Beam Squint and Aperture–Bandwidth Limitations in Wideband RIS-Assisted THz Links Waqas Khalid, Chiew Foong Kwong, David Chieng, and Qianyu Liu
arXiv:2609.14037v1 [cs.IT] 12 Sep 2026
Department of Electrical and Electronic Engineering, Next Generation Internet of Everything Laboratory (NGIoE Lab) University of Nottingham Ningbo China, Ningbo, 315100, China Email: [email protected], [email protected], [email protected], [email protected]
Abstract—This paper analyzes the ergodic rate of a wideband reconfigurable intelligent surface (RIS)-assisted terahertz link under beam squint and finite-resolution phase control. Independent Rayleigh fading on the two RIS-assisted hops produces double-Rayleigh cascaded amplitudes. Exact secondand fourth-order channel moments are derived by incorporating the frequency-dependent array response and phase-quantization errors. Moment matching then provides a Gamma approximation for the received power on each subcarrier and a tractable closedform approximation for the wideband ergodic rate. A first-null bandwidth approximation characterizes the aperture–bandwidth limitation of frequency-flat RIS control. Monte Carlo simulations validate the analytical framework. Index Terms—RIS, terahertz communications, ergodic rate, beam squint, phase quantization.
I. I NTRODUCTION Terahertz (THz) communication can support extremely high data rates because of the abundant spectrum available at submillimeter wavelengths. Nevertheless, THz links experience severe spreading loss, molecular absorption, and sensitivity to blockage, which restrict their coverage [1]. Reconfigurable intelligent surfaces (RISs) can alleviate these limitations by creating controllable reflected paths without conventional radio-frequency chains [2]. The large number of reflecting elements available at THz frequencies can also provide considerable passive beamforming gain. RIS research has further considered simultaneous transmitting and reflecting architectures, active operation, hardware impairments, phasedependent amplitude responses, physical-layer security, and malicious-surface threats [3]–[8]. Most statistical analyses of RIS-assisted links adopt a narrowband model in which one RIS phase configuration coherently combines the reflected signals across the operating band [3], [4], [6]–[8]. This assumption becomes inaccurate for wideband THz transmission. Practical frequencyflat phase shifts designed at the center frequency cannot perfectly align the reflected components on all subcarriers. The resulting frequency-dependent phase progression, known as beam squint, reduces coherent array gain and becomes more severe as the bandwidth or RIS aperture increases [9], [10]. Existing wideband RIS studies have mainly addressed beamforming and phase-shift optimization under deterministic or geometry-based channel models. These studies condition
on a channel realization and optimize an instantaneous objective, whereas statistical performance analysis averages over fading to derive channel distributions or long-term metrics. The present work follows the latter direction: its RIS profile is prescribed by center-frequency alignment, and its main contribution is the derivation of channel-power moments and ergodic-rate expressions rather than a new beamforming algorithm. Stochastic performance analyses have considered fading distributions, phase imperfections, and pointing errors [11]– [14]. However, these research directions do not jointly characterize cascaded small-scale fading, frequency-dependent beam squint, and discrete RIS phases. The analytical difficulty is that the effective channel becomes a frequency-dependent complex sum of double-Rayleigh amplitudes with deterministic beamsquint phases and random phase-quantization errors, whose received-power distribution has no convenient exact form. Accordingly, this paper develops a tractable ergodic-rate analysis for a wideband RIS-assisted THz link. Its main contributions are as follows: Exact second- and fourth-order moments of the effective channel are derived under cascaded Rayleigh fading, beam squint, and finite-resolution phase control. The expressions explicitly incorporate the subcarrier-dependent finite-aperture array factors. • The received power is approximated by a momentmatched Gamma random variable, which yields a closedform approximation for the per-subcarrier and wideband ergodic rates. • A first-null bandwidth condition is derived to reveal how RIS size and propagation geometry limit the bandwidth supported by a frequency-flat phase configuration. Monte Carlo simulations validate the analytical framework. •
II. S YSTEM AND C HANNEL M ODEL Consider the wideband single-input single-output THz system illustrated in Fig. 1, where a source communicates with a destination through an N -element passive RIS using a frequency-flat phase profile. The direct link is unavailable because of severe blockage. The RIS is modeled as a uniform linear array with inter-element spacing dR . For a K-subcarrier
and cross terms; hence, the present model is most accurate for sparse or dominant-path THz channels. Assuming perfect knowledge of the cascaded channel phases at fc , the RIS aligns the reflected contributions at the center frequency; channel-estimation and control overhead are outside the present scope. With B-bit phase control, the residual quantization error is modeled as εn ∼ U[−∆B , ∆B ], where ∆B = π/2B [13]. The errors are mutually independent and independent of the fading coefficients. Because the same RIS profile is applied to every subcarrier, phase alignment is generally lost when fk ̸= fc . After centerfrequency phase compensation, the effective channel is Fig. 1. Wideband SISO THz link assisted by an N -element frequency-flat RIS. The common B-bit phase profile designed at fc produces residual phase nνk + εn on subcarrier k, while the direct link is blocked.
orthogonal frequency-division multiplexing waveform, the frequency of subcarrier k is K −1 fk = fc + k − ∆f, k = 0, . . . , K − 1, (1) 2 where fc is the center frequency, ∆f is the subcarrier spacing, and W = K∆f is the occupied bandwidth. We assume odd K and define the center-subcarrier index as kc ≜ (K − 1)/2. Let sk be the unit-power symbol transmitted on subcarrier k with power pk . The received signal is √ (2) yk = pk Hk sk + nk , where Hk is the RIS-assisted channel and nk ∼ CN (0, σk2 ) is PK−1 additive Gaussian noise. The powers satisfy k=0 pk ≤ P , and σk2 = N0 ∆f , where P and N0 denote the total transmitpower budget and noise power spectral density, respectively. Unless otherwise stated, equal allocation, pk = P/K, is assumed. The normalized comparisons in Section IV impose ρk = ρkc to isolate beam squint. Let d1 and d2 denote the source–RIS and RIS–destination distances. The deterministic cascaded THz amplitude coefficient is c c κabs (fk )(d1 + d2 ) βk = ηk exp − , 4πfk d1 4πfk d2 2 (3) where c is the speed of light, κabs (fk ) is the molecular absorption coefficient, and ηk captures the antenna gains, RIS element response, and reflection efficiency [1], [11]. The normalized small-scale coefficients of the two RIS hops satisfy gn ∼ CN (0, Ωg ), and hn ∼ CN (0, Ωh ), and are independent across elements and between hops [15]. Within each realization, their amplitudes are assumed constant over the OFDM band, while the frequency-dependent inter-element phase is retained through νk . The single-tap assumption isolates beam squint from multipath frequency selectivity. Additional delayed components in practical THz channels would make the hops frequency selective and alter the received-power moments, potentially increasing the Gamma-model mismatch. Extending the analysis would require the tap powers, delays,
Hk = βk
N −1 X
Xn = |gn ||hn |,
Xn exp [j (nνk + εn )] ,
n=0
(4)
where the frequency-dependent phase progression is 2π(fk − fc )dR (sin ϑ1 + sin ϑ2 ) , (5) c and ϑ1 and ϑ2 are the signed incident and reflected angles measured from the RIS broadside. Under the adopted array convention, the cascaded inter-element delay is dR (sin ϑ1 + sin ϑ2 )/c. Thus, νk = 0 at the center frequency and generally increases in magnitude toward the band edges [9], [10]. For subsequent analysis, define νk =
Zk =
N −1 X
2
Xn e
j(nνk +εn )
,
γk = ρk Zk ,
ρk =
n=0
pk |βk |2 . σk2 (6)
The wideband ergodic rate is K−1
R=
1 X E[log2 (1 + ρk Zk )] . K
(7)
k=0
A tractable statistical characterization of Zk is developed in the following section. III. S TATISTICAL AND E RGODIC -R ATE A NALYSIS A. Effective-Channel Moments For convenience, write Sk =
N −1 X
Xn ej(nνk +εn ) ,
Zk = |Sk |2 .
(8)
n=0
Since Xn = |gn ||hn | is the product of two independent Rayleigh amplitudes, it follows a double-Rayleigh cascaded distribution [16]. Its rth-order moment is r , r = 1, . . . , 4. (9) µr ≜ E[Xnr ] = (Ωg Ωh )r/2 Γ2 1 + 2 The characteristic coefficients of the phase-quantization error are sin(ℓ∆B ) qℓ ≜ E[ejℓεn ] = , ℓ ∈ {1, 2}. (10) ℓ∆B Both coefficients approach one as B increases; for continuousphase control, q1 = q2 = 1 by continuity.
The frequency-dependent finite-aperture array factors are Dℓ,k ≜
N −1 X
ejℓnνk (11)
n=0 jℓ(N −1)νk /2 sin(N ℓνk /2)
. sin(ℓνk /2) When ℓνk = 2πm, m ∈ Z, the ratio is interpreted by continuity and Dℓ,k = N . Thus, D1,kc = D2,kc = N , while destructive combining may occur away from the center frequency. For compactness, define =e
ξ = µ2 − α 2 ,
α = µ 1 q1 ,
ζ = µ2 q2 − α2 ,
t = µ3 q1 − αµ2 q2 − 2αµ2 + 2α3 , u = µ4 − 4αµ3 q1 + 2α2 µ2 q2 2
(12)
4
+ 4α µ2 − 3α . These quantities represent the required central mixed moments of Xn ejεn . Lemma 1: The first and second moments of Zk are M1,k ≜ E[Zk ] = N µ2 + α2 |D1,k |2 − N ,
(13)
and M2,k ≜ E[Zk2 ] 4
B. Closed-Form Ergodic-Rate Analysis Using (16), the per-subcarrier ergodic rate admits the following closed-form approximation. Proposition 1: 1 1 0, 1 Rk ≈ G3,1 , (17) Γ(κk ) ln 2 2,3 ρk θk 0, 0, κk where Gm,n p,q (·) denotes the Meijer-G function. The wideband rate follows by substituting (17) into (7). Proof: Substituting (16) into the rate expectation and setting x = z/θk gives Z ∞ 1 Rk ≈ xκk −1 e−x ln(1 + ρk θk x) dx. (18) Γ(κk ) ln 2 0 Expressing the logarithm in Meijer-G form and applying the standard Mellin-convolution identity [17] gives (17). ■ Equation (17) avoids Monte Carlo averaging and requires only standard special functions. If a symbolic Meijer-G routine is unavailable, the preceding integral can be evaluated using Gauss–Laguerre quadrature. Aperture–bandwidth implication: At the band edge, |fk −fc | ≃ W/2, and the first null of (11) occurs when N |νk |/2 = π. Therefore, Wnull ≃
4
2
2
= α |D1,k | + 4N α ξ|D1,k | ∗ + 2α2 ζ Re D2,k (D1,k )2
(14)
+ 4αt|D1,k |2 + N u + ζ 2 |D2,k |2 − N + 2N (N − 1)ξ 2 . Proof: Let Yn = XnP ejεn = α + Yen , where E[Yen ] = 0. Substitution into Sk = n ejnνk Yn and collection of equaland distinct-index terms yield (13). For (14), use E[|Yen |2 ] = ξ, E[Yen2 ] = ζ, E[Yen2 Yen∗ ] = t, and E[|Yen |4 ] = u, and collect the resulting sums through D1,k and D2,k . ■ The exact distribution of Zk is intractable because it is the squared magnitude of a frequency-dependent sum of double-Rayleigh variables. Motivated by related RIS channel approximations [13], [16], we use Zk ∼ ˙ Gamma(κk , θk ), 2 M1,k κk = 2 , M2,k − M1,k θk =
2 M2,k − M1,k
M1,k
2c . N dR |sin ϑ1 + sin ϑ2 |
(19)
Thus, W < Wnull keeps the operating band within the principal coherent lobe. Increasing the RIS aperture improves the center-frequency gain but reduces the bandwidth supported by one frequency-flat phase configuration. When sin ϑ1 +sin ϑ2 = 0, the linear phase progression vanishes and Wnull → ∞. IV. N UMERICAL R ESULTS Monte Carlo simulations validate the proposed analysis. Unless otherwise stated, fc = 0.3 THz, λc = c/fc , K = 129, dR = λc /2, N = 64, B = 2, and W = 30 GHz. The average channel powers are Ωg = Ωh = 1, with ϑ1 = ϑ2 = 10◦ . Each empirical CDF uses 106 independent realizations of the double-Rayleigh channels and phase-quantization errors. A. Accuracy of the Gamma Approximation
(15)
,
where κk and θk are the shape and scale parameters. The resulting PDF is z κk −1 z fZk (z) ≈ exp − , z ≥ 0. (16) Γ(κk )θkκk θk Accordingly, FZk (z) ≈ P (κk , z/θk ), where P (a, x) = γ(a, x)/Γ(a) is the regularized lower incomplete Gamma function. Both Gamma parameters vary across subcarriers through D1,k and D2,k , thereby retaining the statistical effect of beam squint.
Fig. 2 compares the empirical and Gamma-approximated CDFs at k ∈ {64, 96, 128}, representing the center, intermediate, and upper band-edge subcarriers. To preserve the frequency-dependent power variation, all subcarriers are normalized by the center-subcarrier mean: Vk = Zk /M1,kc , where kc = 64. Hence, vM1,kc Γ FVk (v) ≜ P κk , . (20) θk The analytical curves closely follow the Monte Carlo markers. The normalized mean powers decrease from 1 at k = 64 to 0.7772 and 0.3352 at k = 96 and 128, producing the progressive leftward CDF shift. Meanwhile, κk decreases from 24.9984 to 21.3863 and 10.9913, indicating increased relative dispersion toward the band edge.
Fig. 2. Empirical and Gamma-approximated CDFs of Vk = Zk /E[Zkc ] at the center (k = 64), intermediate (k = 96), and upper band-edge (k = 128) subcarriers.
Fig. 3. Wideband ergodic rate versus the nominal center-subcarrier precombining SNR for different RIS phase resolutions. Lines denote analysis and markers denote Monte Carlo simulation.
Fig. 4. Beam-squint-induced wideband rate loss of frequency-flat RIS control relative to subcarrier-dependent phase alignment. Lines denote analysis and markers denote Monte Carlo simulation.
The distributional mismatch is quantified using the Kolmogorov–Smirnov (KS) distance
frequency-flat profile designed at fc and therefore remains affected by beam squint. The analytical curves obtained from (17) closely match the Monte Carlo markers over the entire SNR range. At ρkc = −20 dB, continuous-, two-, and one-bit phase control achieve approximately 4.21, 3.94, and 3.06 bit/s/Hz, respectively. Thus, two-bit control improves the rate by 28.8% over onebit control and remains only 6.6% below continuous-phase control. At 0 dB, the corresponding rates are 10.77, 10.47, and 9.49 bit/s/Hz. Increasing the resolution beyond two bits therefore provides only a modest gain under the considered conditions.
(v) − FVΓk (v) . DKS,k ≜ sup FbVMC k
(21)
v≥0
The KS distances are 0.0077, 0.0081, and 0.0044 for subcarriers 64, 96, and 128, respectively. Their values below 0.01 quantitatively confirm the close agreement shown in Fig. 2. Thus, the Gamma model captures both the power loss and distributional change induced by beam squint. B. Wideband Ergodic Rate and Phase Resolution Fig. 3 shows the wideband ergodic rate versus the nominal center-subcarrier pre-combining SNR, ρkc . We impose ρk = ρkc on every subcarrier to isolate beam squint and phase quantization from frequency-dependent path loss. The continuousphase benchmark eliminates quantization errors but retains the
C. Aperture–Bandwidth Rate Loss Fig. 4 examines the bandwidth sensitivity of frequency-flat RIS control for different RIS sizes. The ideal benchmark independently aligns the RIS phases on every subcarrier and serves only as an upper reference. Both schemes use identical phase
resolution, nominal SNR, geometry, and channel parameters. The beam-squint-induced relative rate loss is Rflat (W ) LR (W ) = 100 1 − %. (22) Rideal (W ) We impose ρk = ρkc = −20 dB on all subcarriers; thus, the comparison isolates beam squint and is not a fixed-total-power bandwidth sweep. The loss approaches zero in the narrowband limit and increases with bandwidth and RIS size. At W = 30 GHz, the losses are approximately 11.0%, 22.2%, and 36.0% for N = 64, 96, and 128, respectively. Equation (19) predicts first-null bandwidths of approximately 54, 36, and 27 GHz. Therefore, at W = 30 GHz, the band edge has crossed the first null for N = 128, but not for the smaller RISs. At W = 60 GHz, the losses reach 41.4%, 49.7%, and 56.2%, respectively. No discontinuity occurs at Wnull because R averages all subcarrier rates, whereas (19) concerns only the bandedge subcarrier. These results confirm the inverse aperture– bandwidth relationship. A larger relative loss indicates stronger beam-squint sensitivity, but not necessarily a smaller absolute rate. V. C ONCLUSION This paper developed a tractable ergodic-rate analysis for wideband RIS-assisted THz links under cascaded Rayleigh fading, beam squint, and finite-resolution phase control. Exact second- and fourth-order channel moments were derived using frequency-dependent finite-aperture factors and phasequantization errors. Moment matching then yielded a Gamma approximation and a closed-form wideband ergodic-rate expression. Monte Carlo results validated the proposed distribution and rate analysis. They also showed that two-bit control captures most of the continuous-phase gain, whereas beam-squint-induced loss grows with bandwidth and RIS size. The derived first-null condition explains the tradeoff between center-frequency array gain and wideband coherent combining. Although analytical, the framework is validated only through Monte Carlo simulations under single-tap Rayleigh fading, perfect center-frequency channel knowledge, and idealized phase-quantization errors. Measurement-based validation is not provided, and the accuracy may change under practical multipath, channel-estimation errors, and hardware impairments. Future work will consider measured multipath THz channels, hardware experiments, planar RIS geometries, and frequency-dependent element responses. R EFERENCES [1] I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Physical Communication, vol. 12, pp. 16–32, 2014. [2] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environments empowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” IEEE Journal on Selected Areas in Communications, vol. 38, no. 11, pp. 2450–2525, Nov. 2020.
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