A Study of Near-Field Direct Antenna Modulation Systems Using Convex Optimization

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1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, WeB8.5 A Stuy of Near-Fiel Direct Antenna Moulation Systems Using Convex Optimization Java Lavaei, Ayin Babakhani, Ali Hajimiri an John C. Doyle Abstract This paper stuies the constellation iagram esign for a class of communication systems known as near-fiel irect antenna moulation (NFDAM) systems. The moulation is carrie out in a NFDAM system by means of a control unit that switches among a number of pre-esigne passive controllers such that each controller generates a esire voltage signal at the far fiel. To fin an optimal number of signals that can be transmitte an emoulate reliably in a NFDAM system, the coverage area of the signal at the far fiel shoul be ientifie. t is shown that this coverage area is a planar convex region in general an simply a circle in the case when no constraints are impose on the input impeance of the antenna an the voltage receive at the far fiel. A convex optimization metho is then propose to fin a polygon that is able to approximate the coverage area of the signal constellation iagram satisfactorily. A similar analysis is provie for the ientification of the coverage area of the antenna input impeance, which is beneficial for esigning an energy-efficient NFDAM system.. NTRODUCTON A vast majority of problems in circuits, electromagnetics, an optics can be regare as the analysis an synthesis of linear systems in the frequency omain. These systems, in the circuit theory, consist of passive elements incluing resistors, inuctors, capacitors, ieal transformers, an ieal gyrators [. Since the seminal work [, there has been remarkable progress in characterizing such passive (issipative) systems using the concept of positive real functions. This notion plays a vital role not only in circuit esign but also in various control problems [, [3, [. The application of control theory in circuit an communication areas eviently goes beyon the passivity concept. nee, the emerging optimization tools evelope by control theorists, such as linear matrix inequalities (LMs) [5 an sum-of-squares (SOS) [, have been successfully applie to a number of funamental problems in these fiels. For instance, the recent paper [7 proposes an LM optimization to check whether a given multi-port network can be realize using a pre-specifie set of linear time-invariant components (namely an inuctor an small-signal moel of a transistor). Moreover, the work [8 formulates the pattern synthesis of large arrays with boun constraints on the sielobe an mainlobe levels as a semiefinite program. t is well-known that a broa class of problems in circuits, electromagnetics, an optics can be formulate as an optimization over the parameters of a multi-port passive network Java Lavaei an John C. Doyle are with the Department of Control an Dynamical Systems, California nstitute of Technology ( s: lavaei@cs.caltech.eu, oyle@cs.caltech.eu). Ayin Babakhani an Ali Hajimiri are with the Department of Electrical Engineering, California nstitute of Technology, ( s: ayin@caltech.eu, hajimiri@caltech.eu). which is obtaine, for instance, via an electromagnetic (EM) simulation. As an example, it is shown in [9 that a strikingly efficient an practical way to eal with certain complex antenna problems is to extract a circuit moel an then search for appropriate values of its parameters. The circuit moel propose in [9 is inee a simple, general moel which coul be consiere the abstract moel of ifferent types of problems. Our recent work [ stuies such circuit problems using the available techniques evelope in the control theory, especially the LM an passivity concepts. The present paper aims to apply our recent results evelope in [ to the problem of constellation iagram esign for a class of communication systems, referre to as near-fiel irect antenna moulation (NFDAM) systems. The objective is to propose a systematic metho to esign an energy-efficient NFDAM system that is able to sen an optimal number of inepenent signals which can be emoulate reliably. The rest of the paper is organize as follows. Some preliminaries on NFDAM systems are provie in Section an the problem is introuce accoringly. The main results are then presente in Section. The efficacy of this work is eluciate in a practical example in Section V. Concluing remarks are rawn in Section V. Finally, a proof is given in the appenix.. PRELMNARES AND PROBLEM FORMULATON Since the invention of the raio in the en of nineteen century, there have been major revolutions in the architecture of the raio but there has been a central component that all of them ha in common. n all of these systems, the information is generate before the antenna an the role of the antenna is only to efficiently transmit the signal. Different schemes are evelope to a information to a carrier signal (e.g. a sinusoial waveform). The act of aing information to a carrier signal, known as moulation, can be achieve by altering some properties of the carrier signal such as its frequency, amplitue, or phase. n a broa range of wireless communication systems, the information is generate in low frequencies (base-ban frequency region), an then upconverte to a carrier frequency (RF) via a mixer that acts as a multiplier. The base-ban ata forms a series of complex numbers that can be separate into real an imaginary parts. The first set is calle the n-phase signal () that is the real part of the complex signal an the secon set is calle the uarature-phase signal () that is the imaginary part of the complex signal. A simple constellation iagram can 5

2 Signal s Moulate Before the Antenna Base-ban Data Oscillator Mixer fc Antenna unesire irection esire irection Fig.. Moulation before the antenna Signal s Moulate After the Antenna Oscillator fc unesire irection Fig.. Moulation after the antenna [9. esire irection be use to represent this complex signal, where each point of the iagram correspons to some information symbol. After moulating an incoming signal, a conventional antenna propagates the moulate signal in almost all irections. This signal can ieally be receive in ifferent irections after some time elay an power attenuation. This implies that the signal cannot be transmitte securely only in some esire irections, an inee the unerlying transmission technique oes not ifferentiate between esire an unesire irections. This fact is illustrate in Figure. A metho is propose in the recent papers [9 an [ for secure wireless transmission, which allows for generating information by varying the characteristics of the antenna itself as oppose to moulating a signal before the antenna. This iea is illustrate in Figure, an will be further elaborate in the next section. This has mae it possible to simultaneously transmit inepenent information to ifferent irections in space. The feasibility of this iea was prove by implementing a silicon-base transmitter leaing to the chip given in Figure 3. An unmoulate carrier signal rives an on-chip ipole antenna in this chip, an 9 switches an reflectors are use to vary the characteristics of the antenna (the near-fiel bounary conitions aroun the antenna). Note that antennas with time-varying bounary conitions are able to transmit inepenent signals to ifferent irections simultaneously. The systems base on this concept are sai to be near-fiel irect antenna moulation (NFDAM) transmitters. The objective of the current paper is to stuy a NFDAM system with the aim of fining the set of points that can be generate on the signal constellation iagram. As a seconary goal, it is esire to investigate the variation of the input impeance of the antenna an ientify the region that the antenna input impeance belongs to. Fig. 3. The first implemente NFDAM transmitter [.. NEAR-FELD DRECT ANTENNA MODULATON Consier an antenna configuration consisting of a main transmitting antenna an a number of reflectors where each reflector is accompanie by certain ports (see Figure for an example of this configuration). The objective is to control the ports in such a way that the antenna acts as a secure wireless transmission evice that is able to transmit ata correctly in one esire irection an scramble ata in z unesire irections (where z N). To esign a controller for the reflectors ports, the first step is to place z receiving ipole antennas at the far file in all esire an unesire irections. Denote with v j the voltage inuce on the ipole antenna j, for every j,,..., z. By convention, assume that v correspons to the voltage at the esire irection. One can run an electromagnetic simulation at a given frequency ω to extract the circuit moel of the system. The equivalent circuit moel, referre to as Circuit, is epicte in Figure, which comprises the following ingreients: v,..., v z enote the voltages at the esire an unesire irections in the far fiel. v z+, v z+,..., v n enote the voltages on the reflectors ports in the near fiel. v n+ is the voltage of the input source connecte to the transmitting antenna. The block Y-parameter matrix correspons to the Y - parameter matrix of the antenna system. The block control unit represents the controller to be esigne for the reflectors ports. With no loss of generality, assume that v n+ =. Denote the Y -parameter matrix of the antenna system at the given frequency ω with Y s. Moreover, let z in an y in represent the input impeance an the input amittance of the antenna system, respectively. Definition : For every arbitrary column vectors x an y of the same imension, efine the norm x + iy as x x + y y (note that i stans for the imaginary unit). Given a moulation (complex) point α, a problem of interest is how to esign the control unit in Circuit so that the receive voltage v at the esire irection becomes equal to α an, in aition, a number of norm constraints

3 Fig.. v v v n+ Y-Parameter Matrix v z+ z in v n- v z v n Control Unit Circuit moeling the antenna problem uner investigation on y in an v,..., v z, namely: y in y in, (a) v j β j j, j =, 3,..., z (b) are satisfie, where β, β 3,..., β z are given complex numbers an,,..., z are prescribe tolerances. Note that: yin is the amittance of the source elivering power to the transmitting antenna. The constraint (a) is intene to match the input amittance of the transmitting antenna to that of the input source in orer to minimize the reflecte power. For every j, 3,..., z, β j is a moulation point, which is sufficiently istant from α. The constraint (b) is impose to ensure that while the correct symbol α is receive at the esire irection, wrong symbols β,..., β z are receive at the unesire irections. The simplest type of control unit that one can think of is likely a switching network, which correspons to Circuit given in Figure 5a. We have shown in our recent work [ that eciing whether or not there exists a switching control unit that makes the above constraints be satisfie is an NP-complete problem. To alleviate this issue, we have prove that ealing with reciprocal passive control units leas to a convex problem in the form of LM. Thus, it is henceforth assume that the control unit being esigne is a general reciprocal (linear) passive network (see Circuit 3 in Figure 5b). An important question arises as to what moulation point α an input impeance y in can be generate via a passive control unit. Aressing this significant problem is the core of the present work. Define D to be the set of all moulation points α that can be prouce by a passive network in Circuit 3 such that the constraints given in () are satisfie. Likewise, efine to be the set of all feasible input amittance y in. Note that each of the sets D an can be ientifie by a planar region, because every complex number has a - representation. The sets D an will be referre to as feasibility constellation region an feasibility amittance region, respectively. t is critical to know the regions D an before esigning an optimal moulation/emoulation scheme. The rest of the paper aims to investigate the shapes of these regions. Decompose the complex-value matrix Y s in a block form as: W W W 3 Y s = W W W 3 () W 3 W 3 W 33 where W C z z, W C (n z) (n z) an W 33 C (note that C enotes the set of complex numbers). Define also e, e,..., e z to be the set of stanar basis vectors of R z. Throughout this paper, the notation will be use to show inequalities in the positive efinite sense. Moreover, the notations Re an m enote the operators returning the real an imaginary parts of a complex number, respectively. The following theorem is require for the evelopment of the present work (see our recent paper [). Theorem : A point α belongs to D if an only if there exist symmetric matrices M, N R (n z) (n z) an vectors ṽ C z, ṽ C (n z) such that ṽ e = α, an: [ ( Re W W W W ) M N N M, (3) ṽ e p β p p, p =,..., z (a) ṽ W 3 + ṽ W 3 + W 33 yin (b) ṽ = ṽ W W 3W (c) ṽ = (W 3 W W 3 )(M + Ni) () f there exist such matrices M, N satisfying the above constraints, then one caniate for the amittance of the passive control unit at the frequency ω is: (M + Ni) W + W W W. (5) Note that if the optimization problem in Theorem is feasible, then ṽ an ṽ turn out to be equal to the two subvectors of the output voltages, i.e.: ṽ = [ v v v z ṽ = [ v z+ v z+ v n () t is worth mentioning that the optimization problem propose in Theorem is inee an LM problem. n what follows, an immeiate corollary of this theorem will be presente. Corollary : The feasibility constellation an amittance regions D an are both planar convex sets. Proof: The proof is a consequence of the fact that the necessary an sufficient conitions provie in Theorem are convex. That D is a convex set has an important practical implication: to esign an optimal moulation scheme, it is require to ientify a maximal set of points in the feasibility constellation region with the minimum point-to-point istance greater than, where is a given positive number. f D were a non-convex set with a complicate shape, fining such a maximal set woul be a highly complex problem. n contrast, the convexity of the set D simplifies the esign problem significantly, as note below. 7

4 Define D u an u to be the feasibility constellation an amittance regions, respectively, in the case when the constraints given in () o not exist. n other wors, D u an u are meant to characterize the sets of all possible v an y in which coul be generate via a passive control unit. Note that D an are containe in D u an u, respectively, an therefore fining D u an u leas to unerstaning how restrictive the impose constraints are. This is particularly important for a practical esign because if the constraints turn out to be too restrictive, the esigner may nee to relax the constraints to better utilize the system. The following lemma will be later use to stuy the shapes of D u an u. Lemma : Given a natural number m an real vectors x, x, y, y R m, consier the set of all complex points α for which there exist symmetric matrices M, N R m m such that: α = (x + x i)(m + Ni)(y + y i) (7) an: [ M N N M (8) This set, enote by G, is an open circle (ball) centere at the origin with raius: ( x + x ) ( y + y ) (9) Proof: The proof is provie in the appenix. Define: K := Re W 3 W W W 3 K := m W 3 W W W 3 L := Re W W e L := m W W e ( ) := Re W W W W o := Re.5(K + K i)(l + L i) + W 3 W e o := m.5(k + K i)(l + L i) + W 3 W e r := ( ) K + K ( L + L ) o := Re.5(K + K i)(k + K i) + W 3 W W 3 W 33 o := m.5(k + K i)(k + K i) + W 3 W W 3 W 33 r := ( ) K + K () where is the unique symmetric positive-efinite matrix whose square is equal to the positive-efinite matrix. Theorem : The closure of the convex set D u is a circle centere at (o, o ) with raius r. Likewise, the closure of the convex set u is a circle centere at (o, o ) with raius r. Proof: t can be inferre from Theorem that a complex point α belongs to D u if an only if there exist symmetric matrices M, N R (n z) (n z) such that: [ M N () N M an: α = (W 3 W W W 3 )(M + Ni)W W e W 3 W e () The constraint () can be re-arrange as: [ M Ñ Ñ M (3) where: M := M Ñ := N () The constraint () can be expresse in terms of M an Ñ as follows: α = (K + K i)(m + Ni)(L + L i) W 3 W ( ) =.5(K + K i) M + + Ñ i e (L + L i) W 3 W e =.5( K + K i)( M + Ñi)( L + L i) + o + o i (5) where: K j := K j, Lj := Lj, j, () By applying Lemma to the constraints (3) an (5), it can be conclue that D u is a circle with the aforementione properties. The proof for the set u can be carrie out in the same line, after noting that: y in = (W 3 W W W 3 )(M + Ni) (W W W 3 W 3 ) W 3 W W 3 + W 33(7) The etails are omitte for brevity. The first part of Theorem states that the feasibility constellation region D u is simply a circle with known raius an center. This result significantly benefits the esign of an optimal moulation scheme, because fining a maximal set within a circle can be performe systematically. Furthermore, the secon part of Theorem says that the feasibility amittance region is again a circle, which is a useful fact for the moulation esign. To be more precise, enote the impeance of the input source an the input impeance of the antenna system with zin an z in, respectively. The reflection coefficient at the input of the antenna, enote by T, is equal to T = z in z in z in + z in (8) 8

5 v v v n+ Y-Parameter Matrix v z+ z in v n- v z v n v v v n+ Y-Parameter Matrix v z+ z in v n- v z v n Passive Network (a) (b) Fig. 5. (a): Circuit with a switching control unit; (b): Circuit 3 with a passive control unit. To maximize the power accepte by the antenna, the norm of this reflection coefficient must be minimize. Since the feasibility amittance region is a circle, the feasibility impeance region, enote by, is also a circle whose raius an center can be obtaine in terms of (o, o ) an r. Now, the minimization of the reflecte power amounts to fining a point z in in the circle such that T is minimum. This problem has a simple analytic solution. n other wors, the optimal input impeance that the antenna system accepts by using a passive control unit can be obtaine routinely. After ientifying the sets D u an u, an obtaining an optimal input impeance, it is require to fin a control unit which makes a number of constraints on the output voltages an the input amittance (i.e. the ones given in ()) be satisfie. This implies that the moulation scheme must be ultimately esigne base on D, rather than D u. t can be seen that even though D is a convex set, it may not be a circle. The question arises as how to reconstruct the feasibility constellation set D (or the set ) in the general case. Due to the convexity of the set D (in light of Corollary ), a polygon approximation of this set can be foun efficiently. Given a positive number, consier the problem of fining a simple (non-intersecting) polygon with its vertices on the bounary of D such that the x-istance (i.e. istance in the x irection) between every neighboring vertices of this polygon is less than or equal to. Note that as tens to zero, the interior polygon being sought converges to the region D. The following algorithm can be use to etermine a polygon satisfying the above-mentione properties. Algorithm : Step : Minimize Rev subject to the constraints (3) an (). f the optimization problem is feasible, enote the optimal value obtaine for v with v min. n case of infeasibility of the optimization problem, halt the algorithm because D is empty. Step : Maximize Rev subject to the constraints (3) an (). Denote the optimal value of v with v max. Step 3: Set j =. Step : f j Revmax, minimize mv subject to the constraints (3), () an Rev = Rev min +j, an enote the optimal value of v with P j. Step 5: f j Revmax v min, maximize mv subject to the constraints (3), () an Rev = Rev min +j, an enote the optimal value of v with P j. Step : f j Revmax v min, increment j by an jump to Step. Step 7: The polygon with the vertices P, P,..., P j, P j,..., P, P satisfies the require properties. v min The above algorithm obtains an approximating polygon after solving at most Revmax v min + LMs. The main ieas behin Algorithm are lai out below: Steps an are intene to obtain two vertical lines x = Rev min an x = Rev max tangential to D between which the region D is confine. Due to the convexity of D, the line x = Rev min +j, j Revmax v min, intersects the region D in two points P j an P j. These points are obtaine in Steps an 5. Algorithm presents an efficient optimization-base metho to approximate the unknown region D by a polygon. A erivative of this algorithm can be use for esigning an optimal rectangular quarature amplitue moulation (AM) scheme. Given a positive constant, consier a constellation iagram J in the from of rectangular AM with the points: (j, j ) j [ ζ, ζ, j [ ζ, ζ, j, j Z (9) where Z represents the set of integer numbers. t is esire to obtain the intersection of D an J, enote by p. Note that p consists of those points in the AM iagram J which can be generate by a passive control unit. The following algorithm can be use for this purpose. Algorithm : Step ) Minimize Rev subject to the constraints (3) an (). f the optimization problem is feasible, enote the optimal value obtaine for v with v min. n case 9

6 of infeasibility of the optimization problem, halt the algorithm because p is empty. Step ) Maximize Rev subject to the constraints (3) an (). Denote the optimal value obtaine for v with vmax. Revmin Step 3) Set j =, = an = max Rev (where an are the ceiling an floor operators). Step ) f <, the set c is empty an therefore exit the algorithm; otherwise, procee to the next step. Step 5) f j, minimize mv subject to the constraints (3), () an Rev = + j, an enote the imaginary part of the optimal value of v with yjmin. Step ) f j, maximize mv subject to the constraints (3), () an Rev = + j, an enote the imaginary part of the optimal value of v with yjmax. Step 7) f j, increment j by an jump to Step 5. Step 8) The set p is equal to: [ p = ( + j, k) k yjmin, yjmax, j [,, k, j Z () 3 x Du myin Port acts as a receiving antenna sampling the raiation pattern of the transmitting antenna at a specific angle in the far fiel. Ports to 5 are intene to change the bounary conition of the transmitting antenna. Port 5 correspons to the transmitting antenna. The circuit moel of the antenna system is extracte at the esire frequency GHz (using localize ifferential lumpe ports) by means of the electromagnetic software E3D [. This moel can be either Circuit or Circuit 3, epening on how the impeances of the parasitic elements are esigne an implemente. Note that n an z are equal to 5 an, respectively, an that vn+ = v5 =. As the first goal, it is esire to obtain all possible values for v an yin in this antenna system uner a general passive control unit use to control ports to 5. Algorithm is eploye to approximate the feasibility constellation an amittance regions Du an u (with equal to.). These regions are epicte in Figure 7. t can be seen that both of the regions are circle, which is in accorance with Theorem. Due to the circular shape of Du, it is easy to... V. S MULATON RESULTS u. 5 Consier the antenna configuration epicte in Figure, which consists of a transmitting ipole antenna, metal reflectors each with 5 ports (antenna parasitic elements), an a receiving ipole antenna locate at the far fiel in the upwar irection. There are 5 ports as follows: Configuration of the antenna problem stuie in Section V. mv Note that Algorithm fins the set p after solving at most + LMs. The importance of these algorithms will be further reveale in the simulations provie in the sequel. Fig.. Rev (a) 5 3 x Reyin 8 3 x (b) Fig. 7. (a): The feasibility constellation set Du ; (b): the feasibility amittance region u. fin an optimal number of moulation points in Du which are far away from each other by a prescribe number. To compare the achievable performances of switching an passive control units, enote with Ds the feasibility region for v uner switching control units. Fining the exact shape of Ds requires computing v for all possible switchings, i.e. 5 combinations. Since this may not be possible, a number of switching networks are generate at ranom an the corresponing values of v are plotte in Figure 8a. t can be seen that even though a passive network has far more free parameters than a switching network, the region Du is a fairly goo approximation of Ds, which can be use for fining the har limits on the switching performance. Recall that the feasibility region for yin is a circle shown in Figure 7b. t is esire to fin the set of all possible v uner a passive control unit when the input amittance is matche to the center of this circle. Since yin is enforce to be fixe, one may speculate that the corresponing feasibility constellation region, enote by D, is noticeably smaller than Du. However, it is interesting to note that the set D 7

7 mv x 3 D u D s mv x 3 D D u myin... u mv x 3 D Du 5 5 Rev x Rev x 3. 8 Reyin x Rev x 3 (a) (b) (a) (b) Fig. 8. (a): The feasibility constellation sets D s an D u ; (b): the feasibility constellation regions D an D u. is only a little smaller than D u, as shown in Figure 8b. This is ue to the high egree of freeom in evising a passive control unit. Notice that D, foun using Algorithm, is a circle espite the fact that imposing a constraint on the input amittance can potentially make the feasibility region non-circular (this will be emonstrate later). Now, assume that the impeance of the input source is equal to the stanar value 5Ω. Since 5Ω is outsie the circular feasibility impeance region, the goal is to fin an input impeance (amittance) for the antenna system that minimizes the reflection factor T. By solving a simple geometric problem base on the circle obtaine for u, the optimal input amittance can be foun as.8 +.i. Since this number is locate on the bounary of u, it is expecte that imposing y in to be equal to.8 +.i leas to a feasibility constellation region much smaller than D u. Suppose that the input amittance y in is permitte to be ifferent from the value.8+.i by at most 3. The corresponing amittance feasibility region is shown in Figure 9a (see the colore area). The feasibility constellation region uner this input amittance constraint, enote by D, is also plotte in Figure 9b. t can be seen that this region covers a big part of D u an that D is a non-circular (but nearly circular) region. V. CONCLUSONS This paper stuies the constellation iagram esign for a recently introuce communication system that is base on the concept of near-fiel irect antenna moulation (NF- DAM). Unlike the conventional architectures, the signal is moulate in a NFDAM system after the antenna by varying the electromagnetic bounary conitions of the antenna via a passive controller. One of the major challenges in esigning an optimal NFDAM system is to fin the coverage area of the signal constellation iagram. t is shown that this coverage area is always a convex region that turns into a circle if no constraints are impose on the parameters of the system. Later on, a linear matrix inequality (LM) optimization metho is propose to approximate the coverage region by a polygon with any prescribe accuracy. A similar analysis is Fig. 9. (a): The feasibility amittance constellation (colore area) uner the constraint that the input amittance is in a circle centere at.8+.i with raius 3 ; (b): the feasibility constellation regions D an D u. performe for the ientification of the coverage area of the antenna input amittance. ACKNOWLEDGMENT This research was supporte by ONR MUR N Scalable, Data-riven, an Provably-correct Analysis of Networks, ARO MUR W9NF Tools for the Analysis an Design of Complex Multi-Scale Networks, an the Army s W9NF-9-D- nstitute for Collaborative Biotechnology. REFERENCES [ N. Nagai, Linear circuits, systems, an signal processing: avance theory an applications, Marcel Dekker, 99. [ O. Brune, Synthesis of a finite two terminal network whose rivingpoint impeance is a prescribe function of frequency, Journal of Mathematics an Physics, vol., pp. 9-3, 93. [3 K. S. Narenra an A. M. Annaswamy, Stable aaptive systems, Dover, 5. [ J. Bao an P. L. Lee, Process control: the passive systems approach, Springer, 7. [5 S. Boy an L. Vanenberghe, Convex optimization, Cambrige University Press,. [ P. A. Parrilo, Structure semiefinite programs an semialgebraic geometry methos in robustness an optimization, PhD issertation, California nstitute of Technology,. [7 J. Harrison, Formal synthesis of circuits with minimum noise figure using linear matrix inequalities, EEE Transactions on Circuits an Systems, vol. 5, no., pp , 7. [8 H. G. Hoang, H. D. Tuan an B. N. Vo, Low-imensional SDP formulation for large antenna array synthesis, EEE Transactions on Antennas an Propagation, vol. 55, no., pp. 7-75, 7. [9 A. Babakhani, D. B. Rutlege an A. Hajimiri, Transmitter architectures base on near-fiel irect antenna moulation, EEE Journal of Soli-State Circuits, vol. 3, no., pp. 7-9, 8. [ J. Lavaei, A. Babakhani, A. Hajimiri an J. C. Doyle, Solving largescale linear circuit problems via convex optimization, in Proceeings of 8th EEE Conference on Decision an Control, Shanghai, China, 9. [ A. Babakhani, D. B. Rutlege an A. Hajimiri, Near-fiel irect antenna moulation, EEE Microwave Magazine, vol., no., pp. 3-, 9. [ E3D electromagnetic simulation an optimization software, Zelan Software nc., 7

8 APPENDX Proof of Lemma : The proof will be carrie out in two steps. First, the goal is to show that the set G is containe in the aforementione circle. To this en, consier a point α G together with its associate matrices M an N satisfying the relations (7) an (8). t can be verifie that: α = [ [ [ x x M N y y N M y y [ M N [ x x [ y y y y N M () On the other han, since the eigenvalues of a Hamiltonian matrix are symmetric, it follows from the inequality (8) that the eigenvalues of the symmetric matrix given in the left sie of (8) are all in the interval (, ). Hence: [ M N () N M Therefore: [ [ y y [ M N y y y y N M y y [ [ y y y y y y = ( y y y + y ) The above inequality can be manipulate to arrive at: [ [ [ M N y y y y N M y y y y [ M N ( y N M + y ) Substituting () into () leas to: (3) () α < [ ( x x y + y ) [ x x = ( x + x ) ( y + y ) (5) This shows that the point α is insie a circle centere at origin with the raius given by (9). t remains to show that this circle is containe in G too. A constructive proof will be provie in the sequel. Let α be an arbitrary point in the unerlying circle. Two symmetric matrices M an N satisfying the constraints (7) an (8) are to be constructe. To present the basic iea, assume for now that x = y =. Observe that Reα = x ( Reαx y x y ) y. This suggests that M be consiere as x Reαy x y. However, the symmetry constraint on M may not be satisfie for this choice of M. To resolve this issue, one can symmetrize this term an then efine: symmetrization step must be somehow nullifie. Define now: M := Reα x y + y x (x y ) x y N := mα x y + y x (x y ) x y (7) t is esire to show that the parameters M an N efine above, along with α, make the constraints (7) an (8) hol, which in turn proves that α belongs to G. t is easy to observe that (7) is satisfie. To show the valiity of (8), it is sufficient to prove that (in light of the inequality α < x y ): x y + y x (x y ) x y (8) Given arbitrary constants ζ, ζ R, one can write: (ζ x + ζ y ) ( x y x y y x + (x y ) ) (ζ x + ζ y ) = ( x y x y ) ( ) x ζ y ζ (9) This shows that the matrix inequality (8) hols (note that the inequality (8) was pre- an post-multiplie by a vector ζ x +ζ y rather than a general vector, because the columns of x y + y x are in the span of the vectors x an y ). So far, it is shown how to construct M an N in the case when x an y are zero vectors. These matrices can be obtaine in the same line for the general case, although the argument is more involve. Note that the staring point for constructing these matrices is to efine the matrix U as: [ [ [ x x U : = Reα mα y y x x mα Reα y y ( x + x ) ( y + y ) (3) The main property of U is the following: [ [ Reα mα x = x mα Reα x x f U can be expresse in the form of: [ M N N M U [ y y y y (3) (3) then the corresponing matrices M an N extracte from U satisfy the constraints (7) an (8). This happens when x = y an x = y. Nonetheless, the matrix U must be symmetrize an then the reunant term be compensate in the general case, similar to what was carrie out above in the special case x = y =. The etails are omitte here for brevity. M = Reα x y + y x x y () t can be seen that the constraint (7) may not hol for this choice of M. As a result, the term ae in the 7