Fundamental bounds on MIMO antennas

Size: px

Start display at page:

Download "Fundamental bounds on MIMO antennas"

Shawn Dixon
5 years ago
Views:

1 1 Fundmentl bounds on MIMO ntenns Csimir Ehrenborg, Student member, IEEE, nd Mts Gustfsson, Member, IEEE rxiv: v3 [physics.clss-ph] 1 Aug 17 Abstrct Antenn current optimiztion is often used to nlyze the optiml performnce of ntenns. Antenn performnce cn be quntified in e.g., minimum Q-fctor nd efficiency. The performnce of MIMO ntenns is more involved nd, in generl, single prmeter is not sufficient to quntify it. Here, the cpcity of n idelized chnnel is used s the min performnce quntity. An optimiztion problem in the current distribution for optiml cpcity, mesured in spectrl efficiency, given fixed Q-fctor nd efficiency is formulted s semidefinite optimiztion problem. A model order reduction bsed on chrcteristic nd energy modes is employed to improve the computtionl efficiency. The performnce bound is illustrted by solving the optimiztion problem numericlly for rectngulr pltes nd sphericl shells. Index Terms MIMO, Physicl bounds, Q-fctor, Semidefinite progrmming, Convex optimiztion I. INTRODUCTION Wireless communiction in modern systems utilize multiple input multiple output (MIMO) networks nd ntenns [1], [2]. These systems consist of two sets of ntenns, one trnsmitting, nd one receiving. Normlly, one of these sets is situted in loction where spce lloction is not n issue, such s bse sttion. However, the other set is usully contined within smll device, such s mobile phone, where design spce is limited [3]. Nturlly, ntenn designs im t mximizing performnce in such n environment. However, there is little knowledge of how the performnce depends on size, Q-fctor nd efficiency restrictions. Hving this knowledge priori would enble designers to optimize their ntenn designs more efficiently. There hs been efforts to bound MIMO ntenns performnce for sphericl surfces [4], [] nd through informtion-theoreticl pproches [6], [7], [8]. In this letter method for constructing performnce bound on cpcity for rbitrry shped MIMO ntenns using current optimiztion is presented. Antenn current optimiztion cn be used to determine physicl bounds for ntenns of rbitrry shpe [9]. These physicl bounds re found by mximizing certin performnce prmeter by freely plcing currents in the design spce. By hving totl control of the current distribution n optiml solution cn be reched. While these currents might not necessrily be relizble they provide n upper bound for the considered problem. Construction of such physicl bounds re mde possible by the bility to formulte convex optimiztion problems [] for the performnce quntity of interest. The performnce of simple ntenns cn be quntified in e.g., the Q-fctor, gin, directivity, nd efficiency [11]. MIMO ntenns, on the other hnd, re more complex nd Csimir Ehrenborg nd Mts Gustfsson re with the Deprtment of Electricl nd Informtion Technology, Lund University, Box 118, SE Lund, Sweden. (Emil: {csimir.ehrenborg,mts.gustfsson@eit.lth.se}@eit.lth.se). single prmeter is insufficient to determine their performnce. As such, it is chllenging problem to construct physicl bounds for MIMO systems. However, it is still possible to utilize ntenn current optimiztion to mximize given performnce quntity, such s cpcity, with restrictions on, e.g., the Q-fctor nd efficiency. In communiction theory MIMO network s cpcity is usully optimized for fixed set of ntenns. The performnce of the ntenns is ccepted s it is nd the upper bound on network performnce is clculted by e.g., wter filling [2]. However, in doing so we forgo n opportunity to gin extr performnce through optimizing the ntenns. In this pper we illustrte how bounds on cpcity of MIMO ntenn cn be determined by ntenn current optimiztion. Considering the chnnel between two sets of ntenns leds to optimiztion for specific scenrios or circumstnces, in this pper we re interested in estblishing generl performnce bounds for MIMO ntenns. As such we focus on one set of ntenns nd idelize the other. The second set of ntenns re chrcterized s the sphericl modes in the fr-field. This leds to n idelized chnnel in terms of sphericl modes [], which cn be thought of s direct line of sight chnnel where ll rdition is received. Considering such chnnel lso hs the benefit of reducing computtionl complexity. This is further reduced by model order reduction of the method of moments (MoM) impednce mtrix chrcterizing the ntenn. The convex optimiztion problem is constrined by the efficiency or Q-fctor. These re expressed s qudrtic forms in the current density, where the stored energy in [12] is used. This leds to convex optimiztion problem tht mximizes the cpcity in terms of spectrl efficiency for fixed signlto-noise rtio (SNR) nd Q-fctor. The convex optimiztion problem is semi-definite progrm [] expressed in the covrince mtrix of the current distribution. II. MIMO MODEL A clssicl MIMO system is modeled s [2] y = Hx + n, (1) where x is N 1 mtrix of the input signls, y is M 1 mtrix of the output signls, n is M 1 mtrix of dditive noise, nd H is the M N chnnel mtrix. The chnnel mtrix models how power is trnsmitted from the input signls to the output signls, this includes the receiving nd trnsmitting ntenns nd the wve propgtion between them [2]. Fig. 1 displys clssicl MIMO setup where two sets of ntenns form chnnel. Anlysis of such systems depend gretly on externl fctors, such s, scttering phenomen, chnnel chrcteriztion, nd ntenn loction [2]. However, to investigte performnce bounds for MIMO ntenns we

2 2 ) b) Ω T J T H H J R Ω R Ω R J R where M denotes the mp from the currents to the sphericl modes. This is direct chnnel between the ntenn current distribution nd the sphericl modes [1]. The cpcity, expressed s spectrl efficiency ( b/(s Hz)), of this chnnel is given by [2] Å C = mx log 2 det ã MP ı M ıh, (4) Tr(ÛRP)=P N 0 where 1 is the M M identity mtrix, nd N 0 is the noise spectrl power density. The noise is modeled s white complex Gussin noise. The optiml energy lloction in this chnnel for cpcity mximiztion is given by the wterfilling solution [2]. Alterntively, the optiml solution for this problem cn be solved by semidefinite optimiztion progrm, Ω T J T mximize log 2 det(1 + γ ı MP ı M H ) Tr( Ù RP) = P () Fig. 1. Illustrtion of the MIMO system model with trnsmitter region Ω T nd receiver region Ω R. Prt () shows the clssicl MIMO setup with sptilly seprted regions. Prt (b) illustrtes the idelized cse when the receiver region entirely surrounds the trnsmitter. The system in (b) is utilized in this pper to determined performnce bounds on MIMO ntenns confined to the region Ω T. must limit the degrees of freedom to single ntenn. This implies tht H in (1) should model the chnnel between n rbitrry ntenn nd n idelized receiver, corresponding to Fig. 1b. The trnsmitting ntenn is modeled with its current distribution using MoM pproximtion [11] such tht ech bsis function corresponds to n element of x. The receiver is modeled with the rdited sphericl modes, where ech mode is n element in y [13], []. This leds to MIMO system of infinite dimension s N increses with mesh refinement nd M increses with the number of included sphericl modes. In numericl evlution N nd M re chosen sufficiently lrge to ensure convergence. The trnsmitted signls re modeled s the MoM current elements I = Tx, where the mtrix T mps the trnsmitted signls x to the current distribution on the ntenn I. The covrince mtrix of the trnsmitted signl is P = E { xx H}, where E { } denotes the temporl verge [2]. With this mtrix we cn clculte the verge trnsmitted power, P = 1 2 E { I H RI } = 1 2 E { x H T H RTx } = 1 2 Tr E { T H RTxx H} = 1 2 Tr(Ù RP), (2) where Ù R = T H RT, nd R is the resistive prt of the MoM impednce mtrix, Z = R + jx [11]. Since we re concerned with connecting the currents on the ntenn structure to the sphericl modes [14] in the idelized receiver we express our chnnel s y = MI + n = MTx + n = ı Mx + n, (3) where the unit trnsmitted power is considered, nd γ = P/N 0 is the totl SNR. Mximizing the cpcity of this chnnel corresponds to focusing the rdition of the ntenn to the orthogonl sphericl modes. The solution to () is unbounded nd increses s mesh refinement nd the number of sphericl modes re incresed. In order to solve this issue the number of degrees of freedom must be reduced. This is done by dding constrints on the losses or Q-fctor of the trnsmitting ntenn [], [16]. The Ohmic losses re clculted s P Ω = 1 2 E { I H R Ω I } = 1 2 E { x H T H R Ω Tx } = 1 2 Tr(Ù R Ω P), (6) where Ù R Ω = T H R Ω T, nd R Ω is the loss mtrix of the ntenn [11]. The stored electric energy is W e = 1 4ω E { I H X e I } = 1 4ω E { x H T H X e Tx } = 1 4ω Tr(Ù X e P), (7) where X Ù e = T H X e T, nd X e is the electric rectnce mtrix [11]. The stored mgnetic energy W m is similrly defined by the mgnetic rectnce mtrix X m s W m = 1 4ω Tr(Ù X m P), where X Ù m = T H X m T. With these constrints in hnd we cn formulte our optimiztion problem. We note tht the solution is independent of the power P, so it is sufficient to consider the cse P = 1 giving mximize log 2 det(1 + γmp ı M ıh ) Tr(( Ù X e + Ù X m )P) 2Q Tr( Ù XP) = 0 Tr( Ù R Ω P) 1 η where η is the ntenn efficiency, nd self-resonnce is enforced. Here, the problem hs been normlized to dissipted power, including losses. The consequence of this is tht the Q-fctor considered includes losses in its clcultion. It is possible, nd sometimes dvntgeous, to normlize to (8)

3 3 different quntities such s the rdited power. Eqution (8) is semi-definite optimiztion problem which hs unique solution []. However, the problem is non-trivil due to the lrge number of unknowns for relistic ntenn problems. For exmple rectngulr plte of size l l/2 discretized into rectngulr elements hs N = 4000 unknowns. This size is not problem for convex optimiztion of type G/Q nd Q [14], [17], [11]. However, the semi-definite relxtion hs close to N 2 /2 = 8 6 unknowns, mking the problem much more computtionlly demnding. Moreover, the logrithm used in the definition of cpcity is more involved thn the simple qudrtic functions in G/Q nd Q type problems [14], [11]. Here, the number of unknowns is reduced by expnsion of the currents in chrcteristic, energy, nd efficiency modes [11], with similr results. The expnsion includes only the dominting modes nd s such constitutes model order reduction. This implies chnge of bsis I UĨ, where U mps between the old nd the new currents. This reduces the number of unknowns to the included modes N 1 N. With this pproximtion the stored energy, for exmple, is clculted s reltive σ l l/2 Fig. 2. The singulr vlues of the chnnel mtrix M Ù for rectngulr plte l l/2 for the wvelength l = 0.21λ reltive σ 1 I H Ù Xe I ĨH U T Ù Xe UĨ = ĨH Xe Ĩ = Tr( X e ĨĨH ) = Tr( X e Y), (9) where Y = ĨĨH, nd X e = U T Ù Xe U. Similrly Ù X m, nd Ù R, re expressed s X m = U T Ù Xm U, nd R = U T Ù RU. These replce the corresponding mtrices in (8), with Y replcing P. This reduces the number of unknowns from pproximtely N 2 /2 to N 2 1 /2. III. NUMERICAL EXAMPLES In the following exmples the optimiztion problem (8) hs been solved for MIMO system resembling Fig. 1b using the Mtlb librry CVX [11], [18]. The logrithm in the optimiztion problem (8) ws replced by root of order M [18]. After the optimiztion hs been crried out the cpcity is clculted s norml with the optimized currents. The energy restriction on the number of trnsmitter modes nd the number of sphericl hrmonic modes in the receiver hve been chosen sufficiently lrge to ensure convergence nd vries from exmple to exmple. Using too mny modes my lso result in the solver filing to solve the problem due to its size nd must therefore be regulted for ech run individully. Since the performnce of MIMO ntenn cnnot be quntified by single prmeter the optimiztion ws run with different constrints. This illustrtes how cpcity is bounded by different requirements on the trnsmitting ntenns. The optimiztion hs lso been run for sphericl shell circumscribing the ntenn. By performing singulr vlue decomposition of the chnnel mtrix ı M we cn see how mny chnnels dominte the informtion trnsfer between the plte nd the sphericl modes, see Fig. 2. Here, we see tht there re only few chnnels tht dominte the rest. This indictes tht so long s our model order reduction preserves these chnnels it produces correct solutions. Fig. 3. The singulr vlues of the chnnel mtrix M Ù for sphericl shell r =, where = 0.6l, for the wvelength l = 0.21λ. In Fig. 4 the cpcity hs been optimized for plte of electricl size l = 0.21λ, nd is depicted s function of the Q-fctor restriction. We see cut-off for Q 12 where the optimiztion problem is unble to relize fesible current distribution for so low Q-fctor, cf., the lower bound on the Q-fctor [17]. For higher SNR the cpcity increses but the cut-off stys the sme, since the SNR does not ffect the Q- fctor. We cn insted regrd the problem with fixed SNR nd investigte how the cpcity vries with ntenn size, see Fig.. Depending on which Q is chosen the solution is only relizble for sizes bove certin cut-off. This cut- 30 γ = 80 Q lb = 12 γ = 0 l/2 γ = 1 γ = 80 γ = Fig. 4. Mximum spectrl efficiency chievble for loss-less rectngulr plte of size l l/2 for the wvelength l = 0.21λ given mximum Q-fctor on the horizontl xis. The dshed lines show the mximum spectrl efficiency chievble for the corresponding circumscribing sphere. l Q

4 4 30 Q = 60 Q = 0 2 Q = 60 Q = 40 Q = Q = Q l/λ Fig.. Mximum spectrl efficiency chievble for loss-less rectngulr plte of electricl size l/λ for mximum Q-fctor with SNR γ = 0, cf., Fig. 4. The dshed lines show the mximum spectrl efficiency chievble for the corresponding circumscribing sphere. 1 Q = 30 l = 0.29λ Q = 40 Q = Q = 30 l = 0.13λ l = 0.21λ Q = Fig. 6. Mximum spectrl efficiency chievble for rectngulr plte of electricl size l/λ for minimum efficiency η. The losses re modeled s resistive sheet with R = 0.2 Ω/. The minimum Q-fctor is set to 30 for the three min grphs nd SNR γ = 0. Solid lines re optimized without enforcing resonnce nd dshed lines re optimized with resonnce. For l = 0.21λ the Q-fctors [, 30, 40] re plotted. off corresponds to the size which hs the chosen Q s its minimum chievble Q. Above this size the cpcity seems to depend linerly on the ntenn size. This is consistent with how cpcity scles with the number of ntenns included in MIMO system [2]. In both Fig. 4 nd the dshed lines show the optimiztion problem solved for sphericl shell circumscribing the plnr region. We see tht the spectrl efficiency chievble by plnr ntenn is much less thn tht of the sphere. Setting n efficiency requirement on the optimiztion my restrict which modes re relizble. Fig. 6 illustrtes how cpcity vries s function of ntenn efficiency. We see tht the cpcity is unffected until some cut-off vlue where the solution is no longer relizble. For electricl sizes l = 0.21λ nd 0.29λ this occurs when ntenn efficiency requirements is high, bove 90%. However, for smller sizes, such s l = 0.13λ, we see tht this cut-off occurs t lower ntenn efficiencies. The optimiztion problem hs been solved both with nd without enforcing resonnce. When resonnce is enforced, showed in dshed lines, we see tht the cut-off occurs t lower efficiencies, this is due to self-resonnt currents being inherently less efficient [19]. For the size l = 0.21λ η Q lb l/λ Fig. 7. Illustrtion of the bounding surfce of spectrl efficiency for loss less rectngulr plte s function of size nd Q-fctor with SNR γ = 0. The red curve shows minimum Q [17] the Q-fctor requirement ws vried s well, leding to slight reduction or increse in cpcity. Close to the cut-off efficiency we see slight decrese in cpcity for ll cses. This corresponds to the requirement on efficiency limiting the optimiztion problem. For lower efficiency requirements other constrints limit the optimiztion nd the cpcity is unffected by the bound on efficiency. In Fig. 7 both the size of the ntenn nd the Q-fctor re vried to crete two dimensionl bounding surfce. This surfce hs shrp cut-off long the minimum Q line [17] seen on the left in Fig. 7. We see tht the increse in cpcity follows the shpe of the minimum Q curve s l/λ nd Q re incresed. This surfce provides bound on the cpcity chievble for MIMO ntenns of different sizes nd with different bndwidth requirements. IV. CONCLUSIONS In this letter we hve presented frmework for constructing performnce bounds for MIMO ntenns. We simplified the chnnel problem often considered in communiction theory to n idelized chnnel consisting of sphericl receiver surrounding the ntenn region. This enbles the formultion of semi-definite optimiztion problem tht gives bounding cpcity for ny ntenn tht cn be constructed within the considered region limited by size, SNR, ntenn efficiency, nd Q-fctor. By utilizing model order reduction bsed on energy nd chrcteristic modes [17] the complexity of the problem is reduced such tht it is solvble. These physicl boundries of MIMO ntenns represent the idel solutions possible given complete freedom of current plcement within the design re. While the shpe of these current distributions re not esily relizble [19], the bounding vlues provide n upper limit to wht is possible for rel ntenn topologies. It remins interesting to investigte how these bounds compre to ntenn designs nd mesurements. 0

5 ACKNOWLEDGMENT The support of the Swedish foundtion for strtegic reserch under the progrm pplied mthemtics nd the project Complex nlysis nd convex optimiztion for electromgnetic design is grtefully cknowledged. APPENDIX A ANTENNA PARAMETERS The impednce mtrix Z = R + jx is determined from MoM description of the ntenn structure. The impednce mtrix is divided into its resistnce R nd rectnce X. Moreover, the rectnce is decomposed into its mgnetic nd electric prts, i.e., Z = R + j(x m X e ), where the stored electric nd mgnetic energies re [12], [] W m 1 8 IH Å X ω + X ω W e 1 8 IH Å X ω X ω ã I = 1 4ω IH X m I, ã I = 1 4ω IH X e I, respectively, nd the dissipted power P d is given by () (b) P d = 1 2 IH RI. (11) The Q-fctor is defined s the quotient between the timeverge stored nd dissipted energies [21], [22], [9] Q = 2ω mx{w e, W m } = mx{ih X e I, I H X m I} P d I H. (12) RI APPENDIX B MAXIMUM EFFICIENCY To motivte the cut-off vlues seen in Fig. 6 the mximum efficiency for given Q-fctors ws investigted. This ws evluted using two optimiztion problems, one to find the minimum efficiency for set Q, minimize Re Tr( Ù R Ω P) Tr(( Ù X e + Ù X m )P) = 2Q Tr( Ù XP) = 0 (13) nd one to find the minimum Q-fctor for certin efficiency, minimize Re Tr(( Ù X e + Ù X m )P) Tr(( Ù X m Ù X e )P) = 0 Tr( Ù R Ω P) = 1 η P 0. (14) These problems cn be reformulted so tht resonnce is not enforced, this results in higher efficiency limit. The optimiztion problems (13) nd (14) for the efficiency lso rise from semi-definite relxtion [] of the corresponding problems formulted in the current I. Semi-definite relxtion is technique to solve qudrticlly constrined qudrtic progrms (QCQP) nd cn pplied to mny ntenn problems [23], [24], [2]. REFERENCES [1] A. F. Molisch, Wireless Communictions, 2nd ed. New York, NY: John Wiley & Sons, 11. [2] A. Pulrj, R. Nbr, nd D. Gore, Introduction to Spce-Time Wireless Communictions. Cmbridge: Cmbridge University Press, 03. [3] Z. Ying, Antenns in cellulr phones for mobile communictions, Proceedings of the IEEE, vol. 0, no. 7, pp , July 12. [4] A. A. Glzunov, M. Gustfsson, nd A. Molisch, On the physicl limittions of the interction of sphericl perture nd rndom field, IEEE Trns. Antenns Propg., vol. 9, no. 1, pp , 11. [] M. Gustfsson nd S. Nordebo, On the spectrl efficiency of sphere, Prog. Electromgn. Res., vol. 67, pp , 07. [6] M. Migliore, On electromgnetics nd informtion theory, IEEE Trns. Antenns Propg., vol. 6, no., pp , Oct. 08. [7] P. S. Tluj nd B. L. Hughes, Fundmentl cpcity limits on compct MIMO-OFDM systems, in IEEE Interntionl Conference on Communictions (ICC), June 12, pp [8] L. Kundu, Informtion-theoretic limits on MIMO ntenns, Ph.D. disserttion, North Crolin Stte University, 16. [9] M. Gustfsson, D. Tyli, nd M. Cismsu, Physicl bounds of ntenns. Springer-Verlg, 1, pp [] S. P. Boyd nd L. Vndenberghe, Convex Optimiztion. Cmbridge Univ. Pr., 04. [11] M. Gustfsson, D. Tyli, C. Ehrenborg, M. Cismsu, nd S. Nordebo, Antenn current optimiztion using MATLAB nd CVX, FERMAT, vol. 1, no., pp. 1 29, 16. [Online]. Avilble: org/rticles/gustfsson-rt-16-vol1-my-jun-00/ [12] G. A. E. Vndenbosch, Rective energies, impednce, nd Q fctor of rditing structures, IEEE Trns. Antenns Propg., vol. 8, no. 4, pp ,. [13] M. Gustfsson nd S. Nordebo, Chrcteriztion of MIMO ntenns using sphericl vector wves, IEEE Trns. Antenns Propg., vol. 4, no. 9, pp , 06. [14], Optiml ntenn currents for Q, superdirectivity, nd rdition ptterns using convex optimiztion, IEEE Trns. Antenns Propg., vol. 61, no. 3, pp , 13. [1], Bndwidth, Q-fctor, nd resonnce models of ntenns, Prog. Electromgn. Res., vol. 62, pp. 1, 06. [16] M. L. Morris, M. Jensen, J. W. Wllce et l., Superdirectivity in MIMO systems, IEEE Trns. Antenns Propg., vol. 3, no. 9, pp , 0. [17] M. Cpek, M. Gustfsson, nd K. Schb, Minimiztion of ntenn qulity fctor, IEEE Trns. Antenns Propg., 17. [18] M. Grnt nd S. Boyd, CVX: Mtlb softwre for disciplined convex progrmming, version 1.21, Apr. 11. [Online]. Avilble: [19] L. Jelinek nd M. Cpek, Optiml currents on rbitrrily shped surfces, IEEE Trns. Antenns Propg., vol. 6, no. 1, pp , 17. [] M. Cismsu nd M. Gustfsson, Antenn bndwidth optimiztion with single frequency simultion, IEEE Trns. Antenns Propg., vol. 62, no. 3, pp , 14. [21] T. Ohir, Wht in the world is Q? IEEE Microw. Mg., vol. 17, no. 6, pp , June 16. [22] J. Volkis, C. C. Chen, nd K. Fujimoto, Smll Antenns: Minituriztion Techniques & Applictions. New York, NY: McGrw-Hill,. [23] Z.-Q. Luo, W.-K. M, A. M.-C. So, Y. Ye, nd S. Zhng, Semidefinite relxtion of qudrtic optimiztion problems, IEEE Signl Process. Mg., vol. 27, no. 3, pp. 34,. [24] B. Fuchs, Appliction of convex relxtion to rry synthesis problems, IEEE Trns. Antenns Propg., vol. 62, no. 2, pp , 14. [2] B. Jonsson, S. Shi, L. Wng, F. Ferrero, nd L. Lizzi, On methods to determine bounds on the Q-fctor for given directivity, rxiv preprint rxiv: , 17.

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl