Design of Near-Field Beamforming for Large Intelligent Surfaces

In this paper, we propose a novel three-dimensional (3D) near-field beamforming (BF) design for Large Intelligent Surface (LIS). We firstly investigate the definitions of near-field and far-field of LIS, and derive the Fresnel near-field region where amplitudes variations are negligible but only phase variations worsen the harvested array-gains. We show that the Fresnel region which covers the majority part of near-field, can be enlarged by a factor of four when considering possible imperfectness from a conventional two-dimensional (2D) far-field BF. Therefore, it is of interest to design an analog 3D-BF that can recover array-gain losses in this region. Secondly, with a decomposition theorem we show that the optimal 3D-BF can be decomposed into a 2D far-field BF and a one-dimensional (1D) near-field BF. The 2D far-field BF compensates phase variations from mismatches in the azimuth and elevation angles, while the 1D near-field BF compensates remaining phases variations caused by distance differences from a user-equipment (UE) to different antenna-elements on LIS. Such a proposed “2D+1D” BF design reduces codebook-size significantly and is compatible with the existing far-field BF in the fifth-generation new-radio (5G-NR) system. Thirdly, we analyze an optimal codebook design for the 1D near-field BF, and show that with a small codebook it can perform close to optimal. Numerical results verify that the proposal is effective to recover array-gains in the near-field of LIS.


I. INTRODUCTION
I N THE fifth-generation new-radio (5G-NR) and sixth- generation (6G) wireless systems [2], [26], [29], novel architectures and network elements have been introduced.Those additions require researchers and engineers to redesign network infrastructures and revisit the fundamentals of wireless communication.Recently, the concept of Large Intelligent Surface (LIS) [3], [4], [5] arises as an evolution of massive multi-input multi-output (MIMO) system, which shows substantial improvements in signal-transmission and terminal-positioning. Similar concepts including reflection intelligent surface (RIS), intelligent reflection surface (IRS), holographic MIMO [6], [7], [8], [9], and others have also been developed and gained emerging interests.In practice, LIS can be implemented in a discrete form following an optimal sampling [3], which is close to an extremely large antennaarray (ELAA) system [10].An important design aspect of LIS is three-dimensional beamforming (3D-BF).Unlike a conventional MIMO system that beams a signal towards a certain direction, LIS can direct the signal towards a point in the 3D space via BF.This property has stimulated extensive interests in using LIS for applications e.g., interferencenulling, wireless energy-transfer, gesture-detection, and many others [8], [11].
Conventionally, radio-wave propagation is classified into two regions, the near-field and the far-field, respectively.When an antenna radiates a signal in free space, the field distribution is uniquely determined by Maxwell's equations [12], and the characterize of propagation varies.In the far-field region, variations of amplitude and phase are both negligible, and the path-loss effect is dominant in determining the received signal strength.While in the near-field region, there are noticeable amplitude and phase variations depending on the distance from a considered user-equipment (UE) to an antenna-surface [13], [14].The distance that splits the nearfield and far-field is called the Fraunhofer or Rayleigh distance, which limits the phase variations to be less than π/8.Moreover, the near-field can be further split into the reactive region and the radiative Fresnel region [13].The reactive nearfield is quite close to the antenna-surface, whose boundary is considered to be only proportional to λ/(2π), where λ is the wavelength.Below this boundary, there are strong inductive and capacitive effects from the currents and charges in the antenna.The Fresnel near-field region, on the other hand, covers the majority part of near-field that spans from λ/(2π) onward to the Fraunhofer distance.
With LIS, definitions of near-field and far-field regions are similar to conventional ones by treating the entire surface as a whole.However, there are also differences.Firstly, even when a UE is in the near-field of LIS, from the perspective of each antenna-element on it, the far-field assumption holds whenever a UE is a few wavelengths away.Secondly, even when the UE is in the far-fields of all antenna-elements, the transmitted signals from different antenna-elements reaching the UE (and vice-visa) can still yield distinctive propagation properties.In which case, the planar-wave assumption of the far-field no longer holds, and it means that the UE enters the near-field of LIS.Lastly but most importantly, with an aperture-antenna the entire physical structure is treated as a whole and there is no freedom to mitigate the near-field impacts, but with LIS each antenna-element can be controlled separately, and near-field 3D-BF can be designed accordingly to resolve those impacts occurred.
It is worth-noting that the conventional near-field is derived when the largest phase-difference is less than π/8 from any two points on an aperture-antenna, which is attained with the center-point and the points at the boundary.However, this only holds when the UE is located on the boresight of the antenna, and when UE is off the boresight, it is no longer true.This observation is important when considering the nearfield region of LIS, or to be more precise, the region where near-field BF can be effective.As we show in this paper, with a two-dimensional (2D) far-field BF that perfectly compensates mismatches from azimuth and elevation angles, an arbitrarily located UE can be equivalently transferred to being as if located on the boresight, and the conventional definition of near-field holds.However, if the 2D far-field BF is imperfect or without any BF, the Fraunhofer distance can be enlarged by a factor of four.Since achieving a perfect far-field BF requires a prohibitive codebook-size, the region where nearfield impacts come into play is relatively large and near-field BF is important.
Although 3D near-field BF has been considered in literature before, e.g., [6], [7], [15], [17], [18], [19], [26], [27], to our best knowledge, it has not been studied from a practical implementation perspective that cooperates with existing farfield BF schemes such as in 5G-NR system.For instance, the near-field BF in [15] for a microphone-array is derived to be equal to the signal propagation vector, which is however, unknown at the transmitter.In [6] and [7], the near-field BF is designed for a RIS that cooperates with a next-generation NodeB (gNB), which shows that the optimal BF weights depend on the distances among gNB, RIS, and UE.The design in [6] uses an estimate of distance via sensing, while in [7] the BF utilizes a codebook based approach with measurement feedbacks from UE.Nevertheless, detailed analysis of the near and far fields are not fully elaborated in those approaches.Moreover, the insight that 3D-BF in the Fresnel region can be decomposed and designed in a much simpler form than it seems, is missing.
In this paper, we propose a novel and practical near-field 3D-BF design for LIS.We firstly analyze the Fraunhofer distance and the distance that determines the Fresnel nearfield region in Sec.II, when UE can be off the boresight.Enlightened by the observation that only phase variations worsen the array-gains, with a decomposition theorem the optimal 3D-BF is decomposed into 2D+1D BF in the Fresnel region.Based on which, we propose a two-step BF scheme to mitigate the near-field impacts in Sec.III.The first Fig. 1.An illustration of the LIS transmission under two cases: a UE is located on the z-axis that is aligned with the boresight; or located off the boresight with an elevation angle φ (0 < φ < π/2) and an azimuth angle ϕ (0 ≤ ϕ < 2π).
step follows a conventional 2D far-field BF to compensate phase variations resulted from azimuth and elevation angles, which is equivalent to rotate the LIS-array such that the UE is transferred onto its boresight.Secondly, a derived 1D near-field BF is applied to compensate remaining phase variations, which is different from a conventional discrete-Fourier-transform (DFT) beam.In Sec.IV, we analyze the optimal codebook design of the proposed 1D near-field BF, and usu a Lloyd-Max iteration algorithm to optimize the quantization and boundary points.It is shown that the meansquare-error (MSE) is strictly reduced over iterations until converge, and a small codebook can perform close to optimal.Simulation results are shown in Sec.V, which demonstrates that dramatic array-gain losses can be observed when UE enters the near-field of LIS.However, those losses can be effectively recovered with the proposed 2D+1D BF scheme.Finally, the summary is in Sec.VI.

A. Model Assumptions
To analyze the near-field region of LIS, we consider a single-antenna UE under a perfect line-of-sight (LoS) scenario.When a UE is equipped with an antenna-array for receiving, it is typically of a small array-size such as 2 × 2 or 4 × 4. Therefore, the propagation differences among these receivingantennas are negligible, and the near-field impacts are identical to all.Once these near-field impacts are compensated at the transmitting LIS, the received signals can be regarded as planar waves when reaching the UE, and applying a conventional 2D far-field BF for receiving is close to optimal.However, when the UE is also equipped with a LIS, the impacts from the receiving LIS should also be considered, and similar principles can apply due to the reciprocity between the uplink and downlink transmissions.
The LIS-array1 is assumed to contain M × N antennaelements and forms a 2D uniform planar-array (UPA).Any two adjacent antenna-elements in each dimension are separated with a distance of half-wavelength λ/2.An illustration is depicted in Fig. 1, where without loss of generality we assume that the LIS-array is located on the x − y plane and centered at the origin point with Cartesian coordinates (0, 0, 0).We further assume an orthogonal polarization in the x and y directions, and the propagation is along the z-axis.The Cartesian coordinates of the (m, n)th antenna-element, i.e., an antenna-element located at the mth (1 ≤ m ≤ M ) row and the nth (1 ≤ n ≤ N ) column of the LIS-array, are denoted as The largest distance between two antenna-elements, attained by those two at corners on a diagonal, equals Conventionally, D = λ 2 √ M 2 + N 2 is used to define the Fraunhofer distance [30], but for array-antenna D can be more precise.Nevertheless, when M and N are large, the difference is small.Furthermore, the UE is assumed to be located at an arbitrary location with coordinates (x, y, d), and the distance to the LIS-center is ( Without loss of generality, we define two mild conditions, That is, the UE is relatively far way from the LIS compared to its radius, and meanwhile, the LIS-array is relatively large.
In what follows, we consider two different cases, referred to as Case-1 and Case-2, respectively: • Case-1: A UE is located on the boresight of LIS.
• Case-2: A UE is located off the boresight of LIS.Note that after a perfect 2D far-field BF, the transmitting angles in both azimuth and elevation directions are aligned between LIS and UE, and the UE can be equivalently treated as located on the boresight of LIS, which is Case-1.Even so, when UE enters the near-field of LIS, the harvested arraygain can be dramatically decreased.With Case-2, the near-field region can be enlarged when UE is slightly off the boresight, which can be, for instance, resulted from an imperfect 2D farfield BF.In the next, we always treat Case-1 firstly, and then generalize the results to Case-2.

B. Near-Field Region With Case-1
When the UE is located on the boresight of LIS at coordinates (0, 0, d), it holds that d = d.The shortest and longest distances, denoted as d 1 and d 2 , respectively, are corresponding to two antenna-elements that are the closest and farthest to the LIS-center and equal to By applying the approximation x is small, it holds that Note that the approximations in ( 5) and ( 7) follow from the conditions in (4).Utilizing the same definition with an aperture-antenna [14] such that the Fraunhofer distance is defined for the largest phase variation to be π/8, it holds from (7) that which yields Therefore, the Fraunhofer distance d F,1 follows a conventional definition and is summarized in Property 1, which linearly increases in the surface-area of LIS and decreases in λ.Property 1: The Fraunhofer distance equals d F,1 = 2 D2 /λ with Case 1, which is the boundary of the near-field and farfield regions of LIS.
Note that although a UE is in the near-field of LIS when (9) holds, seen from a single antenna-element on LIS, the UE can be in its far-field.
Example 1: When λ = 5cm, it holds from (9) that d F,1 is about 48m with M = N = 32, and 198m with M = N = 64, respectively.In both cases, the near-field regions are not so small, especially for indoor deployments.

C. Enlarged Near-Field Region With Case-2
Although d F,1 follows a conventional definition [25], [27], [30], the fact that it is derived under the assumption that UE is perfectly located on the boresight of LIS, is usually overlooked.In practice, a UE can be located off the boresight.Although a 2D far-field BF can be used to compensate the azimuth and elevation angles, it is however, can be imperfect due to practical limitations such as codebook-size and angle precision.As a result, the Fraunhofer distance, below which the near-field impacts deteriorate array-gains, can be larger than Case-1.In fact, even with a small elevation angle φ after Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the 2D far-field BF, the Fraunhofer distance can be enlarged by four times.This observation is stated in Property 2.
Property 2: Assuming the UE is off the boresight of LIS with a small elevation angle φ such that the Fraunhofer distance is enlarged to The distance between the LIS-center and the projection point on the LIS of a UE located at (x, y, d), equals d tan φ as shown in Fig. 1, and we consider two different cases.
The first one is d tan φ ≤ D/2, and the largest distance difference from the UE to different antenna-elements, by assuming D 2d + tan φ is small from (4), equals which happens when the projection point is on top with an antenna-element on the diagonals.For the phase difference to be less than π/8, it is sufficient to have which yields This also implies The second case is d tan φ > D/2.From the triangle inequality, the distance from the projection point of UE to any antenna-element on the LIS belongs to . Hence, the largest distance, which occurs when the projection point is also on a diagonal, equals For the phase difference to be less than π/8, it requires Hence, in both cases it is sufficient to have d ≥ d F,2 , while the angle φ satisfies (10), for the UE to be in the far-field of LIS.Property 2 reveals an interesting fact that when a UE is off the boresight, the Fraunhofer distance can be enlarged by a factor of four, even when the elevation angle φ is small.Hence, the chance for a UE to enter the near-field of LIS, i.e., when near-field BF can be effective to address the nearfield impacts, is higher than it seems from a conventional perspective.When the 2D far-field BF perfectly compensates the phase variations from mismatches in azimuth and elevation angles, Case 2 can be equivalently transferred into Case 1 with φ = 0, and the Fraunhofer distance remains the same as in Property 1.However, since a 2D far-field BF can be imperfect, it is important to apply near-field BF in the enlarged near-field region to recover array-gains.One such example is in below.
Example 2: A UE is located at coordinates Note that the Fraunhofer distance d F,2 is measured as the vertical distance between the UE and LIS, and the distance from UE to the LIS-center is d = d/ cos φ.When φ is small, the difference between them is negligible.On the other hand, when φ is large, d should also increase for the UE to be in the far-field, similar to the case with a conventional apertureantenna.

D. Negligible Amplitude Variations in Fresnel Region
Besides phase variations, there are also amplitude variations in the near-field.With Case 1 and seen from ( 7), the largest distance variation is negligible as long as d ≥ 2 D, which has been shown in [14] and [30].When amplitude variations are negligible, the power attenuations from different antennaelements to the UE are also negligible, and applying an analogy BF is sufficient to harvest the maximal array-gains.However, when UE is in the near-field with d < d F,2 and even with a small elevation angle (after conventional 2D far-field BF) such that tan φ ≤ D 2d , it holds from (12) that the largest distance difference can be enlarged by four times compared to Case 1. Analogous to Case 1, this indicates that it shall hold d ≥ 8 D with Case 2 for amplitudes variations to be also negligible, which is stated in Property 3.
Property 3: When the distance d ≥ d 0 ≜ 8 D, which is proportional to the diameter of LIS, the amplitude variations are negligible for general UE locations when the elevation angle satisfies 0 < tan φ ≤ 1  16 .While for the case that UE is located on the boresight of LIS (φ = 0), it is sufficient to have d ≥ 2 D.
Note that the ratio which is 8.8m and 17.8m for N = 32 and 64, and the ratio d 0 /d F,2 is 18% and 9%, respectively.In other words, in the majority region of near-field, the amplitude variations are negligible and only phase variations need to be compensated with analog phase-shifts.This is also aligned with theoretical analysis in [30], which shows that maximal array-gains can be fully recovered even in the near-field.In the next section, we will provide an effective and low-complexity BF design to attain those theoretical gains.
To summarize the discussions so far, we show that when a UE is in Case 2, the Fresnel near-field, i.e., the region where phase variations deteriorate array-gains, is in the distanceregion (2 D, 8 D2 /λ), while a conventional Fresnel near-field region spans between (2 D, 2 D2 /λ) and derived under an ideal assumption of Case 1.When a UE is in Case 2, in order to have negligible amplitude variations for analog phase compensations to be most effective, the Fresnel nearfield region however, should increase to (8 D, 8 D2 /λ).Further, when the distance is larger than 8 D2 /λ, UE is always in the far-field of LIS, and a conventional 2D far-field BF is sufficient to harvest full array-gains.These observations are illustrated in Fig. 2.

III. OPTIMAL 3D NEAR-FIELD BF FOR LIS
Since the amplitude differences are negligible, the main objective of 3D near-field BF is to recover array-gain losses resulted from phase variations in the Fresnel near-field region.In this section, we derive the optimal 3D near-field BF for LIS-array, and propose a design comprised of two steps.The first step uses a conventional 2D far-field BF to align the boresight of LIS to the UE, i.e., transferring a general UE location in Case 2 to Case 1. Afterwards, we apply the derived 1D near-field BF to compensate remaining phase variations to address the near-field impacts.Although the optimal 1D-BF in the second step depends on the distance between the UE and LIS, since the UE is assumed to be located on the boresight, heuristic approaches without knowing the distance can be applied.For instance, a 1D code-booked based beam sweeping and selection procedure can be applied similar to those in 5G-NR [40].The proposed 3D-BF design is much simpler than existing 3D-BF designs [7], [27], and the rationale behind is proven in Theorem 1.

A. Conventional 2D Far-Field BF
In general, the effective channel, after 2D far-field BF operations at both transmitter and receiver sides, is proportional to the harvested array-gain and can be expressed as where β(ϕ r , φ r ), β(ϕ t , φ t ) denote the steering vectors at the receiver and transmitter sides, respectively, and (•) † denotes the conjugate transpose.With UPA and omitting the subscripts 'r' and 't', both vectors can be decomposed as tensor-products in two dimensions (x and y) as where ϕ and φ are the azimuth and elevation angles in Fig. 1, respectively.Further, the steering vectors are typically DFT beams and equal to2 where (•) T is the transpose operator.
Then, the optimal far-field BF vectors α(ϕ r , φ r ) and α(ϕ t , φ t ) at receiver and transmitter sides are set to (without power-normalization): This yields the highest array-gain that can be harvested from (18).In the Fresnel near-field, however, the steering vector from the LIS is not of the form β(ϕ t , φ t ), but rather it is a vector γ that comprises phases variations e jϵmn that depend on distances.At the UE, as the receiving antenna-array is typically small, the phase variations from the LIS to different receiving antennas are negligible, and both the steering vector β(ϕ r , φ r ) and the receiving BF vector α(ϕ r , φ r ) remain unaltered as if being in the far-field.Hence, the effective channel (18) with the perfect 2D far-field receiving BF becomes However, a conventional 2D-BF vector α(ϕ t , φ t ) at LIS yields array-gain losses in the near-field.By assuming a general 3D-BF vector γ at the LIS, the effective channel becomes and |γ † γ| is the harvested array-gain resulting from an applied BF vector γ that measures the effectiveness of the BF.
In ideal case, when UE-location is known, a perfect 3D near-field BF can be designed such that γ = γ, which attains the full array-gain γ † γ = M N .On the other hand, setting γ = 1 that contains all ones, i.e., without any BF, renders array-gain losses with In order to design the optimal 3D-BF, next we derive the closed-form of γ in the Fresnel near-field region, and analyze its properties.
B. Formulation of 1D Near-Field BF With Case 1 We start from Case-1.The distance between the UE and the (m, n)th element of LIS equals where the approximation holds when UE is in the Fresnel region, and ∆ mn denotes the distance between the LIS-center and the (m, n)th element, which is known and equal to Then, the phase variation in the received signal emitted at the (m, n)th element is The optimal near-field BF needs to compensate these phase variations at different antenna-elements of the LIS.Note that the phase variation from ( 27) can be decomposed into where the variable θ is defined as and the common phase factor η equals Since the first term e jη in ( 28) is identical for all antennaelements, it can be removed from the BF, while the second and third terms correspond to the optimal BF weights in two directions, respectively.From (28), the optimal near-field BF becomes 1D and only depends on θ, which is summarized in Property 4, due to that the UE is assumed to be located on the boresight of LIS with Case-1.From implementation perspective, the 1D near-field BF still contains two dimensions with UPA.
Property 4: For a LIS of size M × N and when UE is located on its boresight, the optimal 1D near-field BF vector is γ(θ) = α 1 (θ) ⊗ α 2 (θ), where α 1 (θ) and α 2 (θ) are the BF vectors in two dimensions and of length M and N , respectively, and equal to (31) Although the optimal 1D near-field BF vector in Property 4 depends on θ, or equivalently, on d, it is a single parameter which can be estimated.Further, even without any knowledge, a codebook based beam-sweeping and selection approach can be implemented on θ at a low lost, similar to the beam management procedure in 5G-NR [40].

C. Design of 3D Near-Field BF With Case 2
With Case-2, the near-field BF becomes 3D, and a direct approach computes all distances d mn in (25) to obtain the correct phase-shifts.This requires to estimate the 3D-location of UE in terms of spherical coordinates ( d, φ, ϕ), or Cartesian coordinates (x, y, d).Note that for a UE located at (x, y, d), the azimuth and elevation angles satisfy Inserting (32) back into (20) yields the optimal far-field BF.Alternatively, the near-field BF can be based on a selection process from a predetermined 3D codebook.Conventional 3D-BF designs [7], [25], [27] mainly fall into these two approaches.However, the drawbacks are that it is not practical to estimate the UE location at the initial access stage [38], [39] when there is no communication link between UE and LIS, and a beam-sweeping on a 3D codebook is rather complex.In 5G-NR system, the 2D far-field BF utilizes a beamsweeping procedure to select the most suitable angles from a predetermined 2D codebook of (φ, ϕ).When considering near-field BF, the distance d also comes into play, and a 3D codebook for beam-sweeping that comprises all three dimensions ( d, φ, ϕ) can however, exponentially increase the codebook-size.
Next we show that the 3D near-field BF can actually be implemented in a simple form with negligible overheads compared to the conventional 2D-BF.Interestingly, we can decompose the 3D codebook into a conventional 2D codebook comprising the two angles (φ, ϕ), and an additional 1D codebook on 1/ d.As a result, the beam-sweeping can firstly apply a 2D far-field BF that compensates the angels (φ, ϕ), followed by the 1D near-field BF derived in Property 4 that only depends on θ.This result is referred to as a "decomposition theorem", and is summarized in Theorem 1.
Theorem 1 (Decomposition Theorem): In the Fresnel near-field region, the optimal 3D near-field BF can be decomposed into two separate processes: a conventional 2D far-field BF and an additional 1D near-field BF.The 2D far-field BF compensates phase variations due to mismatches in the azimuth and elevation angles, which is equivalent to rotate the LIS-array accordingly such that an arbitrarily located UE is transferred onto its boresight.Afterwards, the 1D near-field BF that compensates remaining phase variations caused by distance differences from the UE to different antenna-elements on LIS, can follow Property 4.
Proof: The phase variation at the (m, n)-th antennaelement equals When UE is located in the Fresnel region, the amplitude variations are negligible, and the distance difference can be approximated as On the other hand, it holds that where s 2 mn is the distance between the (m, n)-th antennaelement and the projection point of UE on the LIS, as shown in Fig. 3. Further, it also holds that Inserting ( 36) and (37) back into (35) yields Noting that it holds from (34) that In (40), the first term which are derived in ( 27) that depend on ∆ mn that is known, and the distance d that is unknown.
Combining the above analysis, it shows that an optimal 3D near-field BF in (40) with Case-2 can be decomposed into two parts: a conventional 2D far-field BF in (41) and the 1D near-field BF in (43) following Property 4. The only assumption made is d mn ≈ d in (34), i.e., when the UE is in the Fresnel near-field with negligible amplitude variations, which completes the proof.
From Theorem 1, the optimal 3D-BF vector in ( 23) is of the form where Γ is a diagonal matrix whose diagonal elements are phase shifts e −jϵmn , aiming to compensate the phases variations due to the near-field impacts, and β(ϕ t , φ t ) corresponds to a conventional 2D far-field BF as in (20).The advantage of ( 44) is that it is fully compatible to the existing far-field BF schemes, and the 1D compensation with Γ can be seen as an additional operation.When UE is in the farfield and the distance is sufficiently large, all ϵ mn are close to 0 and Γ becomes an identity matrix, which means that no near-field BF is needed.While when UE is in the Fresnel nearfield region, it holds that γ = γ, and the 3D-BF according to (44) is close to optimal.Nevertheless, it is worth-noting that from a practical perspective it is natural to firstly apply the 2D far-field BF followed by the 1D near-field BF, although the order of them can be switched.

IV. A PRACTICAL CODEBOOK DESIGN FOR 1D NEAR-FIELD BF
The 2D codebook designs of far-field BF can follow conventional designs [7], [16], [20].Similarly, a codebook based 1D near-field BF can be designed with a number of predetermined discrete values of θ.The granularity of θ is subjected to trade-offs between complexities and attained array-gains.Interestingly, we observe that with a small codebook, the 1D near-field BF can perform quite well.This is partly due to that the phases in the formulation (43) are not so sensitive to the distance when UE is in the Fresnel region with d 0 < d ≤ d F,2 .

A. MSE-Minimizing Based 1D Codebook Design
The codebook design of θ is a 1D problem which splits an interval on the boresight following the second BF step in Theorem 1.As we are interested in the Fresnel region, a general codebook design can consider an interval a 0 < d < a K+1 , where a 0 and a K+1 denote the smallest and largest distances for the near-field BF to be effective, and can be set to d 0 and d F,2 , respectively.Note that if a LIS is deployed with a certain height such as on a wall, the minimal distance from a UE located on the ground to the LIS will be larger than that height.
Assuming the interval (a 0 , a K+1 ) is sampled with then the objective is to find the best quantization vector The design concept is illustrated in Fig. 4. For a UE located on the boresight of LIS (e.g, after 2D far-field BF), the optimal 1D near-field BF can be set to e jθ k m(m−M −1) and e jθ k n(n−N −1) on each dimension following Property 4, with when the distance satisfies To find optimal quantization and boundary points, one design principle can be to minimize the MSE of quantization, with the cost function set to where p(x) denotes the probability distribution function (pdf) of a UE located with a distance x.For a LIS that supports 3D UE-deployments, p(x) is proportional to the surface-area 4πx 2 , while for 2D UE-deployments, p(x) is proportional to the circumference 2πx.Hence, we let p(x) = β 1 x 2 and β 2 x to represent the 3D and 2D deployments, respectively, where β 1 and β 2 are normalizing factors for the probability-sum over the considered region to be one.Minimizing J is a classical scalar-quantization problem and can be solved with the Lloyd-Max method [21], [22].Taking the first-order derivative with respect to (w.r.t.) Letting it to be zero gives the optimal solution Similarly, taking the first-order derivative w.r.
Letting it to be zero yields the optimal b k , which is the centroid of 1/x and equal to Hence, (50) and ( 52) forms a classical Lloyd-Max iteration.
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For 3D UE-deployments, the optimal b k equals while for 2D UE-deployments, the optimal b k is The algorithm to optimize the 1D codebook design is summarized in Algorithm 1.Note that the iterations (50) with ( 52) or ( 54) only provides a necessary condition to minimize the MSE.The sufficient conditions for uniqueness of a stationary point have been investigated by Fleisher [23], which shows that the simplest condition consists of that p(x) is differentiable and ln(p(x)) is concave [24,Theorem 4], where "ln" denotes the natural logarithm.In order to transform the MSE-minimization problem (48) into a conventional one, we substitute the variable 1/x with t, and rewrite it as where 'E{•}' takes the expectation, and With (55), the pdf is changed to where n = 4 and 3 are for the 3D and 2D deployments, respectively.However, the function ln(q(t)) is strictly convex in t, rather than concave, and the sufficient condition of a uniqueness stationary point in [24] is not met.Hence, it is not guaranteed that Algorithm 1 can always converge to the global optimum.This is also due to that in contrary to conventional quantization problems whose cost functions are in convex, in our codebook design the cost function is concave.Nevertheless, in the next we show that J is convex in both a and b, and over iterations the MSE is strictly decreased until it converges to a stationary point.To see this, firstly note that the cross-terms ∂ 2 J ∂a k ∂at = 0 and ∂ 2 J ∂b k ∂bt = 0 when k ̸ = t.And to show J is strictly convex in a or b, it is sufficient to prove that the second-order derivatives w.r.t. to each element in a or b is positive.Taking the second-order derivative w.r.t. a k from (49) yields Algorithm 1 Codebook Design for 1D Near-Field BF via Lloyd-Max Iteration Input: (a 0 , a K+1 , K).In addition, adjacent antenna-elements can also be grouped together, e.g., those contained in a square, and share a common phase shift in the BF.
From the facts that b k > b k+1 and a k > 1/b k , it holds from (57) that for 3D deployments, Similarly, for 2D deployments, it holds that Therefore, the cost function J is strictly convex in a.
On the other hand, J is quadratic in b k , and from (51) it holds that Hence, J is also strictly convex in b.Therefore, the MSE is strictly decreased over each iteration with Algorithm 1.
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B. Complexity Reduction With Group Based Phase-Shifts
Although the proposed 3D-BF only requires analog phaseshifts, in order to reduce the cost, group based designs of the near-field BF can be adopted.In which case, the LIS-array is split into a number of subgroups, and each subgroup shares a common phase-shift with the same phase adjustment.Note that from (31) the optimal near-field beam is symmetric and the number of phase-shifts is already half-reduced compared to a DFT beam.In Fig. 5, we illustrate the grouping based approximations of the 1D near-field BF.As the antennaelements on each circles have an identical phase variation with Case 1, it indicates that a circular shaped LIS-array could be more efficient for near-field communications than a UPA.To reduce the number of phase-shifts, one can also group adjacent antenna-elements contained in a square together for sharing the same phase-shift, since variations in d in the same group can be small.

V. NUMERICAL RESULTS
In this section, we present numerical results and compare the attained array-gains with the proposed 2D+1D BF scheme to the state-of-the-art BF schemes [25], [27], [33], [34].A thorough summary and comparisons of existing BF schemes can refer to e.g., [32] and [35].In [33] and [34], DFT and angle-domain based codebooks are designed for 2D far-field BF, while [25] and [27] consider 3D near-field BF.There are also different methods to select an optimal beam with reduced complexities, including hierarchical codebook design [37] and multiple-stage search [36].
We model the channel between the (m, n)th antennaelement of LIS and UE as Rician, according to where K R denotes the Rician factor, and z is a Rayleigh channel whose entries follow an independent and identical (i.i.d.) complex-valued Gaussian distribution with zero-mean and unit-variance. A. Near-Field BF and Array-Gains We start with comparing the beam pattern of the derived 1D near-field beam in Property 4 with a DFT beam of length N = 16, whose elements are of the forms e jθn(n−N −1) and e jθn , respectively.As seen in Fig. 6, compared to a DFT beam, the periodicity of the 1D near-field beam is half reduced to π, due to its symmetry in (31).Further, the side-lobes are also higher, which indicates that the near-field beam is more robust against mismatches.
In Fig. 7, we illustrate the near-field and far-field regions.The array-gains |γ † γ| in ( 23 In Fig. 8, with the same setups we apply the proposed 2D+1D BF under a LoS scenario.As seen, the array-gains are improved significantly when UE enters the Fresnel region.In addition, we also simulate grouping-based approximations with a LIS-array of size 32 × 32, which is split into 8 groups of array-size 4 × 4. In each group, a common phase shift is applied.We consider two grouping methods.With the first method, the mean of 16 phases e jϵmn is used for all antennaelements in the group, while with the second one, the mean of 16 distances d mn are used for deriving a common phase in (27).As seen, both approximations only yield marginal losses in array-gains, while the second method performs slightly better than the first one.This is due to that the amplitude of compensation term can be different from one with the first method, which is suboptimal.Nevertheless, it can been seen that when d becomes very small, the near-field BF is less effective.This is essentially because amplitudes variations are no longer negligible and the approximation in (34) becomes less accurate.To further resolve this, the amplitude variations should also be considered in the BF design.

B. Codebook Design of 1D Near-Field BF
Next we evaluate the codebook design of the 1D near-field BF, and the normalized MSE (NMSE) is shown in Fig. 9, which is defined as for 3D UE-deployments.As seen, Algorithm 1 converges as the number of iterations increases and NMSE is monotonically decreasing under all cases, which is aligned with the analysis.Further, NMSE can be reduced by about 3dB compared to a uniform codebook, i.e., the initialization points.When K increases, the NMSE can be further reduced.By increasing K from 10 to 20 when designing the codebook for an interval (5, 100)m, the NMSE is reduced by about 6dB.However, as we see from simulation results in the next, setting K = 8 is sufficiently good, which means that in practice the codebooksize of 1D near-field BF can be small.An illustration of the optimized codebook compared to a uniform one is depicted in Fig. 10, where we see that the optimized codebook has a higher granularity in the region when distances are small, due to that the phase variation depends on 1/d and changes faster.

C. Array-Gains With Random UE Locations
In Fig. 11, cumulative probability functions (CDFs) of normalized array-gains under a LoS scenario (K R = ∞) are plotted with a LIS-array of size 64 × 64, where we see significant gains with the proposed 2D+1D BF.In this case, d F,1 = 198m, and a UE enters the Fresnel near-field when d > d 0 = 17.8m from Property 3. We generate 2,000 random locations (x, y, d) for a single-antenna UE located randomly and uniformly in a 3D-space in front the LIS spanned by -50m< x, y <50m and d <50m.Further, to investigate the Fresnel region, we consider different constraints on d such that d >5m, 15m, 20m, 25m, and 30m, respectively.As already shown in Fig. 8, the harvested array-gains without BF are very small and therefore not presented.We are interested in comparing the case with only 2D far-field BF with the proposed 2D+1D BF, and show the improvements in arraygains with our proposal when UE is in the near-field of LIS.
Under the constraint d > 15m, the array-gain losses are negligible as the main near-field impacts are phase variations, which can be effectively compensated by the proposed nearfield BF.But under the constraint d > 5m, there are noticeable losses compared to the case with d > 15m.This verifies our definition of enlarged Fresnel region, and also verifies that setting d 0 = 2 D = 4.5m as in [14] is not sufficiently large in practice.Further, the 2D far-field BF under the constraint d > 30m only yields marginal improvements compared to the case with d > 5m, due to that the near-field impacts are not considered.
In Fig. 12 and 13, we simulate the array-gains with a codebook based BF for a LIS with M = N = 64, where a UE is located randomly and uniformly in a 3D-space Fig. 12. CDFs of array-gains with conventional and proposed near-field BF schemes with a codebook based 1D near-field BF, while the size of 2D codebooks for all approaches is N 2 .Fig. 13.CDFs of array-gains with conventional and proposed near-field BF schemes with a codebook based 1D near-field BF, while the size of 2D codebooks for all approaches is 4N 2 . in front the LIS spanned by -25m≤ x, y ≤25m, and 5m≤ d ≤50m.We model the link between antenna-elements to the UE as Rician, with a typical Rician factor K R = 8 [31].The 1D near-field codebook is optimized with K = 8 for the interval (5, 100)m.Moreover, as the UE is in the near-field, we consider perfect beam selections and measurements without noise.In Fig 12, we consider two different codebook designs for 2D far-field BF, namely, the angle-domain based codebook evaluated by International Mobile Telecommunications (IMT) [34], which quantizes the angles φ and ϕ into N/2 and 2N discrete point, and the widely used DFT based codebook [33] with N beams in each dimension.The 2D codebook-size is N 2 for both.Further, the 3D near-field BF in [27] is also evaluated, which jointly searches over the 2D and 1D codebooks for both 2D-BF designs, with a total searching-size of KN 2 .However, with the proposed 2D+1D BF, it firstly finds the best 2D far-field beam, and then searches for the best 1D near-field BF from K discrete points, with the searching-size reduced to N 2 + K.In Fig. 13, the settings are the same, but the 2D codebook-size is doubled in each dimension.Further, the upper-bounds with genie 2D far-field BF, i.e., the 2D-BF is perfect in azimuth and elevation angles, are also presented.
As it can be seen from both figures that DFT based 2D farfield BF performs better than the angle-domain based TMT codebook, which aligns with the observation in [32].However, with the proposed 2D+1D BF, both DFT and angle-domain based 2D far-field BF are significantly enhanced, with the additional 1D near-field BF, for both finite 2D codebooksize and the genie case (i.e., the codebook-size is infinite).Further, our proposal also performs close to conventional 3D near-field BF, but at a much lower complexity.This verifies the effectiveness of our proposal under both general Rician channels and practical codebook based 2D far-field BF methods.Further, it is also interesting to see that the 2D+1D BF with K = 8 performs even better than the unquantified case (K = ∞), which is due to both the imperfectness from Rician fading and also the fact that the distance can be less than d 0 in the tests.

VI. SUMMARY
We have considered the design of 3D near-field BF of LIS, and analyzed the Fresnel near-field region where only phase variations worsen the harvested array-gains but not amplitude variations.Conventional observations show that the Fraunhofer distance and the distance that defines the Fresnel region scale up linearly in the surface-area and the diameter of LIS, respectively.But those observations are made under an ideal assumption that UE is located on the boresight of LIS.Interestingly, we have shown that when UE is slightly off the boresight, which in practice can result from an imperfect 2D far-field BF, the Fraunhofer distance and the Fresnel region can be both enlarged by a factor of four.This justifies the need of applying near-field BF to recover potential array-gain losses in the Fresnel region.
Further, we have derived the optimal analog 3D nearfield BF in the Fresnel region, and proposed a novel 2D+1D BF scheme that decompose the 3D-BF into twosteps with the decomposition theorem.The first step applies a conventional 2D far-field BF that compensates phase variations from azimuth and elevation angles, while the second step utilizes the derived 1D near-field beam to compensate remaining phases variations resulted from distance differences.Simulation results have shown that the proposed 2D+1D BF can effectively recover array-gains degraded by the nearfield impacts.Further, practical BF designs with group based approximations have been evaluated.Moreover, the codebook design of 1D near-field BF is also analyzed, which can perform close to optimal with a small codebook.Note that we have only considered a UE with a small receiving antenna-array, and when a UE also receives with LIS, the near-field impacts at receiving side should also be addressed, which is reciprocal to the transmitting LIS and can follow similar analysis.

4 ,d
and on top of an antenna-element at the upper-right corner of LIS in Fig. 1.This is a special case with d tan φ = D/2.The shortest and longest distances are d 1 = d and d 2 = D2 + d 2 , respectively, and it holds d ≥ d F,2 for UE to enter the near-field of LIS.

Fig. 2 .
Fig. 2. The near-field and far-field regions of a LIS-array with general UE locations.When UE is not located on the boresight of LIS, both d 0 and d F,2 are increased by a factor of four from their conventional definitions.

Fig. 3 .
Fig. 3.An illustration of the decomposition in Theorem 1, where the distance dmn between UE and the (m, n)-th antenna-element on the LIS is represented by the solid blue curve, and the distance d between UE and the LIS-center is used as a reference, and represented by the solid red curve.The phase compensation depends on the difference dmn − d, which can be decomposed into two separate parts that correspond to a 2D far-field BF and a 1D near-field BF in the Fresnel region.

Fig. 5 .
Fig.5.Antenna-elements on different circles have identical phase variations with Case 1, and can share a common phase-shift in the 1D near-field BF.In addition, adjacent antenna-elements can also be grouped together, e.g., those contained in a square, and share a common phase shift in the BF.

Fig. 6 .
Fig. 6.Beam-patterns of the 1D near-field beam and a DFT beam both of length 16.

Fig. 7 .
Fig. 7. Array-gains for a UE with two cases and at different distances without BF.
) are measured with γ = 1 in relation to the distance d with Case 1, and Example 2 with Case 2. With Case 1, the UE is located on the boresight of LIS, and enters the near-field when d < d F,1 , which is 48m with λ = 5cm and M = N = 32.This corresponds to a normalized distance in wavelength of d F,1 /λ ≈ 1000.When the UE is off the boresight as in Example 2, the Fraunhofer distance d F,2 is enlarged four times, which aligns with Property 2.

Fig. 10 .
Fig. 10.The optimized codebook (in blue) compared to a uniform codebook design (in red), where the circles are centers and the crosses are boundary