Orthogonal Beam-Focusing With Planar Arrays for Multiple Access in Near-Field Communications

1 Orthogonal beam-focusing with planar arrays for multiple access in near-field communications Sotiris Droulias and Angeliki Alexiou, Member, IEEE Abstract—Far-field multiple access techniques, such as space division multiple access (SDMA), leverage the angular orthogonality of beams in the far-field to serve multiple users with minimum interference. Recent approaches to extend this idea into the near-field involve the use of uniform planar arrays (UPAs) to generate focused beams that are able to achieve orthogonality only asymptotically, i.e. for infinitely large transmitter sizes. Gaining access to strictly zero beam correlations with transmitters of finite size could therefore lead to the full suppression of interference in the near-field, boosting the multiple access to unprecedented performance levels. In this work, we propose the use of circular, instead of the widely used rectangular, UPAs to generate focused beams that have strictly zero correlations. We demonstrate analytically that such beams satisfy the orthogonality condition for finite-size transmitters, thus offering a multitude of communication modes for multiple access in the near-field. We prove theoretically that focused beam orthogonality, defined in terms of the inner product of beamfocusing steering vectors, guarantees that the maximum of one beam is co-located with nulls of all other beams, similarly to how the angular orthogonality of classical far-field SDMA guarantees interference suppression. Based on this property, we propose codebook designs for near-field SDMA (NF-SDMA), which we evaluate in different scenarios. Our simulation results verify that the spectral efficiency of NF-SDMA schemes based on the proposed beams can be significantly higher than that achieved by focused beams generated with rectangular arrays. I. INTRODUCTION MULTIPLE access offers the advantage of having several users simultaneously served with minimum interference and optimal resource allocation. It relies on the orthogonality of beams, a key element for successfully retrieving the information assigned to different beams, even though they may coexist. For example, in classical far-field space division multiple access (SDMA) [1] it is possible to distinguish multiple users at different angles, due to the angular orthogonality of the beams. Yet, with the gradual shift of wireless connectivity to higher frequencies, communications enter operation in the radiating near-field of antennas, rendering the applicability of wellknown far-field multiple access schemes questionable. In view of the high requirements of future wireless connectivity [2], the utilization of frequency bands, such as the mmWave and THz, is gradually becoming common ground [3], [4], where new electrically large radiating elements are introduced, such as reconfigurable intelligent surfaces (RISs) [5]–[8], large antenna arrays (LAAs) and extremely large-scale antenna The authors are with the Department of Digital Systems, University of Piraeus, Piraeus 18534, Greece (Corresponding author: Sotiris Droulias, email: [email protected]) arrays (ELAAs), equipped with many antenna elements [9]– [11]. With such large radiating apertures, the wavefront of the radiated wave can be tailored to acquire curvature beyond the typical far-field planar form, enabling beams that can focus, bend and even self-heal after encountering blockage [12]–[16]. Beam-focusing, in particular, is constantly gaining ground as a key element for energy-efficient communications [17]– [22] and localization applications [23]–[26]. Because of their unique ability to concentrate the power at desired distances and directions, focused beams offer controlled operation over the range-angle domain. Hence, as wireless connectivity is gradually shifting from the far-field to the radiating near-field of large radiating apertures, the design of orthogonal focused beams could bring the merits of classical far-field SDMA into the radiating near-field. A. Prior works In [27] it was recently shown that beam orthogonality, defined by means of the inner product of steering vectors, is preserved from the transmitter (Tx) to the receiver (Rx), regardless of the exact beam type. A novel technique for near-field SDMA (NF-SDMA) was introduced, which was based on cosine beams, a specially designed family of beams that ensure minimum interference, a necessary feature for serving multiple users equipped with multi-antenna or even single-antenna receivers. In [28] it was demonstrated how to construct communication modes with focused beams, by taking advantage of the narrow extent of the focal area. The proposed orthogonality ensured minimum interference between beams in the angular domain, similarly to how the maxima and minima of beams are treated in the classical far-field SDMA. In [29], a near-field non-orthogonal multiple access (NF-NOMA) scheme was proposed, which was based on beam-focusing, applied to users having different angular directions. It was demonstrated that the proposed NF-NOMA scheme improves the spectral efficiency compared to conventional far-field communications. In [30], [31], the concept of location division multiple access (LDMA) for single-antenna receivers was introduced, employing focused beams. It was shown that zero correlation between focused beams can be achieved only asymptotically, i.e. infinitely large arrays are required. Similarly, in [32], beam-focusing-based near-field multiple access schemes were analyzed, leveraging the angular orthogonality and the asymptotic range orthogonality of focused beams. In [33] it was recently shown that beamfocusing orthogonality is possible with uniform circular arrays (UCAs), i.e. linear arrays folded in a circular shape; however, 2 Table I SUMMARY OF PRIOR WORKS ON FOCUSED BEAM-BASED MULTIPLE ACCESS Work Array type Focused beam orthogonality [28] ULA angular only [29] ULA non-orthogonal beams [30]–[32] ULA/UPA asymptotic [33] UCA for endfire beams only [34] UPA OAM-enabled this work Circular UPA angular and radial all elements and users must be on the same plane and, hence, the waves generated from the back elements are interrupted from the front elements, with which they additionally interfere, rendering the practical and efficient formation of focused beams questionable. Last, in [34] the authors introduced orbital angular momentum (OAM) to achieve the generation of orthogonal focused beams with finite-size transmitters on the broadside. Yet, because the orthogonality derives entirely from the OAM, complex phase profiles are required. Table I summarizes the key aspects of the above prior works. Evidently, with the exception of [27], the design of nearfield multiple access schemes using uniform planar arrays (UPAs) relies primarily on the utilization of focused beams, which are either entirely non-orthogonal [29] or achieve orthogonality only asymptotically [30]–[32]. Hence, a multiple access scheme based on orthogonal beam-focusing with UPAs is still missing. B. Our contributions In this paper, we propose the concept of beam-focusing using circular UPAs, as a means to generate orthogonal beams that can serve as resource elements for near-field multiple access. The contributions are summarized as follows: •We propose the concept of beam-focusing with circular UPAs, as a means to achieve focused beams with zero correlations. Beam correlations are defined as the inner product of the beam-focusing steering vectors. •We derive analytically the correlations of focused beams, generated with rectangular and circular UPAs, and we provide analytical expressions for their orthogonality. •We prove analytically that focused beams generated by circular arrays can be orthogonal for transmitters of finite size, contrary to focused beams generated by rectangular arrays, which are orthogonal only asymptotically, i.e. for transmitters of infinite size. For the former beams, there is a countable set of orthogonal beams, all propagating along the same direction and focusing at different distances from the transmitter. This property does not hold for the latter beams. •We prove theoretically that beam orthogonality, i.e. zero inner product of the beam-focusing steering vectors, guarantees that the maximum of one beam is co-located with a null of all other beams, similarly to how the angular orthogonality of classical far-field SDMA guarantees interference suppression. •Based on the beam orthogonality property, we propose codebook designs for NF-SDMA, using the class of focused beams generated by circular arrays that offer strictly zero correlations for transmitters of finite size, with respect to all associated parameters, such as the beam direction and focal distance. •Our simulation results verify that the spectral efficiency of NF-SDMA schemes based on the proposed beams can be significantly higher than that achieved by focused beams generated with rectangular arrays. C. Organization and notations In Section II we introduce the system model for beamfocusing with rectangular and circular UPAs. In Section III we study the correlations of the two classes of focused beams and we demonstrate the orthogonality of the focused beams generated with circular arrays. In Section IV we propose codebook designs for multiple access in the near-field. In Section Vwe present the performance analysis and in Section VI we summarize the concluding remarks of this work. Notations: In this paper Zdenotes the set of integers. [·]Tand [·]Hdenote the transpose and conjugate-transpose operations, respectively; ⌊·⌋ is the floor function, |·|denotes the norm of its complex argument and the asterisk ∗denotes the complex conjugate. II. SYSTEM MODEL Let us consider a large UPA, centered at the origin of the coordinate system and extending on the xy-plane, as depicted in Fig. 1(a). The UPA consists of Nx×Nyelements with spacing lxand lyalong the xand ydirections, respectively, and has size Lx×Ly, i.e. Lx=Nxlxand Ly=Nyly. The UPA generates beams with tailored amplitude and phase characteristics that propagate in the z>0 semi-infinite space. The E-field profile at the UPA (z=0) has the form E(x,y) = A(x,y)ejφ(x,y),(1) where Ais the amplitude and φthe phase. As, for several applications, it is convenient to have uniform amplitude, throughout this work we will use constant A(x,y) = 1 V/m in all examples. The coordinates x,yare continuous and, hence, only the discrete locations x=nxlx,y=nylywith nx,ny∈Z are implied in (1). For beam-focusing, the UPA elements create spherical wavefronts, which cause all rays from the UPA to converge at the desired focal distance f0, as depicted in Fig. 1(a). To steer the focused beam along the direction θron the xz-plane, the UPA phase must be of the form [13], [14] φ(x,y) = kq(x−f0sinθr)2+y2+(f0cosθr)2,(2) where k=2π/λis the free-space wavenumber and λthe wavelength. An example phase profile is shown in Fig. 1(b), for θr=0◦, f0=10 m. The spherical phase profile has been folded within the (−π,π) range and the amplitude is uniform throughout the 3 (a) (b ) c i r c u l a r f o o t p r i n t r e c t a n g u l a r f o o t p r i n t z = 4 m z = 1 2 m z = 1 6 m z = 2 0 m ( c ) z = 8 m f0 = 1 0 m Figure 1. Focused beam generation with UPAs. (a) Representation of focused beam as rays traveling from the UPA and converging at the focal distance f0. (b) Amplitude and phase of UPA elements, for rectangular and circular beam footprint. (c) Numerical propagation of focused beam with f0=10 m, using all the elements of the UPA (top) and a sub-array selection to form a disk (bottom). The UPA operates at 150 GHz (λ=2 mm) and consists of 500×500 elements, equally spaced by lx=ly=λ/2. entire UPA area. For rectangular UPA, the beam’s footprint extends across the entire rectangular area Lx×Ly; for circular UPA the beam’s footprint is limited within the disk of diameter L≡Lx=Ly. The shape of the footprint only qualitatively determines how the beam power density is redistributed along its propagation, while the focusing location, which is characterized by the parameters θr,f0, is entirely controlled via the phase of the footprint. This is demonstrated in Fig. 1(c), where the beams that correspond to the individual footprints of Fig. 1(b) are propagated numerically (see Appendix VI-A for details). Note how, despite their qualitatively different features, both beams concentrate their power at the intended location (x=0, y=0, z=f0). In this example the UPA consists of 500 ×500 elements. The operation frequency is 150 GHz (λ=2 mm) and the UPA elements are equally spaced by lx=ly=λ/2, throughout all the examples in this work. III. FOCUSED BEAM ORTHOGONALITY To construct codebooks for NF-SDMA, we first need to investigate under which circumstances the maximum of a focused beam can be co-located with nulls of all other focused beams. To this end, we start by analyzing how the power at the Rx location is associated with the beam orthogonality at the Tx, which is defined by means of the inner product of the beam-focusing steering vectors. A. NEAR-FIELD CHANNEL MODELING Consider a Tx equipped with a UPA, consisting of N elements with coordinates (xn,yn,zn),n=1,2,...,N. The Tx is communicating with multiple Rx’s, located at (xm,ym,zm), m=1,2,...,M. The steering vector of the UPA is written as w= [w1,w2,...,wN]T, with entries wn=Anexp(jφn),(3) where Anis the amplitude and φnis the phase of the nth element. As shown in Appendix VI-A, the line-of-sight (LoS) channel between the nth element of the Tx and the mth Rx is hmn =exp(−jkdmn) dmn ,(4) where dmn =p(xn−xm)2+(yn−ym)2+(zn−zm)2is the distance between the nth Tx element and the mth Rx. All things considered, the E-field at the mth Rx, due to the radiated power from the Tx, is expressed as Em=∑ n hmnwn+νm,(5) where νmis the noise at the mth Rx. For steering vectors with uniform amplitude, we may set A1=A2=··· =AN≡A0. Additionally, for operation in the radiating near-field, yet relatively far from the Tx, we may approximate dm1=dm2=···=dmN ≈dRfor the 1/dmn term in (4). In this case, (5) is written in compact form as Em=A0 dR √N√NaH haw+νm,(6) with ah=1 √Nejψ1,ejψ2,...,ejψNT,(7a) aw=1 √Nejφ1,ejφ2,...,ejφNT,(7b) and ψn=kdmn, i.e. ψn=kq(xn−xm)2+(yn−ym)2+(zn−zm)2.(8) Note that, while hmn in (4) is associated with the radiated field of a single array element, Emin (5) expresses the collective response from all elements. Hence, while hmn is agnostic of L(which expresses the number of all elements) and f0,θr(which express their relative phases), the total field Emdepends on all three parameters. B. BEAM CORRELATIONS AND NF-SDMA Depending on the functional form of φn, the steering vector phase, different types of beams can be generated (e.g., focused [13], [14], bending [16], [35], cosine [27]). While φnin (7b) can take any desired form, ψnin (7a) is always given by (8) and, therefore, φnand ψnare generally different. Note, however, that specifically for beam-focusing, the functional form of φnbecomes identical to that of ψn, simply because focusing requires the formation of spherical wavefronts, which 4 are exactly what the term hmn represents. This becomes apparent when comparing ψnin (8) with φn=kq(xn−f0sinθr)2+y2 n+( f0cosθr)2,(9) which is (2), calculated at the Tx element locations. In view of this observation, consider two focused beams, A and B, with properties (θr,A,f0,A) and (θr,B,f0,B), respectively, corresponding to the steering vector phases φn,A and φn,B. To construct the steering vectors aw,Aand aw,B that generate these beams we use (7b) with φn,Aand φn,B, given by (9). Then, the inner product of the steering vectors will be aH w,Aaw,B. Due to the identical functional form of (8) and (9), aH w,Aaw,Bis equivalent to the inner product aH hawin (6), if we set ψn≡φn,Aand φn≡φn,B; this corresponds to calculating the E-field of beam B at location (xm=f0,Asinθr,A,ym=0,zm=f0,Acosθr,A). As a result, because the nulls of aH w,Aaw,Bwill be also nulls of aH haw, zero correlations of the steering vectors aw,Aand aw,Bwill correspond to locations (xm=f0,Asinθr,A,ym=0,zm=f0,Acosθr,A) where (6) yields Em=0. Hence, the focal point of beam A will be at a location where beam B has a null. Using a different beam A leads to different nulls of beam B and, after scanning the entire parametric space, a beamset is formed, containing all beams that are orthogonal to beam B. If all beams belonging to the beamset are mutually orthogonal, this means that at the focal point of any beam all other beams will be null. Consequently, the orthogonality of the beam-focusing steering vectors, defined in terms of their inner product, guarantees that the maximum of one beam will be colocated with a null of another beam, exactly as occurs with classical far-field SDMA in the angular domain (see Appendix VI-B for details); hence the term NF-SDMA. In view of this property, the task of devising NF-SDMA codebooks simply translates into identifying the combinations of beam-focusing steering vector pairs that yield zero inner product. C. BEAM-FOCUSING STEERING VECTOR ORTHOGONALITY To identify the combinations of beam-focusing steering vector pairs that yield zero inner product, consider a focused beam with tunable parameters θr,f0, and a reference beam with fixed parameters θr,ref,f0,ref, as depicted in Fig. 2(a). The inner product of the two beams is expressed as [27] C=RREref(x,y)E∗(x,y)dxdy pRR|Eref(x,y)|2dxdypRR|E(x,y)|2dxdy ,(10) where Eref(x,y) = exp(jφref(x,y)) and E(x,y) = exp(jφ(x,y)) are the footprints of the two beams at the Tx, with φref (involving θr,ref,f0,ref) and φ(involving θr,f0) given by (2). The integration is performed within the area of the UPA, outside of which the footprints are zero, i.e. |x|<Lx/2,|y|<Ly/2 for rectangular UPA of size Lx×Ly, and x2+y2<(L/2)2 for circular UPA of diameter L≡Lx=Ly. Without loss of generality we consider UPAs with equally spaced elements, i.e. lx=ly≡l. To cast (10) in analytical form, (2) can be simplified by ( a ) - 0 . 0 2 0 0 . 0 2 - 0 . 2 0 0 . 2 r e c t a n g u l a r ( b ) 1 / f01 / f0 , r e f ( m - 1 ) s i n  rs i n  r, r e f - 0 . 0 2 0 0 . 0 2 - 0 . 2 0 0 . 2 c i r c u l a r 1 / f01 / f0 , r e f ( m - 1 ) s i n  rs i n  r, r e f 0 1 - 0 . 0 2 0 0 . 0 2 103 102 101 100 C f0= f0 , r e f ( c ) s i n  rs i n  r, r e f - 0 . 0 2 0 0 . 0 2 103 102 101 100 C f0= f0 , r e f s i n  rs i n  r, r e f - 0 . 2 0 0 . 2 103 102 101 100  r=  r, r e f ( d ) 1 / f01 / f0 , r e f ( m - 1 ) C - 0 . 2 0 0 . 2 103 102 101 100 C  r=  r, r e f 1 / f01 / f0 , r e f ( m - 1 ) Figure 2. Correlation of focused beams. (a) System model for UPA communication systems, depicting a xzcross section of two focused beams with parameters (θr,f0) and (θr,ref,f0,ref), respectively. (b) Plots of C, as a function of the detuning parameters wθ/kl =sinθr−sinθr,ref and wf/kl2=1/f0−1/f0,ref, for a UPA with Nx=Ny=500 elements. Left panel: rectangular UPA. Right panel: circular UPA. The color plots are depicted in log scale, to emphasize the details. (c) Cross-section of Cat the characteristic detuning wf=0. (d) Cross-section of Cat the characteristic detuning wθ=0. The cross-sections are marked in (b) with the dashed lines. taking advantage of the relatively large curvature of the phase, required to focus the beam at distances larger than the RIS size. Expanding (2) around the center of the UPA, i.e. at x=0,y=0, and keeping all terms up to 2nd order leads to φ(x,y) = −kxsinθr+kx2cos2θr+y2 2f0 .(11) As we show analytically in Appendix VI-C, the parabolic phase (11) is a valid approximation of the exact spherical phase profile (2) for a wide range of practical cases. In the limit f0→∞, (11) becomes φ(x,y) = −kxsinθr, i.e. approaches the 5 Table II SUMMARY OF DETUNING PARAMETERS Parameter Description Equation wθDetuning of angles θr,θr,ref (13a) wfDetuning of focal distances f0,f0,ref (13b) phase for conventional beamsteering [13], [14]. In essence, the linear term in (11) accounts for steering and the nonlinear term for focusing. Using (11), the correlation function (10) takes the form C(wθ,wf) = 1 SUPA ZZ ejwθ(x l)e−j 2wfh(x l)2δθ+(y l)2idxdy , (12) where wθ=kl(sinθr−sinθr,ref),(13a) wf=kl l f0−l f0,ref ,(13b) are dimensionless parameters that express the angular detuning and focal distance detuning, respectively, between the two beams, and δθ=f0cos2θr,ref −f0,ref cos2θr f0−f0,ref .(14) The detuning parameters wθand wfare summarized in Table II.SUPA is the UPA area, i.e. SUPA =LxLyfor rectangular UPA and SUPA =πL2/4 for circular UPA. The correlation function Cis demonstrated in Fig. 2(b) for relatively small angles, where δθ≈1, as a function of the detuning parameters wθ/kl =sinθr−sinθr,ref and wf/kl2= 1/f0−1/f0,ref. The UPA has Nx=Ny=500 elements, which are all used in the left panel, while a circular sub-array selection is used in the right panel, to form a circular UPA. The color plots are depicted in log scale, to emphasize the details. Clearly, in the examined parametric space, C=0 can be achieved with both UPAs, as long as f0=f0,ref and θr=θr,ref, i.e. if both beams are focused at the same distance, along different directions. This is explicitly shown in the panels of Fig. 2(c), which correspond to the horizontal cross-sections marked with the dashed lines in Fig. 2(b). However, for beams focused along the same direction, i.e. with θr=θr,ref and f0=f0,ref,C>0 for the rectangular UPA, while C=0 is achieved with the circular UPA, at distinct focal distances. This is demonstrated in the panels of Fig. 2(d), which correspond to the vertical cross-sections marked with the dashed lines in Fig. 2(b). Note that, while Cresults from integration on a continuous aperture, for densely spaced elements (e.g., such as l=λ/2, as considered here), summation of the integrand in (12) over the UPA discrete element locations leads to approximately the same result. Hence, in what follows, we use (12) to analytically calculate the beam correlations for the two characteristic cases, namely wθ=0,wf=0 and wθ=0,wf=0, and subsequently substitute Lx→Nxl,Ly→Nylin the result, to account for the discrete UPA character. D. BROADSIDE BEAMS WITH COMMON FOCAL DISTANCE This case refers to beams with wθ=0 and wf=0, for which θr,θr,ref ≈0◦, i.e. δθ≈1. For rectangular UPA, integration of (12) with wf=0 and δθ=1 leads to Crect (wθ,0) =  sinNx 2wθ Nx 2wθ ,(15) where the subscript rect denotes the rectangular shape of the UPA. The solutions of equation Crect (wθ,0) = 0 satisfy wθ=2πq Nx ,(16) where q∈Z, with q=0, to exclude the nulls of the denominator that lead to Crect (wθ,0) = 1. As dictated by (13a), |wθ|≤2kl, which according to (16) leads to |q|≤Nxkl/π, and hence 0 <|q|≤ Nxkl/π. In Fig. 2(c), left panel, the first few nulls of Crect with q=±1,···±5 are visible. For circular UPA, integration of (12) with wf=0 and δθ=1 leads to Ccirc(wθ,0) = 2 J1N 2wθ N 2wθ ,(17) where N≡Nx=Ny,Jα(·)is the Bessel function of the first kind with α=1, and the subscript circ denotes the circular shape of the UPA. The solutions of equation Ccirc(wθ,0) = 0 satisfy wθ=2rq N,(18) where rq,q∈Z, are the roots of J1(rq) = 0. From the solution set (18) only q=0 lifts the orthogonality and, following the same reasoning as with Crect , the nulls of Ccirc are limited within the values 0 <|rq|≤Nkl. The first few nulls of Crect with q=±1,···±5 are visible in Fig. 2(c), right panel. E. BROADSIDE BEAMS WITH COMMON DIRECTION This case refers to beams with wθ=0 and wf=0, for which θr,θr,ref ≈0◦, i.e. δθ≈1. For rectangular UPA, integration of (12) with wθ=0 and δθ=1 leads to Crect (0,wf) = 2π NxNywf × erf1+j 4Nx√wferf1+j 4Ny√wf , (19) where erf(·)is the error function. Because the function erf(r)/r is nonzero, Crect (0,wf)does not have nulls and, hence, focused beams along the same direction cannot be orthogonal. Although this conclusion has been drawn for small angles, this is a general result, and applies to any angles, as shown in Appendix VI-D. Note that, because r→∞⇒erf(r)/r→0, Crect (0,wf) becomes asymptotically zero for large Nx,Ny. Hence, beams focused along the same direction can become asymptotically orthogonal for large UPA size. As an example, in Fig. 3(a), left panel, Crect is presented as a function of the UPA size, and the detuning parameter wf/kl2=1/f0−1/f0,ref. A crosssection at 1/f0−1/f0,ref =0.1 m−1is marked with the dashed 6 0 250 500 750 1000 - 0 . 2 0 0 . 2 1 / f01 / f0 , r e f ( m - 1 ) r e c t a n g u l a r ( a ) Nx=Ny Ny Nx 0 250 500 750 1000 - 0 . 2 0 0 . 2 1 / f01 / f0 , r e f ( m - 1 ) c i r c u l a r N 0 1 N 0 250 500 750 1000 103 102 101 100 1 / f01 / f0 , r e f = 0 . 1 m - 1 ( b ) Nx=Ny C 0 250 500 750 1000 103 102 101 100 1 / f01 / f0 , r e f = 0 . 1 m - 1 N C Figure 3. From asymptotic to full orthogonality. (a) Beam correlations, as a function of the UPA size Nx=Ny≡N, and the detuning parameter 1/f0− 1/f0,ref, for beams propagating along a common direction with θr=θr,ref =0◦. Left panels: rectangular UPA. Right panels: circular UPA. The color plots are depicted in log scale, to emphasize the details. (b) Cross-sections of Cat 1/f0−1/f0,ref =0.1 m−1for both UPAs. The cross-sections are marked in (a) with the dashed lines. line and is shown explicitly in Fig. 3(b), left panel. Note how increasing the UPA size leads gradually to Crect →0, as also demonstrated in [30], [31]. For circular UPA, integration of (12) with wθ=0 and δθ=1 leads to Ccirc(0,wf) =  sinN2 16 wf N2 16 wf .(20) Equation Ccirc(0,wf) = 0 has roots wf=16πp N2,(21) where p∈Z, with p=0, to exclude the nulls of the denominator that lead to Ccirc(0,wf) = 1, hence |p|>0. The first few nulls of Ccirc(0,wf)with p=±1,···±4 can be identified in Fig. 2(d), right panel. In Fig. 3(a), right panel, Ccirc is presented as a function of the UPA size, and the detuning parameter wf/kl2=1/f0− 1/f0,ref. Contrary to Crect , the correlations acquire zeros, which are explicitly shown in Fig. 3(b), right panel, for the crosssection marked with the dashed line. All orthogonality conditions are summarized in Table III. F. ORTHOGONALITY BEYOND THE BROADSIDE In Sections III-D and III-E we demonstrated that circular UPAs support orthogonal, focused beams for relatively small steering angles. Although the orthogonality breaks for generally different angles and different focal distances, there are two characteristic cases for which it can be preserved at larger steering angles, beyond the broadside.  r, r e f = 5 o, 1 5 o, 2 5 o f0 , r e f = 2 0 m ( a ) c o m m o n f o c a l d i s t a n c e ( b ) c o m m o n d i r e c t i o n 0 4 5 6 103 102 101 100  r, r e f = 5 o Cc i r c 5 1 0 1 5 2 0 103 102 101 100  r=  r, r e f = 5 o Cc i r c 0 1 4 1 5 1 6 103 102 101 100  r, r e f = 1 5 o Cc i r c 5 1 0 1 5 2 0 103 102 101 100  r=  r, r e f = 1 5 o Cc i r c 0 2 4 2 5 2 6 103 102 101 100 Cc i r c  r, r e f = 2 5 o  r ( d e g . ) 5 1 0 1 5 2 0 103 102 101 100  r=  r, r e f = 2 5 o f0 ( m ) Cc i r c Figure 4. Focused beam orthogonality with circular UPAs, beyond the broadside. (a) Correlation of beams with common focal distance f0=f0,ref =20 m. The direction of the tunable beam varies within a range of ±1.5◦around the direction of the reference beam, for θr,ref =5◦,15◦,25◦. (b) Correlation of beams with common direction θr=θr,ref =5◦,15◦,25◦. The reference beam is focused at f0,ref =20 m and the tunable beam within the range 1 −20 m. 1) Beams with common focal distance: In this case (14) becomes δθ =cos2θr,ref −cos2θr, i.e. the orthogonality can be preserved at arbitrarily large angles, as long as θr,ref ≈θr. Hence, beams with the same focal distance that are directed towards close-by angles are expected to achieve strictly zero correlations. This is demonstrated in Fig. 4(a), where f0= f0,ref =20 m, and the direction of the tunable beam varies within a range of ±1.5◦around the direction of the reference beam. In these examples, the beam correlations are studied for θr,ref =5◦,θr,ref =15◦and θr,ref =25◦. 2) Beams with common direction: In this case (14) becomes δθ =cos2θr, which is independent of the focal distance. This is demonstrated in Fig. 4(b), where the beams are focused Table III SUMMARY OF FOCUSED BEAM ORTHOGONALITY CONDITIONS UPA Beam Orthogonality Mode detuning condition enumeration rectangular wf=0wθ=2πq Nx: (16) 0 <|q|≤Nxkl/π wθ=0 - - circular wf=0wθ=2rq N: (18) 0 <|rq|≤Nkl wθ=0wf=16πp N2: (21)|p|>0 7 along a common direction, according to the values of the previous example, i.e. θr=θr,ref =5◦,15◦,25◦. The reference beam is focused at f0,ref =20 m and the tunable beam within the range 1 −20 m. As expected, with increasing angle, the orthogonality gradually breaks. Yet, for a large angular range, Ccirc is preserved at relatively low values, giving promise for efficient suppression interference, even at such non-ideal cases. IV. CODEBOOK DESIGN FOR CIRCULAR UPA’S To design codebooks for near-field multiple access, sets of orthogonal beams are required, i.e. beams with properties that correspond to the nulls of Cfor any pair combination. Next, we will show that, for focused beams generated with circular UPAs, all beams with wf=0 and wθgiven by (18) are orthogonal, and that the same property holds for all beams with wθ=0 and wfgiven by (21). A. BEAMSET ORTHOGONALITY For the beamset with wθ=0 and wf=0 we have found that the nulls of Ccirc are given by (21). Using (13b), we can determine all beams that are orthogonal to the reference beam. These beams are characterized by the focal distance f0=1 f0,ref +π kl 16p N2l−1 ,(22) with p∈Z=0. The values of pin (22) defines a beamset, where all beams are orthogonal to the reference beam. Hence, we can select two beams A and B for two different integers pAand pB, so that each is orthogonal to the reference beam, satisfying (21), i.e. kl l f0,A−l f0,ref =16πpA N2,(23a) kl l f0,B−l f0,ref =16πpB N2,(23b) where f0,Aand f0,Bare the individual focal distances, and the beams propagate along the same direction as the reference beam. Eliminating f0,ref from (23a),(23b) leads to kl l f0,A−l f0,B=16π(pA−pB) N2,(24) where pA−pBis also an integer and, therefore, beams A and B satisfy (21). Because beams A and B are different, pA−pB is nonzero, ensuring that the denominator in (20) is nonzero, i.e. Ccirc ≡0. As a result, because the chosen beams can be any pair belonging to the beamset, (21) guarantees that all beams are orthogonal. For the beamset with wθ=0, wf=0 the proof follows similar steps. B. CODEBOOKS AND MODE ENUMERATION Based on the beam orthogonality that characterizes each beamset of Table III, we can construct two sets of relevant codebooks, which we refer to as CB fnand CBθn. Codebook CB fncontains all focused beams with the same focal distance f0=f0,ref ≡fnand different steering angles. The subscript ndenotes the nth version of the codebook, in which all beams are characterized by the same fn. The orthogonal beam directions are found by expressing the solution (18) in terms of θr, using (13a), i.e. θr=arcsin2rq Nkl .(25) Without loss of generality, we have used θr,ref =0◦for the reference beam. The steering angles in (25) provide the orthogonal beam entries of CB fn, i.e. the near-field communication modes with common focal distance fn. The maximum number of codebook entries depends on the maximum steering angle, θr,max, which is chosen at will. To identify all such beam directions we need to solve (25) for |θr| ≤ θr,max. At this point, we may use a 1st order approximation to the roots of J1(rq), which reads as rq≈π(4q+1)/4. Then, the maximum integer, qmax =max(|q|), that satisfies |θr|≤θr,max is expressed analytically as qmax =sin(θr,max)L λ−1 4,(26) yielding a total of 2qmax +1 orthogonal beams (q= −qmax,...,0,...,qmax), i.e. communication modes. The maximum number of codebook CB fnentries is presented in Fig. 5(a), as a function of the UPA size. The orange dot marks the case for N=500 and max(θr) = 2.5◦. Codebook CBθncontains all focused beams with the same steering angle θr=θr,ref ≡θnand different focal distances. The subscript ndenotes the nth version of the codebook, in which all beams are characterized by the same θn. The orthogonal focal distances are found by are found by expressing the solution (21) in terms of f0,ref, using (13b), which leads to (22). Without loss of generality, we may use f0,ref →∞for the reference beam, in which case the orthogonal beam entries of CBθn, i.e. the near-field communication modes with common direction θn, are given by the focal distances f0=kl π N2l 16p.(27) The maximum number of codebook entries depends on the maximum range of focal distances. The maximum f0, i.e. f0,max, is determined by (27) with p=1, i.e. f0,max =N2l/16; the minimum f0is chosen at will and corresponds to a maximum pin (27). The maximum integer, pmax =max(|p|) that satisfies f0,min ≤f0is found by solving (27) in terms of p, and is given by pmax =$1 8L λ2f0,min λ−1%,(28) yielding a total of pmax orthogonal beams (p=1,2,...,pmax), i.e. communication modes. The maximum number of codebook CBθnentries is presented in Fig. 5(b), as a function of the UPA size. In the example marked with the orange dot, where N=500, f0,max =15.625 m, and f0,min =1 m yields pmax =15, i.e. there are 15 communication modes within the range 1 m −15.625m. 8 0 500 1000 0 2 5 5 0 7 5 100 2 1  r, m a x = 5 o  r, m a x = 2 . 5 o (a) f0=f0 , r e f qm a x N0 500 1000 0 2 5 5 0 7 5 100 125 1 5 f0 , m i n = 1 m pm a x (b )  r=  r, r e f f0 , m i n = 0 . 5 m N Figure 5. Maximum number of communication modes, as a function of the circular UPA size. (a) Modes of codebook CB fn, which contains all beams with common focal distance f0=f0,ref ≡fn, directed within the angular range θr∈[−θr,max,+θr,max]. (b) Modes of CBθn, which contains all beams with common direction θr=θr,ref ≡θn, focused within the focal distance range f0∈[f0,min,16/N2l]. The orange dots mark examples for N=500. C. UPA STEERING VECTORS Once the steering angles and focal distances of all beams have been identified, next the steering vectors are constructed. To this end, we discretize the continuous phase on the UPA grid, i.e. we express (2) at points x=nxlx,y=nyly. The discretized version of the UPA phase acquires the form φ(nx,ny) = kq(nxlx−f0sinθr)2+(nyly)2+(f0cosθr)2, (29) which is (9) expressed in terms of nx=−Nx−1 2,−Nx−1 2+1,...,Nx−1 2,(30a) ny=−Ny−1 2,−Ny−1 2+1,...,Ny−1 2.(30b) The steering vector that imposes the phase profile (29) on the elements of the UPA, is given by a(θr,f0) = 1 pNxNy ay(θr,f0)T⊗ax(θr,f0),(31) where the symbol ⊗denotes the Kronecker product, and ax(θr,f0) = hejφ(−Nx−1 2,0),ejφ(−Nx−1 2+1,0),...,ejφ(Nx−1 2,0)i, (32a) ay(θr,f0) = ejφ(0,−Ny−1 2),ejφ(0,−Ny−1 2+1),...,ejφ(0,Ny−1 2). (32b) The full form of (31) provides the steering vector of the rectangular UPA. For the circular UPA only the subarray with n2 x+n2 y≤((N−1)/2)2is considered, where N≡Nx=Ny. The codebook design steps are outlined in Algorithm 1. V. PERFORMANCE ANALYSIS Due to their unique ability to concentrate power in a small area, focused beams are ideal for serving users that are distributed at generally different angles with minimum Algorithm 1 Codebook design Codebook CB fn M: number of users ▷Same number of beams fn: Choose common focal distance of Mbeams q=1: First codeword index q=M: Last codeword index while q≤Mdo Determine steering angles θrusing (25). Create steering vector for qth beam using (31) Fill in the qth codebook entry end while Codebook CBθn M: number of users ▷Same number of beams θn: Choose common steering angle of Mbeams p=1: First codeword index p=M: Last codeword index while p≤Mdo Determine focal distances f0using (27). Create steering vector for pth beam using (31) Fill in the pth codebook entry end while interference. However, when the users reside in the same LoS with the Tx, multiple access becomes challenging, as all beams have to be co-linear. The latter scenario is therefore particularly interesting and will be considered in the following examples. A. NF-SDMA DESIGN AND ASSOCIATED BEAMS Consider the scenario where six users reside within the range 2.5 m−20 m along the direction θr=0◦. To design six orthogonal focused beams in order to simultaneously serve all users, we will use the codebook CBθn, which is relevant for beams with wθ=0 and wf=0. As reference we choose θr,ref = 0◦and f0,ref →∞, and all beams belonging to the codebook are characterized by θr=0◦and focal distances given by (27). For a UPA with Nx=Ny=500 elements, the first six beams are characterized by f0,1=15.63 m, f0,2=7.81 m, f0,3=5.21 m, f0,4=3.91 m, f0,5=3.13 m, f0,6=2.60 m. The beams that correspond to these focal distances are numerically propagated both for the circular UPA and the rectangular UPA, as shown in Fig. 6(a) and Fig. 6(b), respectively, where cross sections along the center of each beam (x=y=0) are presented. The range of interest is divided into six zones (alternate white/gray areas), the boundaries of which are defined as the distances where the shown cross sections of neighboring beams acquire the same power density. As such, each user is located within each one of the six zones, and all users are served individually and simultaneously by the six focused beams. For circular UPA, the maximum of each beam is co-located with minima of all the other beams, which are suppressed by more than 40 dB in all cases, essentially corresponding to power nulls. Note how, with decreasing distance, the minima become less suppressed, as a consequence of the fact that the approximation dm1=dm2=··· =dmN ≈dRin (6) gradually 9 0 5 1 0 1 5 2 0 2 5 - 8 0 - 6 0 - 4 0 - 2 0 0 m o d e 1 2 3 4 5 6 ( b ) z ( m ) PR x ( d B ) 0 5 1 0 1 5 2 0 2 5 - 8 0 - 6 0 - 4 0 - 2 0 0 m o d e 1 2 3 4 5 6 ( a ) z ( m ) PR x ( d B ) f0,6 f0,5 f0,4 f0,3 f0,2 f0 , 1 Figure 6. Local power distribution of focused beams and application to NFSDMA. Six users reside within the range 2.5 m−20 m, which is divided into six zones (alternate white/gray areas). Each user, located within each zone, is served by a dedicated focused beam according to the codebook CBθnwith θn=0◦for (a) circular and (b) rectangular UPA. The striped pattern marks the area outside the range of interest. All beams are normalized to the maximum of beam #6. breaks. Note also that the maximum of each beam does not occur exactly at the corresponding f0, rather it is slightly shifted closer to the Tx. This is a well-known property of focused beams that has been studied analytically [14], [36] and is associated with how well focusing is performed; for larger footprints and/or smaller Tx-Rx distances, the beam maximum practically occurs at f0. Overall, for rectangular UPA, the minima are at substantially high power levels with respect to the maxima. B. SPECTRAL EFFICIENCY To evaluate the performance of NF-SDMA we use the receive signal-to-interference-plus-noise ratio (SINR) per Rx as performance metric, which is given by SINRm=Pm ∑l=mPl+Pν ,(33) where Pmis the received power at the mth Rx and Pνis the noise power. The receive SINR at each Rx is related to its achievable rate as Rm=log2(1+SINRm),(34) which can be used to express the overall spectral efficiency as R=∑mRm. First, the user locations are fixed within their corresponding zone, i.e. user #1 at z=f0,1, user #2 at z=f0,2,..., user #6 at z=f0,6. Using the beams of Fig. 6, the average spectral efficiency, ⟨R⟩, as a function of the SNR is demonstrated in Fig. 7(a), for both circular and rectangular UPAs. For low SNR, the nulls of focused beams generated with circular - 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 2 0 4 0 6 0 8 0 S N R ( d B ) ( a ) c i r c u l a r r e c t a n g u l a r m a x < R > 6 u s e r s a t f0's < R > ( b p s / H z ) 0 2 5 5 0 7 5 1 0 0 z o n e c o v e r a g e ( % ) ( b ) 0 2 0 4 0 6 0 0 1 F ( R ) R ( b p s / H z ) - 1 0 0 1 0 2 0 3 0 4 0 5 0 6 0 0 1 0 2 0 3 0 4 0 5 0 S N R ( d B ) 4 u s e r s 6 u s e r s 8 u s e r s ( c ) 1 0 0 % c o v e r a g e 6 u s e r s 8 u s e r s 4 u s e r s c i r c u l a r r e c t a n g u l a r < R > ( b p s / H z ) m a x < R > Figure 7. Spectral efficiency of the NF-SDMA scheme, presented in Fig. 6. (a) ⟨R⟩as a function of the SNR, when each user is located at each f0, within its dedicated zone. (b) ⟨R⟩as a function of the zone coverage, for SNR = 60 dB. inset: CDF for 100% zone coverage shown in (b). (c) ⟨R⟩for 100% zone coverage, as a function of the SNR. 0 1 0 2 0 3 0 4 0 0 2 0 4 0 6 0 8 0 c i r c u l a r r e c t a n g u l a r S N R = 1 0 d B ( a ) < R > ( b p s / H z )  r ( d e g . ) 0 1 0 2 0 3 0 4 0 S N R = 2 0 d B  r ( d e g . ) 0 1 0 2 0 3 0 4 0 S N R = 3 0 d B  r ( d e g . ) 0 5 1 0 1 5 2 0 2 5 0 1 0 2 0 3 0 4 0 ( b )  r ( d e g . ) z ( m ) b e a m # 1 f0 = f0 , 1 - 2 0 d B 0 d B 0 5 1 0 1 5 2 0 2 5 0 1 0 2 0 3 0 4 0 ( c )  r ( d e g . ) z ( m ) b e a m # 1 f0 = f0 , 1 - 2 0 d B 0 d B Figure 8. Spectral efficiency of the NF-SDMA scheme presented in Fig. 7(a), for users located beyond the broadside. (a) ⟨R⟩as a function of the common direction θrfor SNR = 10, 20, 30 dB. (b),(c) Cross-sections of beam #1 (shown in log scale), generated with circular and rectangular UPA, respectively. UPAs are submerged in the noise background, yielding similar