϕn) +a 1, (13b) Im ( h T nwe iϕn) b 2 Re ( h T nwe iϕn) +a 2, (13c) where a similar approach he same as geing from (5) o (6) is used o ge o (13). The consrain (13a) limis he insananeous power ransmied from each RF chain. The consrains (13b) and (13c) enforce he induced symbol o be wihin he relaxed region while saisfying he imum required SNR a he receiver. As he firs sep o simplify (13), we absorb he desired symbol phase e iϕn ino he channel o ge s.. w H E k w, (14a) Im( ht n w b 1 Re( ht n w +a 1, (14b) Im( ht n w b 2 Re( ht n w +a 2, (14c) where h T n = h T ne iϕn. Then, we can sack he consrain in (14b) and (14c) o ge s.. w H E k w, (15a) Im ( Hw b 1 Re ( Hw +a 1 1, (15b) Im ( Hw b 2 Re ( Hw +a 2 1, (15c) where H = [ h1,..., h n,..., h Nr and 1 is an Nr 1 vecor conaining uni-valued elemens. Using derivaion in (7), we can remove he real and imaginary pars of (15) o ge s.. w T Ẽ k w, (16a) D w d, (16b) ISBN EURASIP

4 Fig. 2. Average consumed power wih respec o N for he proposed and benchmark schemes wih SNR = 10 db, N r = 10, and M = 16. Fig. 4. Average symbol error rae wih respec o he required SNR a he receiver for spaial peak power imizaion precoder designs wih fixed and relaxed phase consrain when N = N r = Fig. 3. Average maximum power among he RF chains wih respec o N for he proposed and he benchmark schemes when SNR = 10 db, N r = 10, and M = 16. where D = [ [ H2 b 1 H1 a1 1, d = H 2 b 2 H1 a 2 1. (17) The spaial peak power imizaion relaxed design in (16) has a linear objecive wih linear and quadraic consrains which is a convex opimizaion problem and can be solved using sandard mehods. To have feasible design problem in boh (10) and (16), we assume ha N N r. IV. SIMULATION RESULTS In his par, we presen simulaions o evaluae he performance of he designed direcional modulaion ransmier. The exaed performance merics include average ransmier s oal power consumpion, average spaial peak power, and average symbol error rae (SER) a he receiver. The average of he menioned merics is carried ou over muliple designed precoders for various daa and channel realizaions. In all simulaions, channels are considered o be quasi saic block Rayleigh fading ones generaed using i.i.d. complex Gaussian random variables wih disribuion CN(0, 1) and remain fixed Fig. 5. he disribuion of he noise-free communicaed symbols in he relaxed spaial peak power imizaion design for N = N r = 500 and 16-PSK modulaion. during an inerval of N r M -PSK symbols ha are communicaed wih he receiver. Also, he noise is generaed using i.i.d. complex Gaussian random variables wih disribuion CN(0,σ 2 ). As he benchmark schemes, we consider he work in [9, which sudies fixed and relaxed precoder designs wih oal ransmi power imizaion. In he firs scenario, we quanify he power consumpion a he ransmier for fixed and relaxed designs wih oal and spaial peak power imizaion crieria in Fig. 2. As we see, he relaxed design reduces he consumed power a he ransmier for a long range of N. This bigges differences beween fixed and relaxed designs are 1.28 db and 0.84 db for oal and spaial power imizaion problems, respecively, for N = N r = 10. As N grows bigger han N r, he sysem degrees of freedom increases and he power consumpion difference beween fixed and relaxed designs decreases. Ineresingly, we see ha he spaial peak power imizaion design wih relaxed phase consumes less power a N = 10 compared o he fixed design wih oal power imizaion ISBN EURASIP

5 objecive. This illusraes ha even wih hardware limiaion, he spaial peak power imizaion design, he relaxed design can resuls in an more energy efficien ransmier compared o he fixed design. For he nex scenario, we presen he average spaial peak power for fixed and relaxed designs wih oal and spaial peak power imizaion objecives. As we see in Fig. 3, he spaial peak power imizaion wih relaxed design resuls in a considerably less spaial peak power compared o he fixed design for a long range ofn. For example, forn = N r = 10, he relaxed design resuls in 1.4 db less average spaial peak power compared o he fixed design. As N increases, he difference beween he average spaial peak power of he fixed and relaxed designs decreases as Furhermore, as N increases, he difference beween he maximum oupu resuls in less maximum oupu power among he RF chains. The disribuion of he communicaed symbols a he receiver is shown in Fig. 5. As we see, he opimal precoder design resuls in many symbols going well above he required SNR, he dashed circle, a he receiver. Hence, by using lower power, i is possible o ge lower SER hanks o opimizing he locaions of he symbols. As he las scenario, we derive he SER a he receiver for fixed and relaxed cases of spaial peak power imizaion designs. SER wih respec o SNR is shown in Fig. 4 for 8- PSK and 16-PSK modulaions. We observe ha he SER of fixed and relaxed designs are close o each oher in relaively low SNR regime and he disance beween hem increases for relaively high SNR regime. In addiion, he SER of he relaxed and fixed designs for 8-PSK modulaion are closer compared o hose of 16-PSK modulaion. The reason is ha he symbol deecion region shrinks as he modulaion order increases and he relaxed design is more likely o creae an error. By comparing Figs. 2 and 4 for SNR = 10 db and 16-PSK modulaion, we see ha he power saving for a range of N is worhy of he loss in SER, especially for N = N r. V. CONCLUSIONS In his work we designed he opimal symbol-level precoder for a relaxed phase M -PSK direcional modulaion ransmier by imizing he maximum ransmied power among he RF chains. In conras o he sae of he ar, we le he communicaed symbols o vary in a predefined relaxed region insead of considering hem o be fixed in a poin. The simulaions demonsraed ha relaxing he phase resuls in less power consumpion a he ransmier, especially for close number of ransmi and receiver anennas. Ineresingly, i was observed ha he relaxed spaial peak power imizaion design consumes less power han he fixed design wih ransmi power imizaion objecive for equal number of ransmi and receive anennas. The resuls showed ha relaxed design resuls in a less maximum ransmied power among he RF chains compared o he fixed design. While he relaxed design reduces he oal power consumpion and spaial peak power noably, he SER incremen due o relaxed design is negligible compared o he power reducion, especially for close values of ransmi and receive number of anennas. The power reducion and SER improvemen in direcional modulaion are derived a he expense of designing he precoder for communicaion of each group of he symbols. REFERENCES [1 Cisco, Cisco visual neworking index: Forecas and mehodology, , Cisco, Tech. Rep., [Online. Available: hp:// provider/visual-neworking-index-vni/complee-whie-paper-c pdf [2 S. Weinsein and P. Eber, Daa ransmission by frequency-division muliplexing using he discree fourier ransform, IEEE Trans. Commun. Technol., vol. 19, no. 5, pp , Oc [3 D. D. Falconer, F. Adachi, and B. Gudmundson, Time division muliple access mehods for wireless personal communicaions, IEEE Commun. Mag., vol. 33, no. 1, pp , Jan [4 Q. Spencer, A. Swindlehurs, and M. Haard, Zero-forcing mehods for downlink spaial muliplexing in muliuser MIMO channels, IEEE Trans. Signal Process., vol. 52, no. 2, pp , Feb [5 N. Sidiropoulos, T. Davidson, and Z.-Q. Luo, Transmi beamforg for physical-layer mulicasing, IEEE Trans. Signal Process., vol. 54, no. 6, pp , Jun [6 A. Babakhani, D. Ruledge, and A. Hajimiri, Transmier archiecures based on near-field direc anenna modulaion, IEEE J. Solid-Sae Circuis, vol. 43, no. 12, pp , Dec [7 M. Daly and J. Bernhard, Direcional modulaion echnique for phased arrays, IEEE Trans. Anennas Propag., vol. 57, no. 9, pp , Sep [8 M. Daly, E. Daly, and J. Bernhard, Demonsraion of direcional modulaion using a phased array, IEEE Trans. Anennas Propag., vol. 58, no. 5, pp , May [9 A. Kalanari, M. Solanalian, S. Maleki, S. Chazinoas, and B. Oersen, Direcional modulaion via symbol-level precoding: A way o enhance securiy, IEEE J. Sel. Topics Signal Process., vol. PP, no. 99, Aug [10 C. Masouros and E. Alsusa, Dynamic linear precoding for he exploiaion of known inerference in MIMO broadcas sysems, IEEE Trans. Wireless Commun., vol. 8, no. 3, pp , March [11, Sof linear precoding for he downlink of DS/CDMA communicaion sysems, IEEE Trans. Veh. Technol., vol. 59, no. 1, pp , Jan [12 M. Alodeh, S. Chazinoas, and B. Oersen, Consrucive muliuser inerference in symbol level precoding for he MISO downlink channel, IEEE Trans. Signal Process., vol. 63, no. 9, pp , May [13 A. A. M. Saleh, Frequency-independen and frequency-dependen nonlinear models of TWT amplifiers, IEEE Trans. Wireless Commun., vol. 29, no. 11, pp , Nov [14 S. K. Mohammed and E. G. Larsson, Single-user beamforg in large-scale MISO sysems wih per-anenna consan-envelope consrains: The doughnu channel, IEEE Trans. Wireless Commun., vol. 11, no. 11, pp , Nov [15 J. Pan and W. K. Ma, Consan envelope precoding for single-user large-scale MISO channels: Efficien precoding and opimal designs, IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp , Oc [16 D. Spano, M. Alodeh, S. Chazinoas, and B. Oersen, Per-anenna power imizaion in symbol-level precoding, in IEEE Global Commun. Conf. (GLOBECOM), Washingon, DC, USA, Dec [17 A. Fehske, G. Feweis, J. Malmodin, and G. Biczok, The global fooprin of mobile communicaions: The ecological and economic perspecive, IEEE Commun. Mag., vol. 49, no. 8, pp , Aug [18 C. Masouros and G. Zheng, Exploiing known inerference as green signal power for downlink beamforg opimizaion, IEEE Trans. Signal Process., vol. 63, no. 14, pp , Jul [19 M. Alodeh, S. Chazinoas, and B. Oersen, Energy-efficien symbollevel precoding in muliuser miso based on relaxed deecion region, IEEE Trans. Wireless Commun., vol. 15, no. 5, pp , May [20 G. Srang, Inroducion o Linear Algebra, 4h ed. Wellesley- Cambridge Press and SIAM, ISBN EURASIP

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