The information and wave-theoretic limits of analog beamforming

The information and wave-theoretic limits of analog beamforming Amine Mezghani and Robert W. Heath, Jr. Wireless Networking and Commnications Grop Department of ECE, The University of Texas at Astin Astin, TX 787, USA Email: {amine.mezghani, rheath}@texas.ed Abstract The performance of broadband millimeter-wave mmwave RF architectres, is generally determined by mathematical concepts sch as the Shannon capacity. These systems have also to obey physical laws sch as the conservation of energy and the propagation laws. Taking the physical and hardware limitations into accont is crcial for characterizing the actal performance of mmwave systems nder certain architectre sch as analog beamforming. In this context, we consider a broadband freqency dependent array model that explicitly incldes incremental time shifts instead of phase shifts between the individal antennas and incorporates a physically defined radiated power. As a conseqence of this model, we present a novel joint approach for designing the optimal waveform and beamforming vector for analog beamforming. Or reslts show that, for sfficiently large array size, the achievable rate is mainly limited by the fndamental trade-off between the analog beamforming gain and signal bandwidth. Index Terms Large antenna array, millimeter-wave, analog beamforming, directivity-bandwidth trade-off. I. INTRODUCTION The millimeter wave mmwave band offers a mch higher available bandwidth which is a key ingredient for enabling high data rates in next-generation mobile celllar systems [] [4]. De to the reqired high nmber of antennas [4] [6] to compensate for the low SNR per antenna element, this technology creates several challenges at the same time, particlarly in terms of hardware complexity. Analog processing based on phase shifters and the more general hybrid architectre [] are widely considered techniqes for redcing the hardware complexity. The objective of having large bandwidth and large antenna gain simltaneosly reqires a carefl performance analysis that is consistent with the physical limitations. In fact, as an important part of sch commnication system is governed by electromagnetic theory and by antenna theory, a pre mathematical treatment of commnication systems withot consistent link to physical qantities sch as radiated power might be qestionable. The importance of sing wave-theoretic or circit based models for antennas arrays has been investigated in some previos and recent works dealing mainly with the narrowband case [7] []. Thereby, the impact of antenna spacing and copling on the information theoretic reslts of mltiple antenna systems has been stdied with a circit based definition of power in [8] []. An insightfl and general connection between electromagnetic wave theory and information theory in terms of nmber of degrees of freedom for the signal waveform is provided in [], []. In State-of-the art research on the performance of mmwave systems with analog beamforming, however, generally lacks methodologies for deriving information theoretic reslts in accordance to wave-theoretic aspects and nder certain hardware restrictions. In fact, it is known in the classical antenna theory that there is a fndamental trade-off between the imal achievable gain and achievable bandwidth [3], [4]. These classical reslts, however, do not consider the effect of analog processing and do not provide a simple information-theoretic interpretation. In this paper, we stdy the fndamental limits of analog transmit beamforming that is common across freqency given a certain radiated power. To this end, we adopt a broadband array model inclding delay shifts between the antenna elements [5]. We define the radiated power by the srface integral of the sqared field over a sphere enclosing the antenna array [4]. The total radiated power plays an important role for the design of sch mmwave systems not only from energy efficiency point of view bt also de to reglatory restrictions and interference isses. As a conseqence, the spatial precoding and the temporal waveform generation are copled and cannot be considered independently. Therefore, we formlate a rate imization problem nder a certain total radiated power constraint assming analog beamforming nder single-path channel condition. The optimization parameters are jointly the spatial beamforming vector and the spectral shape. The combined wave-theoretic and information theoretic analysis reveals a fndamental directivity-bandwidth trade-off limiting the achievable rate with analog beamforming. It shows that, for sfficiently large array size, the imal achievable capacity is mainly limited by the freqency independent analog beamforming rather than the actal nmber of antennas. This finding constittes a clear indication towards maintaining a separate RF chain for each antenna to flly exploit the potential of very large antenna arrays. II. SYSTEM AND CHANNEL MODEL We consider a single-ser mmwave system, where a transmitter and a receiver are commnicating via a single stream sing analog beamforming. We focs in this paper on the trans-

The broadband array model is however more appropriate in the context of mmwave systems as the array size might become electrically larger than the total grop delay. In other words, denoting the signal bandwidth by B and the imal array size by D, the narrowband condition B D c c: speed of light is generally njstified in mmwave systems with large array size and bandwidth of several GHz. Fig.. Analog beamforming architectre at the transmitter. Fig.. Radiation intensity as fnction of the azimth angle and freqency for a circlar array, N {3, 6}. Smaller beamwidth implies smaller bandwidth. mitter side. We assme that the receiver perfectly selects its beam in the dominant line-of-sight LOS or non-los NLOS direction. The beamforming gain at the receiver is then simply considered as part of the channel. The beamforming at the transmitter as illstrated in Fig. is performed with N antenna elements in the analog domain sbject to a certain total radiated power constraint while the temporal signal shaping is done in the digital domain. De to the anglar selectivity of the receiver, the reslting channel transfer fnction inclding the receive beamforming is approximately described in the freqency domain by single dominant LOS or NLOS path from the point hf = aθ c,ϕ c,f, where is the path coefficient inclding path phase and strength, aθ c,ϕ c,ω is the far-field array implse response for the azimth and elevation angles-of-departre AoD θ c and ϕ c in a spherical coordinate system as a fnction of the freqency f. The single-path assmption is made for simplicity only, and is not essential to or prpose of stdying the limitations of analog beamforming. Frther, we adopt a freqency dependent array response, which we refer to as the broadband array model. Note that the terminology narrowband or broadband refers here to the freqency behavior of the antenna response and not to the propagation channel, which is assmed to be flat. Even when the individal antenna response is freqency flat, the freqency dependency of the array response might still reslt from the grop delays between these elements. This fact is often neglected in the literatre, where only phase shifts are taken into accont to describe a freqency flat array response. For a niform linear array ULA of hypothetical isotropic antennas with element spacing d in wavelengths at the center freqency f c and dimension N, the broadband freqency response in the passband assming all the freqencies propagate with the same speed is [5] [ ] af,θ T =,,e jπdcosθn fc, f,e jπdcosθn f fc, where f c is the center freqency of the occpied band [,f ] = [f c B/,f c B/], i.e., f c = f /. The term dcosθ f f c acconts for the time shift between adjacent antennas in the freqency domain and cannot be approximated by jst a phase shift with f/f c if N f /f c as explained earlier. In analog beamforming, the transmitter applies a plse shaping filter p f in the digital domain and a freqency independent beamforming vector b in the analog domain to the data signal. Both yield the following strctred spatialtemporal processing vector bf = b p f. 3 In other words, the analog precoding part is common over the entire bandwidth and cannot be adapted over the freqency. The restriction of the analog beamforming vector b to be freqency independent is for practical reasons and constittes the major constraint in terms of performance as shown later. In addition to the freqency independence, the vector b is sally sbject to a constant modls constraint de to the implementation sing phase shifters. As we are interested in information and wave theoretical performance limits, this design constraint is not taken into accont. Considering a single-carrier system with the channel vector from, then the received signal in the freqency domain can be described as ỹf = af,θ c,ϕ c T bf xf zf, 4 with the information signal xf having nit power spectral density and the noise zf having the constant power spectral density N. The state of the art design of bf has mainly evolved from the standard SISO approach, where the waveform generation throgh p f and the spatial beamforming throgh b are considered separately. In particlar, b is commonly chosen as the conjgate of the array response evalated at the center freqency and the desired anglar direction, i.e., b af c,θ c,ϕ c. This method might be not optimal for broadband large antenna arrays de to freqency selective natre of the antenna array

that leads to a copled temporal and spatial behavior and a trade-off between bandwidth and antenna gain. As example, Fig. shows the reslting total response of a circlar array and its corresponding analog beamformer, i.e., the radiation pattern, af,θ T b designed at 6 GHz and θ c = 9 for sizes N = 3 and N = 6. We observe that the beamwidth and also bandwidth decrease simltaneosly with the nmber of antenna, in accordance to classical reslts from antenna theory. Another important physical qantities is the radiated power. The total radiated power plays an important role for the design of sch mmwave systems not only from energy efficiency point of view, bt also de to reglatory restrictions. Additionally, the radiated power at these freqencies is also limited compared to the sb-6 GHz freqencies becase the implementation of efficient power amplifiers is qite challenging and costly at mmwave. De to conservation of energy, the radiated power is defined by the srface integral of the radiation intensity af,θ,ϕ T b p f over a sphere enclosing the antenna array in the far field [6] 4π π af,θ,ϕ T bf sinθ dϕ dθ df P R. 5 A very common, bt physically not necessarily consistent, definition of radiated power is based on the sqared norm of the beamforming vector bf df. This is eqivalent to the physical definition in 5 only for the narrowband case with exactly half-wavelength antenna spacing [8]. Based on the above facts and considerations, we formlate in the next section the joint digital waveform and analog beamforming optimization in terms of achievable rate. III. ACHIEVABLE RATE MAXIMIZATION UNDER ANALOG BEAMFORMING As a conseqence of the copling between the temporal and anglar response in the broadband array model, the goals of concentrating the signal in space beamforming and freqency plse shaping shold be considered jointly. The joint spatio-temporal spectral confinement is essential to characterize the actal achievable rate of the analog hardware architectres. Therefore, we formlate the following rate imization problem nder a certain total radiated power constraint assming analog beamforming nder the single-path transmission assmption: Other radiation properties sch as the EIRP are also restricted by reglation, which might also limit the imal athorized antenna gain. This will not be taken into accont as we are interested in the physical limitations. bf=b p f s.t. 4π log π N af,θ c,ϕ c T bf df af,θ,ϕ T bf sinθ dϕ dθ df P R. 6 The optimization parameters are the spatial beamforming vectorb and the shaping filter p f. In the following, we restrict the analysis to the ULA case in and we reformlate the problem in terms of anglar-temporal spectrm. Particlarly, we exploit the Vandermonde strctre of the array response in to interpret the qantityaf,θ c T b as the discrete Forier transform DFT transform of the vector elements in b. In other words, we define the power spectrm densitys f after the digital processing and the anglar spectrm Gcosθ c f representing the analog processing part, sing the sbstittions Gcosθ f = af,θ T b, S f = p f, Frther, we assme an infinite nmber of antennas, as we are interested in the performance limits. Having nlimited nmber of antennas with half-wavelength spacing d = /, we can relax the anglar spectral form G to be arbitrarily, bt periodic with period f c and satisfying the Dirichlet Forier series conditions. Ths, we can obtain the asymptotic and simplified formlation with infinite array size bf log Gcosθ c fs f df s.t. N Gcosθ fs fsinθ dθ df P R, Gcosθ f,s f, f, θ. The optimization problem 8 is non-convex de to the bilinear form Gcosθ fs f and difficlt to solve in general. We provide instead the optimal soltion for S f given Gcosθ f and vice-versa. We introdce first the Lagrangian fnction for the case S f > and Gcosθ f > LG,S,µ= µ log 7 8 Gcosθ c fs f df N Gcosθ fs fsinθ dθ df P R, 9 with the Lagrangian variable µ. For fixed Gcosθ f, the capacity-achieving S f obtained by the KKT conditions follows from the well-known water-filling power allocation

strategy over the freqency [7] S f= N µ π Gcosθ Gcosθ fsinθdθ c f, for f f, where µ is determined by the imm power constraint in 8 and a = a,. Next, we consider the reverse case with fixed S f and optimized Gcosθ f. To this end, we rewrite the Lagrangian fnction 9 sing the sbstittions Ω = cos θ f and = cos θ in a different way LG,S,µ= µ f c log GΩGf c Ω Gcosθ c fs f df N Ω min f, min min Ω f, S Ω ddω P R, where we made se of the periodicity of the fnction GΩ and the symmetry of the cosine fnction. The KKT condition corresponding to the imization with respect togcosθ c f is obtained from the differential of as follows αc N S f N Gcosθ c fs f µ µ min fc cosθc f f, min min fc cosθc f, f min f, min min f, S cosθc f S fc cosθc f d d =, which can be solved with respect to Gcosθ c f in closed form. In the following we consider the soltion for some particlar cases in terms of θ c. A. Soltion arond broadside of the ULA If cosθ /f, then cosθ f and f c cosθ f f c = f. Therefore simplifies to αc N S f N Gcosθ c fs f µ f S cosθc f d =. 3 We obtain then the optimal soltion for G given S Gcosθ c f = N µ f S cosθc f d S f. 4 For the particlar case of constant spectrm S f across the entire bandwidth B, we dedce the following preposition. Preposition. If cosθ c /f, then the following anglar and temporal spectral shapes provide a local minimm or a saddle point for the imization 8 S f = P R B, 5 Gcosθ c f =, 6 cosθ c log f for f f, and zero otherwise. In other words, a spatio-temporal shape Gcosθ fs f which is flat over the bandwidth B = f and a certain freqency dependent beamwidth satisfying cosθ c cosθ f cosθ c f is a potential optimal soltion. Proof. Since flat constant S f and Gcosθ f can be shown to satisfy simltaneosly the soltions for the alternating imization and 4, they solves the joint KKT conditions and are therefore potential joint imizers of the achievable rate. Preposition implies that the imm antenna gain obtained with flat spectrm is, except forθ c = ±π/ broadside, finite regardless of the nmber of antennas and can imally reach the vale in 6. As example, consider a base station antenna configration with a given sector size of ±6 arond the broadside operating in the 7.5-8.35 GHz band intended for 5G [], then the ULA gain is given by G,ULA,8 GHz = cos6 log 8.35 7.5.dB. 7 Higher freqency bands with larger bandwidth, for instance at 6 GHz might be limited by even lower imm flat gain. Deploying other antenna configrations sch as planar array can, however, improves this gain sbstantially. B. Soltion in the end-fire direction of the ULA The end-fire direction θ c = is a limiting case that prodces the imal delay between the antennas. We expect therefore a more severe trade-off between antenna gain and bandwidth. In the narrowband case, however, it is known that the antenna gain might scale sperlinearly with the nmber of antennas [8], [8]. This phenomenon called sper-gain occrs at element spacing smaller than half-wavelength and reqires low-loss antennas and narrowband operation [9]. Here, we aim instead at analyzing the broadband case with half-wavelength antenna spacing. To this end, we assme a flat temporal spectrm S f = P R /B across the available bandwidth B = f and solve for θ c = in terms of G. The soltion reads as Gf = BN P R log µ f ffc f, 8

where µ is chosen to satisfy the radiated power constraint in 8. Hence, the reslting radiation pattern is not flat as in the previos case, and leads to the following achievable rate in bit/s R end fire = f log µ log log f ffc f df. 9 In the following section, we consider some nmerical examples to illstrate the behavior of the data rate for both cases and at different freqency bands. Achievable rate in bit/s 4.5 4 3.5 3.5.5.5 7.5-8.35 GHz, c = 7.5-8.35 GHz, =6 c 57-66 GHz, c = 57-66 GHz, =6 c IV. NUMERICAL EXAMPLE We apply or reslts from the previos section to the two widely-considered mmwave bands at 8 GHz with 7.5 GHz f 8.35 GHz, and 6 GHz with 57 GHz f 66 GHz. We choose two possible directions at θ c = 6 3 apart from broadside and θ c = 6 end-fire. For θ c = 6, we have cosθ c /f for both bands and we can apply the reslts from Sb-section III-A, while for θ c = we se the reslts from Sb-section III-B. The achievable rate with analog beamforming and infinite nmber of antennas is depicted in Fig. 3 verss the carrier-to-noise density ratio C/N P R /N. As expected, the achievable rate in the end-fire direction is lower than arond the broadside. More interestingly, the 6 GHz band is more affected by the trade-off between bandwidth and beamwidth particlarly in the low C/N regime and the larger bandwidth cannot be exploited efficiently. In fact, the 6 GHz band performs even worse than the 8 GHz when the entire available bandwidth is sed at low C/N vales. For this reason, we consider the optimization of the achievable rate based on the reslts from Preposition with respect to the bandwidth B = f that shold be sed for the 6 GHz band, given θ c and P R, i.e., B f c cosθc cos θc P R R = Blog BN cosθ c log fcb/ f c B/ The reslts of this optimization are shown in Fig. 4 for θ c = 6 and f c = 6 GHz. The figre illstrates that the optimal bandwidth is sensitive to the C/N level and scales similarly to the rate. These observations apply for other mmwave freqency bands as well. V. CONCLUSION We showed that analog beamforming with common coefficients across the freqency has a limited capacity regardless of the nmber of antennas. This limitation reslts from the fndamental trade-off between bandwidth and beamwidth of the reslting radiation pattern. The analysis reveals that larger bandwidth is not necessary beneficial for the achievable rate de the redced antenna gain attained by analog beamforming. Conseqently, the joint design of temporal and spatial signal shape becomes a key for achieving the best trade-off. As ftre work, we aim at considering hybrid precoding and other antenna configrations to mitigate this limitation.. 7 75 8 85 9 95 c P R /N in db-hz Fig. 3. Achievable rate vs. the carrier-to-noise density ratio C/N with analog beamforming for the 8 GHz and 6 GHz bands. The 6 GHz band has lower achievable rate at small C/N despite the mch larger bandwidth, which is de to the bandwidth-beamwidth trade-off. Optimal B/f c.3....9.8.7.6.5.4 8 8 84 86 88 9 c P R /N in db-hz Rate in Gbit/s 8 6 4 8 6 4 8 8 84 86 88 9 c P R /N in db-hz Fig. 4. Optimal bandwidth and achievable rate for f c = 6 GHz and θ c = 6 vs. the carrier-to-noise density ratio C/N with flat spectral. Large bandwidth is only meaningfl for sfficiently high C/N. ACKNOWLEDGMENT This research was partially spported by the U.S. Department of Transportation throgh the Data-Spported Transportation Operations and Planning D-STOP Tier University Transportation Center and a gift by Hawei. REFERENCES [] T. Rappaport, J. R. W. Heath, R. C. Daniels, and J. Mrdock, Millimeter Wave Wireless Commnications, st ed. Prentice-Hall, 4. [] F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovski, Five disrptive technology directions for 5G, IEEE Commnications Magazine, vol. 5, no., pp. 74 8, Febrary 4. [3] T. Bai, A. Alkhateeb, and R. W. Heath, Coverage and capacity of millimeter-wave celllar networks, IEEE Commnications Magazine, vol. 5, no. 9, pp. 7 77, September 4. [4] A. L. Swindlehrst, E. Ayanogl, P. Heydari, and F. Capolino, Millimeter-wave massive MIMO: the next wireless revoltion? IEEE Commnications Magazine, vol. 5, no. 9, pp. 56 6, September 4. [5] T. L. Marzetta, Noncooperative Celllar Wireless with Unlimited Nmbers of Base Station Antennas, IEEE Transactions on Wireless Commnications, vol. 9, no., pp. 359 36, November.

[6] E. G. Larsson, O. Edfors, F. Tfvesson, and T. L. Marzetta, Massive MIMO for next generation wireless systems, IEEE Commnications Magazine, vol. 5, no., pp. 86 95, Febrary 4. [7] S. Loyka, On the relationship of information theory and electromagnetism, in IEEE 6th International Symposim on Electromagnetic Compatibility and Electromagnetic Ecology, 5., Jne 5, pp. 4. [8] M. T. Ivrlač and J. A. Nossek, Toward a Circit Theory of Commnication, IEEE Transactions on Circits and Systems I: Reglar Papers, vol. 57, no. 7, pp. 663 683, Jly. [9], The Mltiport Commnication Theory, IEEE Circits and Systems Magazine, vol. 4, no. 3, pp. 7 44, 4. [] T. Laas, J. Nossek, S. Bazzi, and W. X, On Reciprocity of Physically Consistent TDD Systems with Copled Antennas, in th International ITG Workshop on Smart Antennas, March 7, pp. 6. [] M. Franceschetti, On Landa s Eigenvale Theorem and Information Ct-Sets, IEEE Transactions on Information Theory, vol. 6, no. 9, pp. 54 55, Sept 5. [], Wave Theory of Information. Cambridge, UK: Cambridge University Press, 7. [3] R. F. Harrington, Effect of antenna size on gain, bandwidth, and efficiency, J. Res. Nat. Brea Stand., vol. 64D, p., Febrary 96. [4] R. C. Hansen, Fndamental limitations in antennas, Proceedings of the IEEE, vol. 69, no., pp. 7 8, Feb 98. [5] J. H. Brady and A. M. Sayeed, Wideband commnication with highdimensional arrays: New reslts and transceiver architectres, in 5 IEEE International Conference on Commnication Workshop ICCW, Jne 5, pp. 4 47. [6] A. Balanis, Antenna Theory, nd ed. Hoboken, NJ: Wiley, 997. [7] R. G. Gallager, Information Theory and Reliable Commnication. New York: JohnWiley and Son, 968. [8] S. A. Schelknoff, A mathematical theory of linear arrays, Bell Syst. Tech. J., vol., no., pp. 8 7, Jan 943. [9] M. T. Ivrlač and J. A. Nossek, High-efficiency sper-gain antenna arrays, in International ITG Workshop on Smart Antennas WSA, Feb.