Bessel Beam Optimization for Near-Field THz Communications under UE Location Uncertainty Aditya Jolly∗† , Vitaly Petrov† , Gábor Fodor∗† and Emil Björnson†
∗ Ericsson Research, Sweden
† KTH Royal Institute of Technology, Sweden
arXiv:2607.07069v1 [eess.SP] 8 Jul 2026
Email: {aditya.jolly, gabor.fodor}@ericsson.com, Abstract—To achieve the desired coverage and capacity levels, future terahertz (THz) wireless systems are envisioned to utilize extremely large antenna arrays. At THz frequencies, the combination of short wavelengths and large array apertures often makes many of the conventional far-field assumptions invalid in practice. As a result, many UEs operate in the radiative nearfield zone, where novel near-field beam synthesis methods become viable. This paper studies phase-only Bessel-like near-field beam configurations for downlink THz multiple-input multiple-output links under imperfect UE location knowledge. We first formulate a spectral efficiency maximization problem with respect to the “Bessel cone angle”. We then derive low-complexity closed-form approximations for the optimal Bessel beam configuration for: (i) deterministic UE location; (ii) Gaussian and (iii) uniform error in the UE location. Finally, through extensive simulations across multiple signal frequencies, UE locations, and array sizes, we show that our proposed simple closed-form approximations closely match (under 0.1% difference) the best performance achieved via exhaustive search, while simultaneously reducing the configuration complexity down to as low as O(1).
I. I NTRODUCTION Terahertz (THz) (0.3–3 THz) communications can theoretically support up to terabit-per-second data rates by leveraging the large available bandwidth [1]. However, severe propagation losses make practical mobile THz deployments challenging and necessitate the use of steerable high-gain antennas (e.g., large-scale antenna arrays) to maintain a sufficient link budget [2]. As wavelengths shrink and large-scale arrays become more prevalent, the so-called radiative near-field zone can extend to several hundred meters [3]. As a result, a substantial fraction of user equipments (UEs) may operate in this zone, where conventional far-field beamforming based on planewave assumptions can incur significant performance losses [4]. Motivated by this, recent work has increasingly focused on near-field-specific beams, such as beam focusing [3], which concentrates array energy at a spatial focal point rather than only an angular direction [5]. Prior works have quantified its achievable gains, developed phase-optimization algorithms and near-field codebooks [4]–[9], among others. However, beyond single-point energy concentration offered by beam focusing, near-field operation also enables a much broader class of beams with distinct propagation characteristics, including Airy beams [10], Weber/Mathieu beams [11], Hermite–Gaussian beams [12], and vortex beams [13], among others. In this context, Bessel-like beams [14] (hereafter Bessel beams) have attracted significant attention for applications in mobile THz links. This is primarily due to two key properties of Bessel beams. First, their “non-diffracting behavior” [15],
{vitalyp, emilbjo}@kth.se
meaning that they sustain axial power over a finite range and can therefore provide robustness to UE location uncertainty [1], [16], [17], and second, their “self-healing capability” [18], meaning that they can reconstruct the signal after partial link occlusion and thereby improve resilience to smallscale blockage [1], [17], [19]. Recent work spans efficient beam generation and hardwareoriented synthesis to experimental demonstrations showing that Bessel beam-based transmission can enhance spectral efficiency (SE) in ultra-wideband THz systems [14], [20], [21]. However, the use of Bessel beams for practical mobile THz communications raises an important question: “How to best configure a THz Bessel beam for a given setup?”. Unlike configuring far-field beamforming, which is mainly determined by the angle towards the UE in line-of-sight, and near-field beam focusing, which depends on both the angle and the distance to the UE when known, Bessel beams introduce an additional design parameter: the “Bessel cone angle”. This makes the overall task of finding the optimal beam configuration non-trivial. Prior works typically either: (i) adopted certain heuristic parameter choices [16], [20], which may be suboptimal, or (ii) performed exhaustive search over available configurations [21], which is computationally expensive and may lead to control signaling overheads between the access point (AP) and the UE. Hence, neither of the two approaches is suitable for prospective practical THz deployments, motivating the need for an efficient yet low-complexity approach to optimize the THz Bessel beam configuration. The problem is further complicated by the fact that, in practice, UE location estimates are typically imperfect [8]. Hence, the optimal Bessel beam configuration depends jointly on the estimated UE location, the uncertainty distribution, and the performance metric of interest (e.g., received power, signal-tonoise ratio (SNR), or SE), making real-time adaptation with existing approaches challenging. Latest prior studies in this field (e.g., [17] alongside other related works) derive certain boundaries for Bessel beam configuration and performance limits in different conditions for linear arrays under perfect knowledge of the UE location. Still, to the best of the authors’ knowledge, low-complexity techniques for optimizing Bessel beam configuration with realistic planar arrays and under given UE location uncertainty remain under-explored. In this paper, we address this gap by deriving approximate closed-form expressions that map near-optimal Bessel beam configurations to available UE location statistics. We particularly optimize the THz Bessel beam configuration such that
the expected SE (and, consequently, capacity) of the AP–UE THz wireless link is maximized. Through extensive computer simulations, we demonstrate that the proposed expressions closely track the best SE obtained via exhaustive search of available configurations, across a wide range of array sizes, frequencies, UE locations, and uncertainty regimes. These closed-form approximations facilitate: (i) low-complex (down to O(1)) yet (ii) efficient (negligible difference in SE compared to exhaustive search) Bessel beam configuration for the design and evaluation of future near-field THz systems. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
Deterministic UE location
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𝜀#
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Gaussian UE location error Uniform UE location error
(a) System model with UE mean location (µ), and error distribution (εz )
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max
(0, 0, 0)
𝑧
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UE
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𝑦 '' 𝜙$,&
θ
where the scalar r(i,j)→(a,b) ∈ R+ denotes the propagation distance between the corresponding antenna elements. Let µ ∈ R+ denote the estimated UE location along the z-axis obtained from any positioning technique. Then the true distance between the antenna elements lying at the center of the AP and UE array is modeled as p = µ+εz , where εz is an additive error assumed identical for all UE antennas and confined to the z-axis (to isolate range uncertainty while excluding lateral misalignment to study the relationship of distance errors and near-field beam configuration). The resulting propagation distance is given by q r(i,j)→(a,b) = (xa,b − xi,j )2 + (ya,b − yi,j )2 + p2 , (2)
𝜇 𝜎!
𝜀# ~𝒩(0, 𝜎!$ )
…
We adopt a line-of-sight (LoS), non-uniform spherical-wave near-field channel model [23]. We operate in the radiative nearfield region of the array such that dRA < z ≤ dFA . Here, dRA and dFA denote the standard reactive and radiative near-field boundary, respectively [24]. The channel coefficient between the (i, j)-th AP antenna element and the (a, b)-th UE antenna element is denoted by h(a,b)→(i,j) , where (i, j) and (a, b) denote the row and column indices of the antenna elements on the AP and UE UPAs, respectively, and i, j ∈ {1, . . . , NAP } and a, b ∈ {1, . . . , NUE }. Then h(a,b)→(i,j) is given by 2π λ exp −j r(i,j)→(a,b) , (1) h(a,b)→(i,j) = 4πr(i,j)→(a,b) λ
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A. Near-field channel model
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We consider a single-user downlink near-field THz multipleinput multiple-output (MIMO) system, where an AP with an NAP ×NAP uniform planar array (UPA) serves a UE with an NUE×NUE UPA. Each antenna element is modeled as an ideal isotropic, linearly polarized point source (i.e., polarization mismatch and cross-polarization are neglected). Both UPAs use λ/2 element spacing, and mutual coupling is neglected. The AP transmits with total power Ptx (mW) and applies a conic phase profile to generate a Bessel beam (see Fig. 1b), while the UE performs coherent receive combining [22]. Both UPAs are parallel to the xy-plane, with the AP’s UPA centered at the origin. The UE is located on the AP boresight and displaced only along the z-axis. We study three UE location scenarios: (i) deterministic location, (ii) Gaussiandistributed errors along the z-axis, and (iii) uniformlydistributed errors along the z-axis, as illustrated in Fig. 1a.
min
(b) Bessel beam field intensity generated by conic phase profile (ϕBB i,j ) Fig. 1. Downlink near-field THz AP–UE MIMO system utilizing a Bessel beam generated by a conic phase profile (ϕBB i,j ) with “cone angle” θ.
where xi,j ≜ (i−(NAP +1)/2) λ2 , yi,j ≜ (j −(NAP +1)/2) λ2 and xa,b ≜ (a − (NUE + 1)/2) λ2 , ya,b ≜ (b − (NUE + 1)/2) λ2 . For the scenario with a deterministic UE location, the UE is located at µ with no positioning error, i.e., εz = 0. Under the uniform positioning error model, the UE’s location error is modeled as εz ∼ U[−δU , δU ], where the variance of εz is 2 2 /3. While, for the Gaussian positioning error σU = δU model, 2 the UE’s location error is modeled as εz ∼ N 0, σG . In both cases, the UE location is symmetrically distributed around µ (i.e., εz is zero-mean), as shown in Fig. 1a. B. Bessel beam generation To generate a Bessel beam using a phase-only technique, we apply a phase difference of ΦBB ∈ RNAP ×NAP to the AP UPA [16]. The per-element phase difference applied is denoted by ϕBB i,j (assumed to be continuous), and defined as 2π q 2 2 ϕBB xi,j + yi,j sin θ. (3) i,j = λ This phase-based technique applies a conic phase profile with “cone angle” θ at the transmitting aperture to generate a Bessel beam (see Fig. 1b). Its non-diffracting behavior is characterized by Zmax , given by Zmax = R/ tan θ, where
R is the effective aperture radius. We adopt the circumscribed √ radius R = DAP / 2, with DAP = (NAP − 1)λ/2 denoting the side length of the AP UPA. For a fixed frequency and aperture, θ fully determines the Bessel beam’s characteristics.
θ∗ = arg max E[S(θ)] . θ
(4)
(5)
The problem in (5) can be solved via exhaustive search over a discrete set of candidate “cone angle” θ, yielding a benchmark for each UE location scenario. However, this is computationally expensive. Hence, in the following section, we develop closed-form approximations to solve (5), under the considered UE location scenarios. III. O PTIMAL B ESSEL B EAM C ONFIGURATION In this section, we present the exhaustive search benchmark and derive closed-form Bessel beam configuration rules for three UE location scenarios: deterministic location, Gaussian uncertainty, and uniform uncertainty. We first treat the deterministic UE case using a geometric scaling argument, and validate the scaling law by quantifying its dependence on frequency and array size. We then extend the solution to the Gaussian and uniform uncertainty cases via first-order corrections and calibrate the associated correction factors. A. Exhaustive search benchmark For each UE operating point (µ, σ), with σ ∈ {0, σG , σU } denoting deterministic location, Gaussian, and uniform location uncertainty, respectively, we evaluate a discrete candidate set Θ ∈ [θmin , θmax ]. Since θ is the only design variable, (5) is solved via a one-dimensional grid search. For each θ ∈ Θ, the SE is computed over a discrete axial grid z ∈ [zmin , zmax ], and the expected SE is obtained by approximating the expectation via numerical averaging over the location distribution centered at µ. The exhaustive search baseline is θ∈Θ
z!,# 𝑧
+=
.
Let nF denote the post-combining noise power. The resulting SNR is PUE /nF , and the SE becomes S = log2 1+PUE /nF . Our design objective is to choose θ that maximizes the expected SE under the UE location statistics:
θ∗ (µ, σ) = arg max Ez∼L(µ,σ) [S(θ; z)] ,
𝑥
2
BB Ptx h(a,b)→(i,j) ejϕi,j NAP i=1 j=1
Inner rings Mid rings Outer rings
...
a=1 b=1
√ N AP N AP X X
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PUE =
N UE N UE X X
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We formulate Bessel beam configuration as the maximization of the expected SE under the given location statistics. With matched-filtering, the post-combining received signal power equals the squared ℓ2 -norm of the per-antenna received signal:
((, *) AP element
AP
𝑦
...
...
C. Problem formulation
2"#(%!,# )
(6)
where L(µ, σ) denotes the corresponding location distribution (degenerate for σ = 0). The cost of (6) is high because each evaluation of S(θ; z) requires a full spherical-wave near-field channel computation at distance z, with distance-dependent coefficients. Repeating this over many θ ∈ Θ and z samples to numerically average over L(µ, σ) makes real-time operation
...
-$% 2
...
-$% Fig. 2. AP antenna elements within an annular band of radius ri,j and width 2∆w(ri,j ), which add quasi-coherently at zi,j . Inner and mid rings (green/red) are untruncated, and their element count grows with radius, whereas near-boundary rings (blue) are truncated and contain fewer elements.
prohibitive. In our implementation, this burden is partially mitigated using a GPU-accelerated, vectorized PyTorch inhouse simulator with optimized precision and tiled processing. B. Closed-form approximation: Deterministic UE location We first consider a single-antenna UE and perfect location knowledge (deterministic) at the transmitter. To reveal the geometry governing the axial power profile, we apply a stationary-phase approximation to the overall phase term in (4). That is, we assume that the dominant contribution at a given z comes from antenna elements whose phase derivative is approximately zero. For q an AP antenna element (i, j) with radial coordinate 2 , this condition yields the constructive (stari,j = x2i,j + yi,j tionary) axial distance zi,j = ri,j / tan θ, ∀ i, j. Consequently, the discrete square UPA can be interpreted approximately as a set of “rings”: elements with similar ri,j contribute quasicoherently around the corresponding zi,j (see Fig. 2). To obtain a tractable scaling law, we model the coherent set at radius ri,j as an annular band of thickness 2∆w(ri,j ). We assume that over this narrow neighborhood, the propagation distance and phase curvature vary slowly, so the contributions remain nearly phase-aligned. For a finite square aperture with effective radius R, the band is truncated by the boundary, which we capture by ∆w(ri,j ) = min(ri,j , R − ri,j ).
(7)
With sufficiently dense sampling and d = λ/2, the number of elements in this band is approximated via the area density as Nring (ri,j ) ≈
2πri,j ∆w(ri,j ) . d2
(8)
Assuming coherent summation within the annular band, the field amplitude scales as Nring (ri,j ) and the received power scales as Nring (ri,j )2 , with an additional free-space decay 1/z 2 . Since the annular band at radius ri,j contributes pri-
42.6
RMSE(SDCF, SDExh), [bit/sec/Hz]
10°2
Axial power
f = 200 GHz f = 300 GHz f = 600 GHz f = 1000 GHz
10°1
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ni s
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UE
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(a) Effect of aperture and frequency on (10)
De
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Inward correction
CF , (12) (b) Effect of aperture and frequency on SD
Fig. 4. Axial power profile of Bessel beams true − z (10) )/Z Fig. 3. Validation for deterministic UE location case, (a) Heatmap of ∆η = (zpeak max for deterministic (black) and uncertain (red) UE peak locations, depicting the peak-alignment principle. Exh and S CF (log-scaled) vs N across f and NAP , (b) RMSE between SD AP , µ = 10m and NUE = 1. D
marily around z = zi,j = ri,j / tan θ, we evaluate the proxy along this mapping by setting ri,j = zi,j tan θ, yielding 2 Nring (zi,j tan θ) ∝ tan2 θ (∆w(zi,j tan θ))2 . PUE (zi,j ) ∝ 2 zi,j (9) Based on (7), we notice that ∆w(zi,j tan θ) grows linearly with zi,j tan θ until zi,j tan θ = R/2 and then decreases linearly due to truncation (see Fig. 2). Therefore, PUE (zi,j ) increases for zi,j ≤ R/(2 tan θ) and decreases for zi,j > R/(2 tan θ), predicting the dominant axial peak at zpeak = R/(2 tan θ) = Zmax /2.
(10)
We note that (10) relies on a standard first-order approximation that captures the dominant scaling via coherent element density and 1/z 2 spreading. However, (10) is not intended to exactly characterize the peak location since contributions from elements outside the dominant annular region (depending on aperture and frequency) are neglected. This aligns with our goal of exposing the governing geometry while quantifying residual effects empirically. Using (10), we obtain a closed-form Bessel beam configuration for deterministic UE location based on a peak-alignment principle: choose the “cone angle” θ so that the dominant axial maximum of the finite-aperture beam aligns with the UE location. Since the downlink SNR (and thus SE) is maximized when the received power peaks at the UE position, this gives a simple, physically interpretable design rule (see Fig. 4): µ = zpeak = Zmax /2
(11)
and yields the closed-form configuration ∗ θDeterministic = tan−1
! DAP √ . 2 2µ
(12)
Fig. 3 validates (10)–(12) and quantifies how the axial peak depends on the aperture and frequency. Specifically, we sweep f ∈ [100, 1000] GHz (including sub-THz frequencies) with NUE = 1, vary NAP ∈ [400, 700] (corresponding to practical apertures from 6 cm at 1 THz to 1.05 m at 100 GHz), and scan
θ ∈ [0.1◦ , 10◦ ]. For each configuration, our near-field simulator computes the on-axis received power, from which we extrue tract the true axial peak zpeak and define the normalized ratio (10) true η ≜ zpeak /Zmax . Fig. 3a then reports ∆η = η − zpeak /Zmax over (f, NAP ), showing deviations tightly centered around zero (i.e., E[η] = 0.497 and E[∆η ] ≈ 5.3 × 10−3 ) with only a weak residual dependence on (f, NAP ). Fig. 3b quantifies the impact of this weak dependence on SE by reporting the root-mean-squared error (RMSE) (log-scaled Exh ) (6) y-axis) between the exhaustive-search benchmark (SD CF and the closed-form expression SE (SD ) in (12). We vary the array size NAP (x-axis) for f ∈ {200, 300, 600, 1000} GHz while fixing µ = 10 m and NUE = 1. As seen in Fig. 3b, the trends remain mild, with the maximum RMSE on the order of 10−1 bits/s/Hz. Overall, the peak-alignment rule zpeak ≈ Zmax /2 is robust, and the residual aperture/frequency effects are small, so (10)–(12) remain accurate in the deterministicUE-location case. Moreover, for an array-equipped UE with coherent combining, (12) remains a valid first-order rule since zpeak is governed primarily by the transmit-side geometry. C. Closed-form approximation: Uncertain UE location Under location uncertainty, the peak-alignment principle is extended from a single-point target to a distribution-dependent target region. For Gaussian uncertainty, the “cone angle” is chosen to align the dominant axial maximum with the highprobability region of the UE location distribution rather than the mean location alone. Exploiting the smooth axial variation of the Bessel beam power around zpeak , we model the resulting shift via a first-order correction, µ − ασG = zpeak = Zmax /2. Where α captures the effective inward correction induced by Gaussian uncertainty. The same principle applies to the uniform uncertainty case, yielding an analogous first-order correction, µ−βσU = zpeak = Zmax /2. Where β captures the corresponding effective inward correction under uniform uncertainty. As shown in Fig. 4, the inward bias concentrates axial energy and moves the dominant peak toward the region maximizing the expected SE. Although Zmax decreases, the average SE improves under uncertainty since the near-peak power gain outweighs the roll-off beyond
Exh), [bits/sec/Hz] RMSE (SaCF, SG
0.3
0.2
NAP= 500, f = 400GHz 10-2 NAP= 500, f = 500GHz NAP= 500, f = 600GHz -3 NAP= 500, f = 300GHz 10 NAP= 300, f = 300GHz NAP= 800, f = 300GHz -4 10
RMSE across the full test set. Hence, the proposed coefficients robustly link UE uncertainty statistics to the Bessel beam configuration. Substituting the values obtained for α and β yields the following closed-form approximations ! DAP ∗ −1 √ , (13) θGaussian = tan 2 2 (µ − 0.22 σG ) ! D AP ∗ √ . (14) θUniform = tan−1 2 2 (µ − 0.40 σU )
𝜶 = 𝟎. 𝟐𝟐 𝛃 = 𝟎. 𝟒𝟎
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With (α, β) fixed, (13)–(14) fully specify the proposed closedform configurations under Gaussian and uniform location errors, while (12) gives the corresponding closed-form approximation for deterministic UE location. IV. N UMERICAL A NALYSIS This section evaluates the proposed closed-form Bessel beam configurations through numerical simulations. We first establish agreement with exhaustive search by (i) comparing the closed-form prediction of the optimal cone angle θ∗ to the exhaustive-search benchmark and (ii) quantifying the resulting accuracy via empirical CDFs of the SE error over a broad set of system parameters. With this validation in place, we then use the closed-form rules to study SE trends and robustness versus frequency, UE mean location, and location uncertainty.
Exh and S CF for Gaussian error, and S Exh and Fig. 5. RMSE between SG α U SβCF for uniform error, averaged over (a) [µ, σG ], and (b) [µ, σU ].
Zmax . The coefficients α and β thus quantify the strength of this uncertainty-induced inward shift. We obtain α (Gaussian) and β (uniform) by calibrating the closed-form rules against the exhaustive-search benchmark in (6). The exhaustive search is performed over a Θ ∈ [0.1◦ , 10◦ ] with a granularity of 0.05◦ , and z ∈ [dRA , dF A /5] with a granularity of 10 cm. For calibration, we sweep candidate coefficients from α, β ∈ [0.1, 1] and, for each coefficient, compute the predicted “cone angle” and the resulting expected SE, denoted by SαCF (µ, σG ) for Gaussian errors and SβCF (µ, σU ) for uniform errors. We Exh Exh define SG (µ, σG ) and SU (µ, σU ) as the expected SE attained by (6) for Gaussian and uniform errors, respectively. In Fig. 5, we then measure the mismatch via RMSE, averaged over (µ, σG ) and (µ, σU ) grids with σG ∈ [1, 5] m and σU ∈ [0.57, 2.89] m. For illustration, we present the results for three AP sizes NAP ∈ {300, 500, 800} at f = 300 GHz and three frequencies f ∈ {400, 500, 600} GHz at NAP = 500. Fig. 5 reports the RMSE versus α (Gaussian) and β (uniform); the insets zoom into the low-RMSE region and use a log-scale to resolve small differences among minima. The minima occur at slightly different values across configurations, but show no systematic trend with either f or NAP . We set the coefficients to the mean of the per-configuration minimizers, yielding α = 0.22 and β = 0.40, which provide near-minimal
A. Closed-form approximation vs exhaustive search We begin by examining the behavior of the optimal “cone angle” θ∗ . Fig. 6 compares θ∗ obtained via exhaustive search (6) with the values predicted by the closed-form expressions (12), (13), and (14) for the three UE location scenarios. We fix f = 300 GHz and NUE = 1, and vary the AP array size by sweeping NAP ∈ [400, 700] antenna elements. For the uncertain-location cases, we set σG , σU = 2 m and choose µ = 0.5(dRA + dFA /10) to represent a nominal UE distance and uncertainty level. Note that exhaustive search is performed over the same Θ and z grids as in Section III-C. Fig. 6 shows that the closed-form solutions closely track exhaustive search under both Gaussian and uniform uncertainty. In contrast, the deterministic-location reference consistently Optimal “cone angle”, q § [deg]
(b) Uniform error in UE location
Closed-form, Uniform Exhaustive search, Uniform Closed-form, Gaussian Exhaustive search, Gaussian Closed-form, Deterministic Exhaustive search, Deterministic
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Fig. 6. Optimal “cone angle” θ∗ for closed-form vs exhaustive search.
0.057%
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% (a) CDFs of |∆SD |, deterministic location
0.00471%
% (b) CDFs of |∆SG |, Gaussian location error
% (c) CDFs of |∆SU |, uniform location error
% CF , for three UE location scenarios. Fig. 7. CDFs of absolute percentage error in SE |∆SDist | showing the impact of varying NUE and f , on accuracy of SDist
7.85% NAP= 500, NUE=20, µ = 10m NAP= 300, NUE=20, µ = 10m
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Fig. 8. SE trends under deterministic, Gaussian, and uniform UE-location scenarios. (a) SE versus frequency for NAP ∈ {300, 500} and NUE ∈ {1, 20} at µ = 10 m. (b) SE versus mean UE location for fixed σG , σU ∈ {1, 4} m. (c) SE versus location uncertainty for fixed µ ∈ {10, 14} m. For (b)–(c), NAP = 500, NUE ∈ {5, 10}, and f = 300 GHz.
yields a smaller “cone angle”, highlighting the importance of the proposed correction factors for uncertain UE location. Having verified that the closed-form expressions accurately capture the optimal configuration variable θ∗ , in Fig. 7, we next evaluate its impact on the achievable SE performance. We fix the AP size to NAP = 500 and consider frequencies f ∈ {100, 300} GHz to demonstrate that the proposed framework applies to (sub-)THz regimes. The UE mean location is selected within the radiative near-field region for all three UE models, while the uncertainty ranges are swept as σG , σU ∈ [1, 5] m for the Gaussian and uniform error scenarios. The accuracy of the closed-form approximation is quan% tified through the percentage SE error ∆SDist , for Dist ∈ {D, G, U}, representing deterministic location, Gaussian error, and uniform error in location: S Exh (µ, σ) − S CF (µ, σ) % ∆SDist (µ, σ) ≜ 100 × Dist Exh Dist . (15) SDist (µ, σ)
% | on a log-scaled x-axis; crosses empirical CDFs of |∆SDist mark the 95th-percentile errors (the 0.95-CDF intercepts) for each (NUE , f ). In all three UE location scenarios, the CDFs are clustered with no long tails, indicating consistently high accuracy without any rare large-error events. For deterministic location (see Fig. 7a), the CDFs also exhibit the mild frequency dependence predicted in Section III-B: errors are slightly smaller at 300 GHz than at 100 GHz, yet both are extremely small. The maximum 95th-percentile errors across all tested configurations are only 0.057% (deterministic), 0.00722% (Gaussian), and 0.00471% (uniform), confirming close tracking of the exhaustive-search optimum. Hence, validating that our approximations accurately capture the dominant first-order variations that affect the SE performance for the three UE location scenarios.
Exh CF Here, SDist (·) and SDist (·) denote the exhaustive search and closed-form spectral efficiencies, respectively. The resulting CDF curves provide a rigorous statistical comparison between the proposed approximations and the optimal benchmark across varying uncertainty levels and system parameters. We assess scalability with varying UE array size by com% for NUE ∈ {1, 5, 10}. Fig. 7, shows the puting ∆SDist
Fig. 8 examines the absolute SE performance and robustness of the proposed closed-form approximations. Throughout this subsection, we use Ptx = 0 dBm and compute the noise power as nF = −174 + 10 log10 (B) + N F = −82 dBm, with B = 100 MHz and noise figure N F = 12 dB, representative of THz receivers. We first consider the deterministic UE case in Fig. 8a, where operating frequency is swept from 300 GHz
B. Performance dynamics of THz links using Bessel beams
to 675 GHz, and the SE is evaluated at mean UE location µ = 10 m. We consider NUE ∈ {1, 20} and NAP ∈ [300, 500]. Fig. 8a shows the resulting SE versus f plot for deterministic UE location. We observe that increasing NUE or NAP provides a consistent SE gain across all f , showing that increasing aperture delivers the dominant array gain in this regime. As the frequency increases, we see a gradual decrease in achieved SE. This is because, with fixed antenna element spacing and NAP , increasing f reduces the physical aperture and Zmax , reducing the array gain and tightening the axial region, resulting in the gradual SE decrease. Next, we analyze the impact of uncertain UE location in Fig. 8b–8c on achieved SE. We sweep σG , σU ∈ [1, 5] m and µ ∈ [9.25, 14.75] m fixing f = 300 GHz, NAP = 500, and report results for NUE ∈ {5, 10}. Fig. 8b plots SE versus µ for fixed σ ∈ {1, 4} m (deterministic location reference), and Fig. 8c plots SE versus σ for fixed µ ∈ {10, 14} m. In Fig. 8b, increasing µ yields a gentle SE decay consistent with the extended Zmax depth of Bessel beams; the deterministic-uncertain gap shrinks slightly at larger distances, and larger NUE reduces the uncertainty penalty via an averaging effect observed due to coherent combining. While in Fig. 8c, all three models nearly coincide at small uncertainty, with the Gaussian and uniform curves collapsing to the deterministic location baseline, i.e., smooth convergence to (12). As uncertainty grows, the stochastic cases diverge gradually but remain smooth and monotonic. Overall, Fig. 8 highlights that the SE increases consistently with larger NAP and NUE and decreases gradually with f for the deterministic case. Under uncertain UE location SE decays gently as µ increases, with the uncertainty penalty shrinking at larger distances. Meanwhile, increasing the standard deviation σ leads to a performance degradation and a growing divergence from the deterministic UE reference. V. C ONCLUSION This paper studied phase-only Bessel beam configuration for radiative near-field (sub-)THz downlink MIMO links under deterministic and uncertain UE location knowledge. We formulated the Bessel phase configuration as an expected SE maximization problem and approximated the solution using the properties of the axial power profile of Bessel beams. Our analytical contribution is a set of O(1)-complexity closedform expressions that map the near-optimal Bessel beam configuration directly to available UE location information. Our numerical study confirmed that the derived configurations closely track exhaustive search across a range of array sizes, frequencies, and UE location scenarios. The evaluations also highlighted that the deterministic–uncertain SE gap shrinks at larger distances but gets larger as uncertainty in the UE location grows. These closed-form expressions and the presented analysis facilitate the design of future practical nearfield THz systems (e.g., codebook design), as well as accurate yet analytically tractable system-level studies of near-field THz networks, e.g., capacity, coverage, and interference modeling.
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