Arbitrary Beam Synthesis of Different Hybrid Beamforming Systems
aa r X i v : . [ c s . I T ] A p r Arbitrary Beam Synthesis of Hybrid BeamformingSystems for Beam Training
Kilian Roth,
Member, IEEE,
Josef A. Nossek,
Life Fellow, IEEE
Abstract —For future millimeter Wave (mmWave) mobile com-munication systems, the use of analog/hybrid beamforming isenvisioned to be an important aspect. The synthesis of beams is akey technology to enable the best possible operation during beamsearch, data transmission and MultiUser MIMO (MU MIMO)operation. The method for synthesizing beams developed in thiswork is based on previous work in radar technology consideringonly phase array antennas. With this technique, it is possible togenerate a desired beam of any shape with the constraints of thedesired target transceiver antenna frontend. It is not constraintto a certain antenna array geometry, and can handle 1D, 2D andeven 3D antenna array geometries, e.g. cylindrical arrays. Thenumerical examples show that the method can synthesize beamsby considering a user defined trade-off between gain, transitionwidth and passband ripples.
Index Terms —millimeter Wave, hybrid beamforming, beamsynthesis.
I. I
NTRODUCTION
To satisfy the ever increasing data rate demand, the useof the available bandwidth in the mmWave frequency rangeis considered to be an essential part of the next generationmobile broadband standard [1]. To attain a similar link budget,the effective antenna aperture of a mmWave system must becomparable to current systems operating at a lower carrierfrequency. Since the antenna gain, and thus the directivityincreases with the aperture, an antenna array is the onlysolution to achieve a high effective aperture, while maintaininga ◦ coverage.The antenna array combined with the large bandwidthis a big challenge for the hardware implementation as thepower consumption limits the design space. Analog or hybridbeamforming are considered to be possible solutions to reducethe power consumption. These solutions are based on theconcept of phased array antennas. In this type of systemsthe signal of multiple antennas are phase shifted, combinedand afterwards converted into the analog baseband followedby an A/D conversion. If the signals are converted to onlyone digital signal we speak of analog beamforming, otherwisehybrid beamforming is used. For the transmission the digitalsignal is converted to a analog baseband signal, followed bya up-conversion. Afterwards, the signal is split into multiplesignals, separately phase shifted, ampflied and then transmittedat the antennas. K. Roth is with Next Generation and Standards, Intel Deutschland GmbH,Neubiberg 85579, Germany (email: { kilian.roth } @intel.com)K. Roth and J. A. Nossek are with the Department of Electrical and Com-puter Engineering, Technical University Munich, Munich 80290, Germany(email: { kilian.roth, josef.a.nossek } @tum.de)J. A. Nossek is with Department of Teleinformatics Engineering, FederalUniversity of Ceara, Fortaleza, Brazil To utilize the full potential of the system, it is essential thatthe beams of Tx and Rx are aligned. Therefore, a trial anderror procedure is used to align the beams of Tx and Rx [2],[3]. This beam search procedure does either utilize beams ofdifferent width with additional feedback or many beams of thesame width with only one feedback stage [4]. In both cases thebeams with specific width, maximum gain and flatness needto be designed.Based on requirements on the beam shape, this work formu-lates an optimization problem similar to [5], [6]. Afterwardsthe optimization problem is solved numerically. This workincludes the specific constraints of hybrid beamforming andlow resolution phase shifters. In [4], the authors approximatea digital beamforming vector by a hybrid one. We generateour beam by approximating a desired beam instead.The superscript s and f are used to distinguish between sub-array and fully-connected hybrid beamforming. Bold small a and capital letters A are used to represent vectors and matrices.The notation [ a ] n is the n th element of the vector a . Thesuperscript T and H represent the transpose and hermitianoperators. The symbol ◦ is the Hadamard product.II. O PTIMUM B EAM S YNTHESIS
In the following we will develop a strategy to synthesisarbitrary beams based on the formulation an optimizationproblem. Furthermore, we show how different constraints canbe used to model the restrictions of different systems.
A. Objective function
The array factor A ( u , a ) of an antenna array is defined as A ( u , a ) = a T p ( u ) , [ p ( u )] n = e j πλ x n ( u ) , (1)where a is the beamforming vector, u is the spatial directioncombining the azimuth and elevation angle. The scalar x n ( u ) is the distance from the location of antenna element n to theplane defined by the normal vector u and a reference point. Acommon choice for the reference point is the position of thefirst antenna, in this case x ( u ) = 0 .The objective of synthesizing an arbitrary beam pattern canbe formulated as a weighted L p norm between the desiredpattern D ( u ) and the absolute value of the actual array factor | A ( u , a ) | f ( a ) = (cid:18)Z W p ( u ) || A ( u , a ) | − D ( u ) | p d u (cid:19) p , (2)where W ( u ) is the weighting. This objective function itselfis convex over its domain, but the constraints on a shown in (a) M C signalsplitter M C signalsplitterdigitalbaseband RFchain1RFchain M RFE (b) M signalsplitter M signalsplitterdigitalbaseband RFchain1RFchain M RFE
Fig. 1. System model of hybrid beamforming transmitter with M antennas and M RFE
RF-chains for the sub-array (a) and the fully-connected (b) case. the following subsections lead to a non-convex optimizationproblem. This problem formulation ignores the phase of thearray factor, since we require only the magnitude to be of aspecific shape.By only optimizing over the array factor we don’t take thepattern of the antennas into account. As described in [5] toaccount for an antenna pattern it is only necessary to divide D ( u ) and W ( u ) by the pattern of the antenna elements. B. Constraints
We consider two different hybrid beamforming designs.These are the systems currently considered in literature [4],[7]. In the first case, all M antennas are divided into groups ofsize M C . Each subgroup consists of one Radio Frequency (RF)chain, an M C signal splitter followed by a phase shifter anda Power Amplifier (PA) at each antenna (see Figure 1 (a)). Intotal there are M RFE
RF chains. This restricts the beamformingvector a to have the form a = W s α s = w s · · · w s . . . ... ... . . . ... · · · w sM RFE α s α s ... α sM RFE , (3)where α s ∈ R M RFE × and the vectors w si models the analogphase shifting of group i and therefore has the form w si = h e jθ s ,i e jθ s ,i · · · e jθ sMC,i i T . (4)In the second case, each of the RF chain is connected to an M signal splitter followed by a phase shifter for each antenna(see Figure 1 (b)). At each antenna, the phase shifted signalfrom each RF chain is combined and then amplified by aPA followed by the antenna transmission. With this systemarchitecture the beamforming vector a can be decomposed into a = W f α f = h w f w f · · · w fM RFE i α f = e jθ f , e jθ f , · · · e jθ f ,M RFE e jθ f , e jθ f , · · · e jθ f ,M RFE ... ... . . . ... e jθ fM, e jθ fM, · · · e jθ fM,M RFE α f α f ... α fM RFE , (5)with α f ∈ R M RFE × .To limit the maximum output power of the PAs, we needto include the following constraints [ a ] m ≤ ∀ m = { , , · · · , M } . (6)It is important to keep in mind that this restriction is afterthe hybrid beamforming, therefore, it is a nonlinear constraintrestricting output-power of the PA. Another way to bound theoutput power is a sum power constraint of the form || a || ≤ . (7)It is also possible that the resolution of the phase shiftersis limited. This means that the values of θ si,j are from a finiteset of possibilities θ si,j = − π + k i,j πK ∀ i, j and k i,j ∈ { , , · · · , K − } , (8)where K is the number of possible phases. A possible phaseshift in the digital domain needs to be taken into account. Inthe case without quantization, this phase shift is redundantwith the analog phase shift. Therefore, in addition to thescaling α f or α s , we need to take a phase shift ξ f or ξ s intoaccount. For the case of sub-array hybrid beamforming withlimited resolution RF phase shifters the beamforming vector a takes the form a = W s ( α s ◦ ξ s ) , (9)where ξ s are the digital phase shifts defined as ξ s = [ e jξ s , e jξ s , · · · , e jξ sM RFE ] T . (10)The formulation for the fully-connected case does also containaddition phase shifts in the digital baseband signals. gain Fig. 2. Illustration of the trade-off associated with the beam pattern synthesis.
C. Problem Formulation
Combining the objective function with the constraints as-sociated with the hardware capabilities lead to the followingoptimization problem min f ( a ) s.t. g ( a ) ≤ , h ( a ) = , (11)where g ( a ) and h ( a ) are the constraints modelling the desiredhardware capabilities. It is important to mention that beam syn-thesis is a similar procedure as digital filter design, thereforewe us the terminology of digital filter design. The weighting W ( u ) , the desired pattern D ( u ) and the choice of p in f ( a ) ,determine which point in the trade-off gain, passband rippleand transition width is going to be targeted as shown in Fig.2. III. N UMERICAL RESULTS
To compare the designed beams we need to first define somemetrics to quantify the difference between them. Some of thesemetrics are similar to the ones defined in [8]. The first one isthe average gain in the desired direction. Directly connectedto the average gain is the maximum ripple of the array factor inthe desired directions. For more reliable results, the transitionregion is excluded from the search of the maximum ripple. Avery important criteria to evaluate the performance of a beamfor initial access is the overlap of adjacent beams of the samewidth. Here we evaluate the area at which the distance betweentwo beams is less than 5 dB relative to the total area of onebeam. The last measure is the maximum side-lobe relative tothe average gain in the desired directions.In the following, beams synthesized by the describedmethod are shown. For all systems, the transmitter is equippedwith M RFE = 4
RF-chains, connected to 64 Antenna elements,forming an Uniform Linear Array (ULA) with half-wavelengthinter-element spacing. Since the antenna array is one dimen-sional, it is sufficient to look at only one spatial direction. Allplots refer to angle ψ = λ sin( φ ) , where φ is the geometricangle between a line connecting all antennas and the directionof a planar wavefront.For each system, three beams of width b = π, π/ , π/ aresynthesized. But it is important to mention that the beams inFig. 3 and 4 are not designed to be used simultaneously. Incontrast, the beams in Fig. 5 and 6 can be simultaneously used. For an ULA, the spatial direction u is fully represented by ψ ,therefore W ( u ) , D ( u ) and A ( u , a ) depend only on ψ . Sincethe magnitude of each element of a is less or equal to one,if a perfect flat beam without sidelobes could be constructed,it would have the array-factor D max = p N π/b . As alsodescribed in [5], such a beam cannot be realized, therefore D ( ψ ) is equal to βD max at the desired directions and equal tozero, elsewhere. The parameter β ensures the feasibility of asolution. The weighting of different parts of the beam pattern W ( ψ ) is uniformly set to 1, except for a small transitionregion enclosing the desired directions. For all systems, weset p = 4 in the objective function to ensure equal gain andside lobe ripples. The integral of the objective function over allspatial directions in the objective function is approximated by afinite sum. To ensure a sufficient approximation, the interval issplit into 512 elements. As described in [5], the computationalcomplexity can be significantly reduced by reformulating theproblem to use FFT/IFFTs to calculate A ( ψ, a ) and thederivatives of the objective function.For each system, the optimization process was started byconsidering several initializations. Since the used NonLinearPrograming (NLP) and Mixed Integer Non-Linear Programing(MINLP) solvers only guarantee to find a local minimumfor a non-convex problem, the results were compared andthe implementation leading towards the minimum objectivefunction was selected. The metrics to compare the performanceof different beams is shown in Table I alongside a referenceto the respective Fig..In Fig. 3 and 4 the synthesized beams for sub-array andfully-connected hybrid beamforming are shown. For (a), (b)and (c) the gain penalty β was selected to be 3 dB, 2 dBand 2 dB, respectively. Compared to the fully-connected case,sub-array hybrid beamforming is characterized by more gainripples and higher sidelobe energy, while having the sametransition width.In Fig. 5 and 6 fully-connected hybrid beamforming withquantized phase shifters was applied. The beams are designedwith the method described in Fig. 6. The beam in both figuresis optimized to simultaneously transmit us both shown beamsat each stage (a), (b) and (c). The power constraint for this caseis also different, in this case only the sum power is constraintto be less or equal to 1. For our evaluation we used the sameconstraints.In Fig. 6, and, especially in (a) there are multiple pointswhere both beams almost overlap. In these directions anestimation of the link quality achieved with both beams isgoing to be very similar. This can possibly lead to a wrongdecision and, in its turn, to large errors in a multi-stage beamtraining procedure. On the contrary, the solution evaluatedin Fig. 5 offers a sharper transition. The stop directionsattenuation is also close to uniform to enable a predictableperformance. The only disadvantage is the larger ripples insidethe center main beam.The shortcomings which are observed in Fig. 6 are in-troduced during the generation of a . As described in [4]this method approximates a version of a d generated withthe assumption of full digital beamforming. Since for a lownumber of RF-chains this vector cannot be well approximated, ◦ ◦ ◦ ◦ ◦ ◦− − − − gain [dBr] (a) ◦ ◦ ◦ ◦ ◦ ◦ (b) ◦ ◦ ◦ ◦ ◦ ◦ (c)Fig. 3. Beams of different width of a sub-array hybrid beamforming array. ◦ ◦ ◦ ◦ ◦ ◦− − − − gain [dBr] (a) ◦ ◦ ◦ ◦ ◦ ◦ (b) ◦ ◦ ◦ ◦ ◦ ◦ (c)Fig. 4. Beams of different width of a fully-connected hybrid beamformingarray. the resulting beam pattern does not correspond well to thedesired one. It is also important to mention that there is noone-to-one mapping between the error in approximating a d and the errors of the corresponding beam. As shown in [4],the method works well if a d can be well approximated by alarger number of RF chains.IV. C ONCLUSION
The developed approach can synthesize any beam-patternfor hybrid-beamforming systems. The numerical examplesshowed that a sufficient solution to the underlying optimizationproblem can be found with reasonable computational complex-ity. The numeric examples also demonstrated that it is possibleto adapt the approach to any type of constraint arising in thecontext of hybrid beamforming and wireless communication. ◦ ◦ ◦ ◦ ◦ ◦− − − gain [dB] (a) ◦ ◦ ◦ ◦ ◦ ◦ (b) ◦ ◦ ◦ ◦ ◦ ◦ (c)Fig. 5. Beams of different width optimized for sidelobe attenuation and with 2bit quantization of the phase shifters of a fully-connected hybrid beamforming. ◦ ◦ ◦ ◦ ◦ ◦− − − gain [dB] (a) ◦ ◦ ◦ ◦ ◦ ◦ (b) ◦ ◦ ◦ ◦ ◦ ◦ (c)Fig. 6. Beams of different width of fully-connected hybrid beamforming arraywith phase quantization according to [4]. TABLE IC OMPARISON OF THE DESIGNED BEAMS .Beam avg.gain dB max rip-ple dB overlapin % max side-lope dBFig. 3 (a) 18.2 4.00 2.44 -17.4Fig. 3 (b) 21.7 2.89 3.22 -16.2Fig. 3 (c) 26.3 2.76 7.21 -16.3Fig. 4 (a) 18.2 2.04 2.63 -22.6Fig. 4 (b) 22. 0 2.10 2.63 -22.8Fig. 4 (c) 24.8 2.35 5.26 -23.3Fig. 5 (a) 2.52 3.90 7.66 -10.3Fig. 5 (b) 5.50 3.01 6.54 -10.1Fig. 5 (c) 8.23 1.47 6.63 -12.7Fig. 6 (a) 2.22 8.82 34.4 -2.16Fig. 6 (b) 5.04 7.25 8.20 -4.04Fig. 6 (c) 8.02 1.49 14.4 -8.97
If we compare the beams synthesized with the methodintroduced in this method to the ones in [4] we can achievea significant smaller overlap 7.66 %, 6.54 % and 6.63 %compared to 34.4 $, 8.20 %, and 14.4 %. This beams aredesigned for a hierarchical beam search, thus the max sidelope is a especially important criteria. Here our result of -10.3dB, -10.1 dB and -12.7 dB is also significantly better than-2.16 dB, -4.04 dB and -8.79 dB.A
CKNOWLEDGMENT
The research leading to these results received funding fromthe European Commission H2020 programme under grantagreement no 671650 (5G PPP mmMAGIC project).R
EFERENCES[1] F. Boccardi et al., “Five disruptive technology directions for 5G,”
IEEECommun. Mag. , vol. 52, no. 2, pp. 74–80, Feb. 2014.[2]
IEEE Standard for Information technology–Telecommunications andinformation exchange between systems–Local and metropolitan areanetworks–Specific requirements-Part 11: Wireless LAN Medium AccessControl (MAC) and Physical Layer (PHY) Specifications Amendment 3:Enhancements for Very High Throughput in the 60 GHz Band , Std., Dec2012.[3] K. Oteri et al., “IEEE 802.11-16/1447r1 further details on multi-stage,multi-resolution beamforming training in 802.11ay,” Nov. 2016.[4] J. Palacios et al., “Speeding up mmWave beam training through low-complexity hybrid transceivers,” in
Annu. Int. Symp. on Personal, Indoor,and Mobile Radio Commun. (PIMRC) 2016 , Valencia, Spain, Sept. 2016.[5] D. P. Scholnik, “A parameterized pattern-error objective for large-scalephase-only array pattern design,”
IEEE Trans. Antennas Propag. , vol. 64,no. 1, pp. 89–98, Jan. 2016.[6] A. F. Morabito et al., “An effective approach to the synthesis of phase-only reconfigurable linear arrays,”
IEEE Trans. Antennas Propag. , vol. 60,no. 8, pp. 3622–3631, Aug. 2012.[7] W. Roh et al., “Millimeter-wave beamforming as an enabling technologyfor 5G cellular communications: theoretical feasibility and prototyperesults,”
IEEE Commun. Mag. , vol. 52, no. 2, pp. 106–113, Feb. 2014.[8] D. De Donno et al., “Hybrid analog-digital beam training for mmWavesystems with low-resolution RF phase shifters,” in