Fundamental Bounds on MIMO Antennas Casimir Ehrenborg, Student Member, IEEE, and Mats Gustafsson, Member, IEEE

1 Fundamental bounds on MIMO antennas Casimir Ehrenborg, Student member, IEEE, and Mats Gustafsson, Member, IEEE

Abstract—Antenna current optimization is often used to ana- single parameter is insufficient to determine their performance. lyze the optimal performance of antennas. Antenna performance As such, it is a challenging problem to construct physical can be quantified in e.g., minimum Q-factor and efficiency. bounds for MIMO systems. However, it is still possible to The performance of MIMO antennas is more involved and, in general, a single parameter is not sufficient to quantify it. utilize antenna current optimization to maximize a given Here, the capacity of an idealized channel is used as the main performance quantity, such as capacity, with restrictions on, performance quantity. An optimization problem in the current e.g., the Q-factor and efficiency. distribution for optimal capacity, measured in spectral efficiency, In communication theory a MIMO network’s capacity is given a fixed Q-factor and efficiency is formulated as a semi- usually optimized for a fixed set of antennas. The performance definite optimization problem. A model order reduction based on characteristic and energy modes is employed to improve the of the antennas is accepted as it is and the upper bound on computational efficiency. The performance bound is illustrated network performance is calculated by e.g., water filling [2]. by solving the optimization problem numerically for rectangular However, in doing so we forgo an opportunity to gain extra plates and spherical shells. performance through optimizing the antennas. In this paper Index Terms—MIMO, Physical bounds, Q-factor, Semidefinite we illustrate how bounds on capacity of a MIMO antenna can programming, Convex optimization be determined by antenna current optimization. Considering the channel between two sets of antennas leads to optimization for specific scenarios or circumstances, in this I.INTRODUCTION paper we are interested in establishing general performance Wireless communication in modern systems utilize multiple bounds for MIMO antennas. As such we focus on one set of input multiple output (MIMO) networks and antennas [1], antennas and idealize the other. The second set of antennas are [2]. These systems consist of two sets of antennas, one characterized as the spherical modes in the far-field. This leads transmitting, and one receiving. Normally, one of these sets to an idealized channel in terms of spherical modes [5], which is situated in a location where space allocation is not an can be thought of as a direct line of sight channel where all issue, such as a base station. However, the other set is usually radiation is received. Considering such a channel also has the contained within a small device, such as a mobile phone, benefit of reducing computational complexity. This is further where design space is limited [3]. Naturally, antenna designs reduced by a model order reduction of the method of moments aim at maximizing performance in such an environment. (MoM) impedance matrix characterizing the antenna. However, there is little knowledge of how the performance The convex optimization problem is constrained by the depends on size, Q-factor and efficiency restrictions. Having efficiency or Q-factor. These are expressed as quadratic forms this knowledge a priori would enable designers to optimize in the current density, where the stored energy in [12] is used. their antenna designs more efficiently. There has been efforts to This leads to a convex optimization problem that maximizes bound MIMO antennas performance for spherical surfaces [4], the capacity in terms of spectral efficiency for a fixed signal- [5] and through information-theoretical approaches [6], [7], to-noise ratio (SNR) and Q-factor. The convex optimization [8]. In this letter a method for constructing a performance problem is a semi-definite program [10] expressed in the bound on capacity for arbitrary shaped MIMO antennas using covariance matrix of the current distribution. current optimization is presented. Antenna current optimization can be used to determine II.MIMO MODEL physical bounds for antennas of arbitrary shape [9]. These physical bounds are found by maximizing a certain perfor- A classical MIMO system is modeled as [2] mance parameter by freely placing currents in the design y = Hx + n, (1)

arXiv:1704.06600v4 [physics.class-ph] 24 Jul 2019 space. By having total control of the current distribution an optimal solution can be reached. While these currents might where x is a N × 1 matrix of the input signals, y is a M × 1 not necessarily be realizable they provide an upper bound matrix of the output signals, n is a M × 1 matrix of additive for the considered problem. Construction of such physical noise, and H is the M ×N channel matrix. The channel matrix bounds are made possible by the ability to formulate convex models how power is transmitted from the input signals to optimization problems [10] for the performance quantity of the output signals, this includes the receiving and transmitting interest. The performance of simple antennas can be quantiﬁed antennas and the wave propagation between them [2]. in e.g., the Q-factor, gain, directivity, and efﬁciency [11]. Fig.1a displays a classical MIMO setup where two sets of MIMO antennas, on the other hand, are more complex and a antennas form a channel. Analysis of such systems depend greatly on external factors, such as, scattering phenomena, Casimir Ehrenborg and Mats Gustafsson are with the Department of Electri- cal and Information Technology, Lund University, Box 118, SE-221 00 Lund, channel characterization, and antenna location [2]. However, Sweden. (Email: {casimir.ehrenborg,[email protected]}@eit.lth.se). to investigate performance bounds for MIMO antennas we 2

a) H where M denotes the map from the currents to the spherical modes. This is a direct channel between the antenna current distribution and the spherical modes [15]. The capacity, ex- ΩT ΩR pressed as spectral efficiency ( b/(s Hz)), of this channel is J T given by [2] J R Å ã 1 H C = max log2 det 1 + MPı Mı , (4) N0 b) Tr(RPÛ )=P where 1 is the M × M identity matrix, and N0 is the noise ΩR power. The noise is modeled as white complex Gaussian noise. H The optimal energy allocation in this channel for capacity J R maximization is given by the water-filling solution [2]. Alter- natively, the optimal solution for this problem can be solved by a semidefinite optimization program,

ΩT H maximize log2 det(1 + γMPı Mı ) J T subject to Tr(RPÙ ) = 1 (5) P 0,

where the unit transmitted power is considered, and γ = P/N0 is the total SNR. Maximizing the capacity of this channel corresponds to focusing the radiation of the antenna to the orthogonal spherical modes. Fig. 1. Illustration of the MIMO system model with transmitter region The solution to (5) is unbounded and increases as mesh ΩT and receiver region ΩR. Part (a) shows the classical MIMO setup with spatially separated regions. Part (b) illustrates the idealized case when the refinement and the number of spherical modes are increased receiver region entirely surrounds the transmitter. The system in (b) is utilized if the SNR is scaled with the number of channels in Mı [2]. in this paper to determined performance bounds on MIMO antennas confined Here, we consider the case of a fixed SNR where the solution to the region ΩT. only depends on the SNR [16], [17]. The solution to (5) can be made more realistic by adding constraints on the losses must limit the degrees of freedom to a single antenna. This or Q-factor of the transmitting antenna [5], [18]. The Ohmic implies that H in (1) should model the channel between an losses are calculated as arbitrary antenna and an idealized receiver, corresponding to 1 H 1 H H PΩ = E I RΩI = E x T RΩTx = Tr(RÙΩP), Fig.1b. The transmitting antenna is modeled with its current 2 2 distribution using a MoM approximation [11] such that each (6) H basis function corresponds to an element of x. The receiver is where RÙΩ = T− RΩT, and RΩ is the loss matrix of the modeled with the radiated spherical modes, where each mode antenna [11]. The stored electric energy is is an element in y [13], [5]. This leads to a MIMO system of 1 H 1 H H 1 We = E I XeI = E x T XeTx = Tr(XÙeP), infinite dimension as N increases with mesh refinement and 4ω 4ω 2ω M increases with the number of included spherical modes. In (7) H numerical evaluation N and M are chosen sufficiently large where XÙe = T XeT, and Xe is the electric reactance ma- to ensure convergence. trix [11]. The stored magnetic energy Wm is similarly defined 1 The transmitted signals are modeled as the MoM current by the magnetic reactance matrix Xm as Wm = 2ω Tr(XÙmP), H elements I = Tx, where the matrix T maps the transmitted where XÙm = T XmT. signals x to the current distribution on the antenna I. The With these constraints in hand we can formulate our opti- 1 H covariance matrix of the transmitted signal is P = 2 E xx , mization problem. We note that the solution is independent of where E {·} denotes the temporal average [2]. With this matrix the power P , so it is sufficient to consider the case P = 1 we can calculate the average transmitted power, giving H 1 1 maximize log2 det(1 + γMPı Mı ) P = E IHRI = E xHTHRTx 2 2 subject to Tr((XÙe + XÙm)P) ≤ 2Q 1 H H = Tr E T RTxx = Tr(RPÙ ), (2) Tr(XPÙ ) = 0 2 (8) Tr(RÙΩP) ≤ 1 − η where RÙ = THRT, and R is the resistive part of the MoM impedance matrix, Z = R + jX [11]. Since we are concerned Tr(RPÙ ) = 1 with connecting the currents on the antenna structure to the P 0, spherical modes [14] in the idealized receiver we express our where η is the antenna efficiency, and self-resonance is en- channel as forced. Here, the problem has been normalized to dissipated y = MI + n = MTx + n = Mxı + n, (3) power, including losses. The consequence of this is that 3

100 the Q-factor considered includes losses in its calculation. It 2 4 6 8 10 12 is possible, and sometimes advantageous, to normalize to different quantities such as the radiated power. Equation (8) 1 is a semi-deﬁnite optimization problem which has a unique 10− a solution [10]. However, the problem is non-trivial due to the σ ` large number of unknowns for realistic antenna problems. For `/2

relative 2 example a rectangular plate of size ` × `/2 discretized into 10− 64 × 32 rectangular elements has N = 4000 unknowns. This size is not a problem for convex optimization of type G/Q and 3 Q [14], [19], [11]. However, the semi-definite relaxation has 10− close to N 2/2 = 8·106 unknowns, making the problem much more computationally demanding. Moreover, the logarithm Fig. 2. The singular values of the channel matrix MÙ for a rectangular plate used in the definition of capacity is more involved than the ` × `/2 for the wavelength ` = 0.21λ. simple quadratic functions in G/Q and Q type problems [14], 100 [11]. Here, the number of unknowns is reduced by expan- 2 4 6 8 10 12 14 16 18 20 sion of the currents in characteristic, energy, and efficiency modes [11], with similar results. The expansion includes only the dominating modes and σ as such constitutes a model order reduction. This implies a 1 change of basis I ≈ U˜I, where U maps between the old and 10−

relative a the new currents. This reduces the number of unknowns to the included modes N1 N. With this approximation the stored energy, for example, is calculated as

H H T H I XÙeI ≈ ˜I U XÙeU˜I = ˜I X‹e˜I ˜˜H Fig. 3. The singular values of the channel matrix MÙ for a spherical shell = Tr(X‹eII ) = Tr(X‹eY‹), (9) r = a, where a = 0.56`, for the wavelength ` = 0.21λ.

H T where Y‹ = ˜I˜I , and X‹e = U XÙeU. Similarly XÙm, and RÙ, are expressed as T , and T . These X‹m = U XÙmU R‹ = U RUÙ our model order reduction preserves these channels it produces replace the corresponding matrices in (8), with Y‹ replacing correct solutions. P. This reduces the number of unknowns from approximately In Fig.4 the capacity has been optimized for a plate of N 2/2 to N 2/2. 1 electrical size ` = 0.21λ, and is depicted as a function of the Q-factor restriction. We see a cut-off for Q ≤ 12 where the III.NUMERICALEXAMPLES optimization problem is unable to realize a feasible current In the following examples the optimization problem (8) has distribution for so low Q-factor, cf., the lower bound on the been solved for a MIMO system resembling Fig.1b using Q-factor [19]. For higher SNR the capacity increases but the the Matlab library CVX [11], [20]. The logarithm in the cut-off stays the same, since the SNR does not affect the Q- optimization problem (8) was replaced by a root of order factor. M [20]. After the optimization has been carried out the We can instead regard the problem with a fixed SNR and capacity is calculated as normal with the optimized currents. investigate how the capacity varies with antenna size, see The energy restriction on the number of transmitter modes and the number of spherical harmonic modes in the receiver γ = 50 have been chosen sufficiently large to ensure convergence γ = 80 30 and varies from example to example. Using too many modes a may also result in the solver failing to solve the problem due to its size and must therefore be regulated for each γ = 110 20 γ = 80 run individually. Since the performance of a MIMO antenna (s Hz)] / γ = 50 cannot be quantified by a single parameter the optimization [ b was run with different constraints. This illustrates how capacity C a is bounded by different requirements on the transmitting 10 antennas. The optimization has also been run for a spherical ` `/2 shell circumscribing the antenna. Qlb = 12 Q By performing a singular value decomposition of the chan- 20 40 60 80 100 nel matrix Mı we can see how many channels dominate Fig. 4. Maximum spectral efficiency achievable for a loss-less rectangular the information transfer between the plate and the spherical plate of size ` × `/2 for the wavelength ` = 0.21λ given maximum Q-factor modes, see Fig.2. Here, we see that there are only a few on the horizontal axis. The dashed lines show the maximum spectral efficiency channels that dominate the rest. This indicates that so long as achievable for the corresponding circumscribing spherical shell. 4

C [ b/(s Hz)] 100 25 30 Q = 60 Q = 10

Q = 60 80 20 Q = 40 Q = 20 20 Q = 10 60 15 (s Hz)] Q /

10 10 [ b

40 C

`/λ 5 20 Qlb 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0 Fig. 5. Maximum spectral efficiency achievable for a loss-less rectangular 0.1 0.2 0.3 0.4 0.5 plate of electrical size `/λ for maximum Q-factor with SNR γ = 50, cf., `/λ Fig.4. The dashed lines show the maximum spectral efficiency achievable for the corresponding circumscribing spherical shell. Fig. 7. Illustration of the bounding surface of spectral efficiency for a loss C [ b/(s Hz)] less rectangular plate as a function of size and Q-factor with SNR γ = 50. Q = 30 ` = 0.29λ The red curve shows minimum Q [19]. 15 Q = 40 ` = 0.21λ Q = 20 occurs at lower efficiencies, this is due to self-resonant currents 10 Q = 30 ` = 0.13λ Q = 30 being inherently less efficient [21]. For the size ` = 0.21λ the Q-factor requirement was varied as well, leading to a slight reduction or increase in capacity. Close to the cut-off 5 efficiency we see a slight decrease in capacity for all cases. This corresponds to the requirement on efficiency limiting η the optimization problem. For lower efficiency requirements other constraints limit the optimization and the capacity is 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 unaffected by the bound on efficiency. Fig. 6. Maximum spectral efficiency achievable for a rectangular plate of In Fig.7 both the size of the antenna and the Q-factor are electrical size `/λ for minimum efficiency η. The losses are modeled as a varied to create a two dimensional bounding surface. This resistive sheet with R = 0.2 Ω/. The minimum Q-factor is set to 30 for surface has a sharp cut-off along the minimum Q line [19] the three main graphs and SNR γ = 50. Solid lines are optimized without enforcing resonance and dashed lines are optimized with resonance. For ` = seen on the left in Fig.7. We see that the increase in capacity 0.21λ the Q-factors [20, 30, 40] are plotted. follows the shape of the minimum Q curve as `/λ and Q are increased. This surface provides a bound on the capacity achievable for MIMO antennas of different sizes and with Fig.5. Depending on which Q is chosen the solution is different bandwidth requirements. only realizable for sizes above a certain cut-off. This cut- off corresponds to the size which has the chosen Q as its IV. CONCLUSIONS minimum achievable Q. Above this size the capacity seems to depend linearly on the antenna size. This is consistent with In this letter we have presented a framework for constructing how capacity scales with the number of antennas included in performance bounds for MIMO antennas. We simplified the a MIMO system [2]. channel problem often considered in communication theory In both Fig.4 and5 the dashed lines show the optimization to an idealized channel consisting of a spherical receiver problem solved for a spherical shell circumscribing the planar surrounding the antenna region. This enables the formulation region. We see that the spectral efficiency achievable by a of a semi-definite optimization problem that gives a bounding planar antenna is much less than that of the sphere. capacity for any antenna that can be constructed within the Setting an efficiency requirement on the optimization may considered region limited by size, SNR, antenna efficiency, restrict which modes are realizable. Fig.6 illustrates how and Q-factor. By utilizing a model order reduction based on capacity varies as a function of antenna efficiency. We see energy and characteristic modes [19] the complexity of the that the capacity is unaffected until some cut-off value where problem is reduced such that it is solvable. the solution is no longer realizable. For electrical sizes ` = These physical boundaries of MIMO antennas represent the 0.21λ and 0.29λ this occurs when antenna efficiency require- ideal solutions possible given complete freedom of current ments is high, above 90%. However, for smaller sizes, such placement within the design area. While the shape of these as ` = 0.13λ, we see that this cut-off occurs at lower antenna current distributions are not easily realizable [21], the bound- efficiencies. The optimization problem has been solved both ing values provide an upper limit to what is possible for real with and without enforcing resonance. When resonance is antenna topologies. It remains interesting to investigate how enforced, showed in dashed lines, we see that the cut-off these bounds compare to antenna designs and measurements. 5

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