# A beamforming frequency to mitigate the beamsquint effect in sub-THz

Florian Polster-Prieto, E L Hassani, Inbar Fijalkow

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Florian Polster-Prieto, E L Hassani, Inbar Fijalkow. A beamforming frequency to mitigate the beamsquint effect in sub-THz. IEEE Conference on Standards for Communications and Networking (CSCN 2025), IEEE, Sep 2025, Bologne, Italy. hal-05250569v2

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Submitted on 10 Oct 2025

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# A beamforming frequency to mitigate the beamsquint effect in sub-THz

Florian POLSTER--PRIETO, Hajar EL HASSANI, Inbar FIJALKOW

ETIS, UMR 8051

CY Cergy Paris University, ENSEA, CNRS Cergy, France
{florian.polster-prieto,hajar.el-hassani,inbar.fijalkow}@ensea.fr

**Abstract**—In this paper, we propose to adapt the beamforming frequency for ultra massive multiple-input multiple-output uniform linear arrays allowing a maximum array gain while minimizing the angle deviation induced by the beamsquint effect. This original adapted frequency deviates from the center frequency $f_c$. It paves the way for the beamforming frequency to be a degree of freedom. Our simulations show that the proposed analytical solution is close to the optimal numerical solution with very low computational complexity.

**Index Terms**—beamsquint, sub-THz, beamforming, massive MIMO

## I. INTRODUCTION

One current direction of 6G wireless communications to expand spectrum opportunities is to increase the transmission speed by drastically increasing the bandwidth (and necessarily the frequency) [1]. This leads to an increased signal path loss [2], and ultra massive multiple-input multiple-output (UM-MIMO) systems are widely considered to increase the array gain [3], [4]. Digital precoding UM-MIMO at sub-THz frequencies and above induces a huge energy consumption cost [5]. Hybrid precoding has been considered as an enabler to implement UM-MIMO technologies with phase shifters based beamforming at sub-THz frequencies. The idea of hybrid precoding is to decompose the high-dimensional digital precoder into two parts, with a fully digital precoder followed by a fully analog precoder, thus lowering the dimensionality of the system. The analog beamformer, when implemented with phase shifters, suffers from the beamsquint effect [1], [6]. True-time-delay (TTD) has been considered to bypass the beamsquint effect [7], [8], yielding near to optimal results. They can also be combined with phase shifters as in [9]. However, due to their high insertion loss and substantial hardware and energetic costs, they can not fully replace phase shifters [9].

Therefore, we focus in this paper on reducing the beam deviation induced by the beamsquint effect in phase shifters-based beamforming systems. The optimized beamforming frequency is derived by minimizing the angle deviation between the target angle and the angle achieving the maximum array gain for each deviated beam. We make sure that the beamformer frequency is compatible with the existing phase shifter beamforming design.

This work was supported by the ANR under the France 2030 program, grant NF-SYSTERA : ANR-22-PEFT-0006.

## II. PRELIMINARIES

### A. System model

We consider a uniform linear array (ULA) of $N$ antennas with uniform spacing $d$. The phases of the phase shifters are assumed to be continuous without quantization. We define the angle $\theta$, spanning the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$, as the angle of departure (AoD) or the angle of arrival (AoA) when the ULA functions respectively as a transmitter or a receiver. The response vector of the ULA is expressed as [10]

$$ \mathbf{a}(\theta, f) = [1, e^{j2\pi\lambda^{-1}d\sin(\theta)}, \dots, e^{j2\pi\lambda^{-1}(N-1)d\sin(\theta)}]^T, \quad (1) $$

where $\lambda = \frac{c}{f}$ is the operational wavelength. The phase shifters are modeled by the beamforming vector $\mathbf{w}$ as

$$ \mathbf{w} = [e^{j\beta_1}, \dots, e^{j\beta_N}]^T, \quad (2) $$

where $\beta_n$ is the phase of the n-th phase shifter connected to the n-th antenna of the ULA [1]. In the narrowband case, the optimal setting of the beamformer is defined as

$$ \beta_n(\theta_t) = \frac{2\pi}{\lambda_0}(n-1)d\sin(\theta_t) = \frac{2\pi d}{c}(n-1)f_0\sin(\theta_t), \quad (3) $$

where $n \in \{1, \dots, N\}$, and $\theta_t \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ denotes the target beamforming angle. While this is not necessarily the optimal beamforming architecture in wideband systems (as optimizing directly $\beta_1, \dots, \beta_N$ as in [11], [12], is also an option), it is still widely used in hybrid beamforming for wideband systems (see [1], [6], [13] for example). In all works using (3), the beamforming frequency $f_0$ is set to the central frequency $f_c = (f_{max} + f_{min})/2$ of the fixed band [$f_{min}, f_{max}$]. In this paper, we consider the case where the beamforming frequency $f_0$ is not constrained to equal $f_c$. With this difference in mind, the array gain is defined as in [1]

$$ \frac{1}{\sqrt{N}} \mathbf{w}^H \mathbf{a}(\theta, f) = \frac{1}{\sqrt{N}} \sum_{n=1}^{N} e^{j[2\pi\lambda^{-1}(n-1)d\sin(\theta) - \beta_n(\theta_t)]} \quad (4) $$

and its magnitude can be expressed as

$$ \left| \frac{1}{\sqrt{N}} \sum_{n=1}^{N} e^{j(n-1)\pi \frac{2d}{c} (f \sin(\theta) - f_0 \sin(\theta_t))} \right| = \left| \frac{1}{\sqrt{N}} \frac{1 - e^{j\pi N \frac{2d}{c} (f \sin(\theta) - f_0 \sin(\theta_t))}}{1 - e^{j\pi \frac{2d}{c} (f \sin(\theta) - f_0 \sin(\theta_t))}} \right| \quad (5) $$
$$
\begin{align*}
&= \left| \frac{1}{\sqrt{N}} e^{j\pi(N-1)\frac{2d}{c}} (f \sin(\theta) - f_0 \sin(\theta_t)) \right. \\
&\quad \left. \times \left( \frac{e^{j\pi\frac{N}{2}\frac{2d}{c}} (f \sin(\theta) - f_0 \sin(\theta_t)) - e^{-j\pi\frac{N}{2}\frac{2d}{c}} (f \sin(\theta) - f_0 \sin(\theta_t))}{2j} \right) \right. \\
&\quad \left. \times \left( \frac{2j}{e^{j\pi\frac{d}{c}} (f \sin(\theta) - f_0 \sin(\theta_t)) - e^{-j\pi\frac{d}{c}} (f \sin(\theta) - f_0 \sin(\theta_t))} \right) \right| \\
&= \left| \frac{\sin\left(\frac{N\pi}{2f_c}(f \sin(\theta) - f_0 \sin(\theta_t))\right)}{\sqrt{N} \sin\left(\frac{\pi}{2f_c}(f \sin(\theta) - f_0 \sin(\theta_t))\right)} \right|
\end{align*}
$$

resulting from a geometric sum and Euler's formula as in [14]. The antenna spacing *d* is conventionally set to $\frac{\lambda_c}{2} = \frac{c}{2f_c}$ [1], and is considered to be fixed.

The magnitude of the array gain can therefore be expressed as a function of

$$x = \frac{f}{f_c} \sin(\theta) - \frac{f_0}{f_c} \sin(\theta_t) \quad (6)$$

and is denoted by

$$|g(x)| = \left| \frac{\sin\left(\frac{N\pi}{2}x\right)}{\sqrt{N} \sin\left(\frac{\pi}{2}x\right)} \right|. \quad (7)$$

Note that $x$ in (6) is slightly different from the state-of-the-art, since $f_0$ can differ from $f_c$, but the array gain in (7) is the same as in [1].

For fixed angles $\theta_t$ and $\theta$, the array gain is maximum only for the specific frequency satisfying $x = 0$, so that $\frac{f}{f_c}\sin(\theta) - \frac{f_0}{f_c}\sin(\theta_t) = 0$. Equivalently, for a fixed target angle $\theta_t$, to achieve the maximum array gain $|g(x)|_{max} = \sqrt{N}$ over the entire bandwidth $B = f_{max} - f_{min}$ at sub-THz frequencies, the angle $\theta$ deviates from $\theta_t$ over $B$. This is called the beamsquint effect. Indeed, this effect does not occur in lower frequency bands when $B \ll f_0$. In what follows, we investigate the beamforming frequency $f_0$ optimization to minimize the average angle deviation of $\theta$ from $\theta_t$, achieving the maximal gain. To our best knowledge, existing works always consider a fixed beamforming frequency set as the central frequency, i.e., $f_0 = f_c$. In the sequel, we also propose a novel design of the beamforming frequency to mitigate the beamsquint effect.

## B. Beamsquint evaluation metrics

To evaluate the impact of the beamsquint effect on the proposed designs, we introduce the following metrics.

The relative angle deviation $\Delta\theta \in [0, 1]$ measures the gap between the chosen $\theta_t$ and $\theta$ for specific values of $\theta_t$, $\theta$, $f$, $f_0$ and $N$. We also average the relative angle deviation over the bandwidth $B$ to assess the impact of the beamsquint effect over the whole frequency range for specific values of $\theta_t$, $\theta$, $f_0$ and $N$. The two metrics are defined as the angle deviation expressed as

$$\Delta\theta(x) = \left( \frac{\theta_t - \theta}{\pi} \right)^2, \quad (8)$$

and the average angle deviation

$$\langle \Delta\theta(x) \rangle = \frac{1}{B} \int_{[B]} \Delta\theta(x) df. \quad (9)$$

We also use the average array gain in the target direction $\theta = \theta_t$, defined as

$$\langle g \rangle = \frac{1}{B} \int_{f_{min}}^{f_{max}} \left| \frac{1}{\sqrt{N}} \mathbf{w}^H \mathbf{a}(\theta_t, f) \right| df \quad (10)$$

which is also used in [15] to evaluate the impact of the beamsquint effect. In the sequel, we investigate the impact of the beamforming frequency $f_0$ on the previously introduced metrics. We study different beamforming designs to minimize the beamsquint effect with respect to the angle deviation. More specifically, we search for the beamforming frequency minimizing the average angle deviation while ensuring that the maximal array gain is achieved, for a specific beamforming angle.

## III. BEAMFORMING FREQUENCY DESIGN

In order to ensure a fair comparison between the beam-forming frequency's designs, we fix the frequency band to $[f_{min}, f_{max}]$. For the derivations, we will use either $f_{min}$ and $f_{max}$, or equivalently $f_c = \frac{f_{min}+f_{max}}{2}$ and $B = f_{max} - f_{min}$.

### A. Central frequency design

In all the state-of-the-art, $f_0$ is set at the center of the band, i.e., $f_0 = f_c$, see for instance [1], [6], [13]. We investigate in this subsection the impact of the choice $f_0 = f_c$.

The optimal beamforming angle $\theta^*$ maximizing the array gain at each frequency $f$ is obtained when solving $x = 0$. Two cases occur:

* when $\frac{f_c}{f}|\sin(\theta_t)| \le 1$, i.e., $f \ge f_c|\sin(\theta_t)|$, solving $x = 0$ gives an optimal beamforming angle $\theta^*$ that can be expressed as

$$\theta^* = \arcsin\left(\frac{f_c}{f} \sin(\theta_t)\right). \quad (11)$$

This solution allows an optimal gain $|g(0)|_{max} = \sqrt{N}$ but induces the beamsquint effect as the angle deviation $\theta^* - \theta_t$ equals 0 only when $f = f_c$;

* when $\frac{f_c}{f}|\sin(\theta_t)| > 1$, i.e., $f < f_c|\sin(\theta_t)|$, solving $x = 0$ is not possible as the arcsine function is only defined on $[-1, 1]$. For frequencies $f \in [f_{min}, f_c|\sin(\theta_t)|]$, the maximum array gain $\sqrt{N}$ can not be achieved. We can set the optimal beamforming angle as $\theta^* = \frac{\pi}{2} \text{sign}(\theta_t)$, where $\text{sign}(x)$ is the function returning the sign of $x$. This case occurs when $B > 2f_c(1 - |\sin(\theta_t)|)$ corresponding to a large bandwidth as in sub-THz or a high angle $\theta_t$ (or both).

We refer to this first approach as the "usual design" or the "central frequency design". It must be noted that the second case is never mentioned in the literature.

<footnote n="1">¹Note that $x$ can also be equal to $x = 2k, \forall k \in \mathbb{Z}$ as $|g(x)|$ is periodic.</footnote>
## B. Leftmost frequency design

The central beamforming frequency design is not satisfactory when $f < f_c |\sin(\theta_t)|$. Hence, we now introduce a design in which this condition can never be met. As $|\sin(\theta_t)| \in [0, 1]$, we propose to set $f_0 = f_{min}$ at the leftmost frequency, so that $f \in [f_0, f_0 + B]$. Note that the band remains unchanged, i.e., $[f_{min}, f_{max}]$ as previously defined.

In this case, the optimal beamforming angle $\theta^*$ given by

$$ \theta^* = \arcsin\left(\frac{f_{min}}{f} \sin(\theta_t)\right) \qquad (12) $$

allows to achieve a maximum gain $\sqrt{N}$ regardless of other parameters.

However, using $\theta^*$ from (11) with the central frequency induces a smaller angle deviation than using (12). This behaviour motivated us to search for a novel design with a variable beamformer frequency $f_0$, adapted to the angle $\theta_t$, as a trade-off between the previous ones with respect to the frequency.

# IV. ADAPTED FREQUENCY DESIGN

## A. Principle

We propose next to adapt the beamforming frequency $f_0$ of the n-th phase shift $\beta_n(\theta_t) = \frac{2\pi d}{c}(n-1)f_0 \sin(\theta_t)$, to try to combine the best of the previously proposed designs by optimizing $f_0$ with respect to the average angle deviation $\theta_t - \theta^*$. As we can see in Fig. 1, we hope that minimizing the angle deviation should lead to an improvement of the gain in the direction $\theta_t$, thus weakening the beamsquint effect.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Example of the angle deviation induced by the beamsquint effect. The black beam represents the target beam at $\theta_t$ and the red beam represents the beam at the same $\theta_t$ for a frequency $f \ge f_0$, in polar coordinates. The circles represent the relative gain in dB.</figcaption></figure>

More formally, our goal is to find the optimal beamforming frequency satisfying

$$ f_0^* = \arg \min_{f_0} \langle \Delta\theta(x) \rangle. \qquad (13) $$

To ensure an achievable maximum gain for each frequency, we impose the constraint $\frac{f_0}{f_{min}} |\sin(\theta_t)| \le 1$ allowing $\theta^*$ to satisfy (11), so that $x = 0$ is achievable. This constraint

ensures that the array gain can reach its maximum value of $\sqrt{N}$. Hence, the optimization problem can be written as

$$ \min_{f_0} \langle \Delta\theta(0) \rangle, \quad \text{s.t.} \quad \frac{f_0}{f_{min}} |\sin(\theta_t)| \le 1. \qquad (14) $$

Solving this optimization problem analytically is challenging because the equations depend on both $\theta$ and $\sin(\theta)$. To provide an analytical solution, we propose to introduce a surrogate relative angle deviation function using the sinus of the angles

$$ \Delta\tilde{\theta}(x) = \left( \frac{\sin(\theta_t) - \sin(\theta)}{2} \right)^2 \in [0, 1]. \qquad (15) $$

We can then solve the relaxed optimization problem

$$ \min_{f_0} \langle \Delta\tilde{\theta}(0) \rangle, \quad \text{s.t.} \quad \frac{f_0}{f_{min}} |\sin(\theta_t)| \le 1, \qquad (16) $$

which can be rewritten as

$$ \begin{aligned} \min_{f_0} \langle \Delta \tilde{\theta}(0) \rangle = & \min_{f_0} \frac{1}{4B} \int_{f_{min}}^{f_{max}} \left( \sin(\theta_t) - \frac{f_0}{f} \sin(\theta_t) \right)^2 df \\ & \text{s.t.} \quad \frac{f_0}{f_{min}} |\sin(\theta_t)| \le 1. \end{aligned} \qquad (17) $$

To solve (17), we use the Karush-Kuhn-Tucker (KKT) conditions [16]. These KKT conditions require both a convex objective function and a convex constraint function on the interval of interest $f_0 \in [f_{min}, f_{max}]$, which are proven next.

**Proof of convexity.** The second-order derivative of the objective function equals

$$ \frac{d^2}{df_0^2} \langle \Delta \tilde{\theta}(0) \rangle = \frac{\sin(\theta_t)^2}{2B} \left( \frac{1}{f_{min}} - \frac{1}{f_{max}} \right). \qquad (18) $$

The objective function is convex as it satisfies $\frac{d^2}{df_0^2} \langle \Delta \tilde{\theta}(0) \rangle \ge 0$, as $f_{max} > f_{min} > 0$.

The constraint in (17) is linear since it can be written as $f_0 |\sin(\theta_t)| - f_{min} \le 0$, and thus it is convex. This concludes the proof. $\square$

**Proof of the constraint optimization.** The Lagrangian can be written as following, with $\lambda \ge 0$,

$$ L(f_0, \lambda) = \langle \Delta\tilde{\theta}(0) \rangle + \lambda(f_0 \sin(\theta_t) - f_{min}), \qquad (19) $$

as the constraint has to be written as $f_0 \sin(\theta_t) - f_{min} \le 0$. According to the KKT conditions, the optimal solution $f_0^*$ satisfies:

$$ \frac{d}{df_0} L(f_0^*, \lambda) = 0 \qquad (20) $$

and

$$ \lambda(f_0^* \sin(\theta_t) - f_{min}) = 0 \qquad (21) $$

Equation (21) implies that for $\lambda > 0$, $f_0^*$ is given by

$$ f_0^* = \frac{f_{min}}{|\sin(\theta_t)|} \qquad (22) $$
In the case of $\lambda = 0$, the constraint ensures that $f_0^* \sin(\theta_t) - f_{min} \le 0$, and finding $f_0^*$ involves solving (20):

$$ \frac{d}{df_0} L(f_0, \lambda) = \frac{\sin(\theta_t)^2}{4B} \left[ -2 \ln(f) - 2 \frac{f_0}{f} \right]_{f_{min}}^{f_{max}} = 0, \quad (23) $$

resulting in

$$ f_0^* = \ln \left( \frac{f_{max}}{f_{min}} \right) \frac{f_{max} f_{min}}{f_{max} - f_{min}} \quad (24) $$

Since both the objective function and constraint are convex, solving the KKT conditions gives a unique solution

$$ f_0^* = \begin{cases} \tilde{f}_0 = \ln\left(\frac{f_{max}}{f_{min}}\right) \frac{f_{max} f_{min}}{f_{max} - f_{min}} & \text{if } \tilde{f}_0 |\sin(\theta_t)| \le f_{min} \\ \frac{f_{min}}{|\sin(\theta_t)|} & \text{otherwise} \end{cases} \quad (25) $$

We have

$$ f_{min} \le \tilde{f}_0 \le f_c. \quad (26) $$

The proof can be found in the Appendix.

Additionally, $\tilde{f}_0$ can be approximated by $f_c$ when $B \ll f_c$, which proves the coherence of the adapted design with the usual design for frequency band lower than the sub-THz.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Graphical representation of the proposed solution</figcaption></figure>

The proposed solution in (25) corresponds to the expected trade-off between the central and leftmost designs. It can be used in the phase shifters architecture as defined in (3) without any difficulty. We call this proposed design the adapted frequency design.

## B. Equivalent phase shifter adaptation

An alternative to adapting the beamforming frequency design is to optimize the phase shifters as

$$ \beta_n(\theta_t) = \frac{2\pi d}{c} (n - 1) f_c \left(\sin(\theta_t) + \varphi(\theta_t)\right). \quad (27) $$

In this case, we have $x = \frac{f_c}{f} \sin(\theta) - \sin(\theta_t) - \varphi(\theta_t)$, leading to $\theta^* = \arcsin\left(\frac{f_c}{f} \left(\sin(\theta_t) + \varphi(\theta_t)\right)\right)$. When we search for $\varphi(\theta_t)$ minimizing the average angle deviation, we find using the same methodology as in IV-A:

$$ \varphi(\theta_t)^* = \begin{cases} \tilde{\varphi} = \sin(\theta_t) \frac{\tilde{f}_0}{f_c} - \sin(\theta_t) & \text{if } -1 \le \frac{f_c}{f_{min}}(\sin(\theta_t) + \tilde{\varphi}) \le 1 \\ \operatorname{sign}(\sin(\theta_t)) \frac{f_{min}}{f_c} - \sin(\theta_t) & \text{otherwise} \end{cases} \quad (28) $$

Replacing $\varphi(\theta_t)^*$ in the phase shifter expression (27) gives the exact same expression as the proposed adapted beamforming frequency $f_0^*$ in the phase shifter (25). This shows that the proposed solution is equivalent to relaxing the phase shifter expression (3).

# V. SIMULATIONS

## A. Simulation settings

In the sequel, we show the impact of the proposed adapted beamforming frequency $f_0^*$ with respect to the target angle $\theta_t$ on the averaged angle deviation and average array gain.

We evaluate the different designs over the same bandwidth $[f_{min}, f_{max}]$. The central frequency is set at $f_c = 300$ GHz with a bandwidth $B = 30$ GHz and $N = 100$ antennas, as in [6], unless otherwise stated.

## B. Validation of the surrogate optimization

Figs. 3(a) and 3(b) represent the objective functions $\langle \Delta\tilde{\theta}(0) \rangle$ and $\langle \Delta\theta(0) \rangle$ for different values of $\theta_t$. In the case of a small value of $\theta_t$, $\langle \Delta\theta(0) \rangle$ and $\langle \Delta\tilde{\theta}(0) \rangle$ have almost the same minimum. Fig. 3(a) shows this behaviour for $\theta_t = 0.1\pi$, where both objective functions have the same minimum at $f_0^*$. For $\theta_t = 0.3\pi$ in Fig. 3(b), we are in the case where $\tilde{f}_0$ is close to $\frac{f_{min}}{|\sin(\theta_t)|}$, i.e. $\frac{\tilde{f}_0}{f_{min}}|\sin(\theta_t)| \approx 1$. This case is the worst case scenario as the adapted frequency $f_0^*$ is the farthest from the optimal beamforming frequency computed with $\langle \Delta\theta(0) \rangle$. For higher values of $\theta_t$ (close to $\pm \frac{\pi}{2}$), both objective functions give the same result $f_0^* = \frac{f_{min}}{|\sin(\theta_t)|}$. This validates the use of the surrogate function as $f_0^*$ stays close to the optimal beamforming frequency while achieving very low computational complexity, specifically $O(1)$ for the closed form solution in equation (25).

<figure><img src="image_4.png" /><figcaption>Fig. 3: Plot of $\langle \Delta\tilde{\theta}(0) \rangle$ and $\langle \Delta\theta(0) \rangle$ versus $f_0$ for $\theta_t = 0.1\pi$ rad. 2(a) case where $\theta_t$ is small, i.e., $\theta_t = 0.3\pi$ rad. 2(b) case where $\tilde{f}_0 |\sin(\theta_t)| \approx f_{min}$</figcaption></figure>

## C. Beamsquint mitigation

Fig. 4 represents the magnitude of the array gain $|g(x)|$ as a function of the angle $\theta$ for a fixed $\theta_t = 0.45\pi$ greater than $|\arcsin(\frac{f_{min}}{f_c})|$. The dashed lines represent the beams for $f = f_{max}$ and the full lines represent the beams for $f = f_{min}$. In this case, the usual design is not able to reach a maximal array gain of $\sqrt{N}$ for $f < f_c |\sin(\theta_t)|$ (i.e. $\frac{f_c}{f} |\sin(\theta_t)| > 1$), as the optimal beamforming angle $\theta^* = \arcsin(\frac{f_c}{f} \sin(\theta_t))$ is not defined in this case. Both our adapted design and the leftmost frequency design are able to achieve the maximal array gain $\sqrt{N}$ for each frequency. Moreover, the adapted design has the least average angle deviation $\langle \Delta\theta(0) \rangle$, thus encountering less beamsquint. We can also observe that the usual design $f_0 = f_c$ is not able to achieve the maximal gain, resulting in a degradation of the array gain of 26.063% for $f_{max}$.
<figure><img src="image_5.png" /><figcaption>Fig. 4: Array pattern versus $\theta$ for $B = 30$ GHz, $f_c = 300$ GHz, $N = 100$ and $\theta_t = 0.45\pi$ rad, for fixed frequency $f$. The beams representing $f_{min}$ are in full lines and the beams representing $f_{max}$ are in dashed lines.</figcaption></figure>

### D. Impact on the array gain

Table I summarizes the evaluation of the different designs for $B = 30$ GHz, $f_c = 300$ GHz, $N = 100$ and different values of $\theta_t$. In this case, we set $\theta = \theta_t$. The metric $\langle g \rangle$ represents the magnitude of the array gain $|g(x)|$ averaged over the whole band, in a similar fashion to equation (9). Note that the number of antennas $N$ has no impact on the average angle deviation $\langle \Delta\theta(0) \rangle$, but still impacts the average gain $\langle g \rangle$.

<table><caption>TABLE I: Average angle deviation and array gain for $B = 30$ GHz, $f_c = 300$ GHz, $N = 100$ and $\theta = \theta_t$</caption><thead><tr><th>θ = θ<sub>t</sub></th><th colspan="3">0.1π</th></tr></thead><tbody><tr><td>f<sub>0</sub></td><td>f<sub>c</sub> and [12]</td><td>f<sub>min</sub></td><td>f<sub>0</sub><sup>*</sup></td></tr><tr><td>⟨Δθ(0)⟩</td><td>8.959 × 10<sup>-6</sup></td><td>3.373 × 10<sup>-5</sup></td><td><b>8.925 × 10<sup>-6</sup></b></td></tr><tr><td>⟨g⟩</td><td><b>7.252</b></td><td>4.378</td><td>7.246</td></tr></tbody><thead><tr><th>θ = θ<sub>t</sub></th><th colspan="3">0.3π</th></tr></thead><tbody><tr><td>f<sub>0</sub></td><td>f<sub>c</sub> and [12]</td><td>f<sub>min</sub></td><td>f<sub>0</sub><sup>*</sup></td></tr><tr><td>⟨Δθ(0)⟩</td><td>1.634 × 10<sup>-4</sup></td><td>5.404 × 10<sup>-4</sup></td><td><b>1.612 × 10<sup>-4</sup></b></td></tr><tr><td>⟨g⟩</td><td>3.598</td><td>2.146</td><td><b>3.601</b></td></tr></tbody><thead><tr><th>θ = θ<sub>t</sub></th><th colspan="3">0.45π</th></tr></thead><tbody><tr><td>f<sub>0</sub></td><td>f<sub>c</sub> and [12]</td><td>f<sub>min</sub></td><td>f<sub>0</sub><sup>*</sup></td></tr><tr><td>⟨Δθ(0)⟩</td><td>Non Achievable<sup>2</sup></td><td>4.149 × 10<sup>-3</sup></td><td><b>3.089 × 10<sup>-3</sup></b></td></tr><tr><td>⟨g⟩</td><td><b>3.108</b></td><td>1.848</td><td>2.767</td></tr></tbody></table>

The results show that the leftmost frequency design is outperformed in every case. The adapted frequency design is able to achieve the best average angle deviation while having an average array gain similar to the usual design for $\theta_t$ equal to $0.1\pi$ and $0.3\pi$.

In the case of the average array gain, our proposed design does not allow to have a consistent improvement compared to the usual design. As $f_0$ in (24) will deviate from $f_c$ when the beamsquint effect increases, the impact of our proposed solution will also increase, in the sense of the optimized metrics $\langle \Delta\theta(0) \rangle$, compared to the usual design.

The adapted frequency $f_0^*$ outperforms the usual design in terms of angle deviation. It has also a positive effect in terms of the antenna gain for $\theta_t = 0.3\pi$. For $\theta_t = 0.45\pi$, the usual design achieves a better antenna gain but with more beamsquint.

## VI. CONCLUSIONS AND PERSPECTIVES

To mitigate the beamsquint effect, we proposed different beamforming frequency designs, in particular to minimize the average angle deviation. The proposed adapted design offers a nearly optimal solution with low computational cost in $O(1)$.

To further improve the performances, an interesting path would be to relax the constraint on the maximum gain to allow a more flexible solution.

## VII. APPENDIX

*Proof of (26).* We first show $\tilde{f}_0 \le f_c = \frac{f_{min} + f_{max}}{2}$

$$ \text{We have } 2 \ln \left( \frac{f_{max}}{f_{min}} \right) f_{min} f_{max} - f_{max}^2 + f_{min}^2 \le 0 \quad (29) $$

As $f_{max} > 0$, we can divide it by $-f_{max}^2$ and rewrite it as

$$ 2 \ln(x) x + 1 - x^2 \ge 0 \quad (30) $$

with $x = \frac{f_{min}}{f_{max}} \in (0, 1]$ ($x = 1$ when the bandwidth is 0). We set $g(x) = 2 \ln(x)x + 1 - x^2$, thus giving $g'(x) = 2 \ln(x) + 2 - 2x$ and $g''(x) = \frac{2}{x} - 2$.

For $x \in (0, 1]$, as $g''(x) \ge 0$ and $g'(1) = 0$, we have $g'(x) \le 0$. This implies that $g(x)$ is decreasing, and as $\lim_{x \to 0^+} g(x) = 1$ and $g(1) = 0$, we have $g(x) \ge 0$, allowing to prove that $\tilde{f}_0 \le f_c$.

To prove $f_{min} \le \tilde{f}_0$, we use the fact that $g'(x) \le 0$:

$$ \begin{align} & 2 + 2 \ln \left( \frac{f_{min}}{f_{max}} \right) - \frac{f_{min}}{f_{max}} \le 0 \tag{31} \\ & f_{min} f_{max} - f_{min}^2 \le \ln \left( \frac{f_{max}}{f_{min}} \right) f_{min} f_{max} \nonumber \\ & f_{min} \le \ln \left( \frac{f_{max}}{f_{min}} \right) \frac{f_{max} f_{min}}{f_{max} - f_{min}} = \tilde{f}_0 \nonumber \end{align} $$

for $f_{max} > f_{min} > 0$. This concludes the proof that $f_{min} \le \tilde{f}_0 \le f_c$

<figure><img src="image_6.png" /></figure>

## REFERENCES

[1] M. Cai, K. Gao, D. Nie, B. Hochwald, J. N. Laneman, H. Huang, and K. Liu, "Effect of wideband beam squint on codebook design in phased-array wireless systems," in *2016 IEEE Global Communications Conference (GLOBECOM)*, pp. 1–6, 2016.

[2] J. M. Jornet and I. F. Akyildiz, "Channel modeling and capacity analysis for electromagnetic wireless nanonetworks in the terahertz band," *IEEE Transactions on Wireless Communications*, vol. 10, no. 10, pp. 3211–3221, 2011.

[3] A. Faisal, H. Sarieddeen, H. Dahrouj, T. Y. Al-Naffouri, and M.-S. Alouini, "UltraMassive MIMO systems at terahertz bands: Prospects and challenges," *IEEE Vehicular Technology Magazine*, vol. 15, no. 4, pp. 33–42, 2020.

<footnote n="2">²In the case of $\theta_t = 0.45\pi$, $f_c/f = |\sin(\theta_t)| > 1$ is true for a fraction of the band, thus making it impossible to solve $x = 0$ (implying that the maximum gain $\sqrt{N}$ can not be achieved for this fraction of the band). In this case, as $\theta^* = \frac{\pi}{2} \sin(\theta_t)$, the average angle deviation $\langle \Delta\theta(0) \rangle$ can not be computed properly.</footnote>
[4] A. Mezghani, D. Plabst, L. A. Swindlehurst, I. Fijalkow, and J. A. Nossek, "Sparse linear precoders for mitigating nonlinearities in massive mimo," in *2021 IEEE Statistical Signal Processing Workshop (SSP)*, pp. 391–395, 2021.

[5] H. Halbauer and T. Wild, "Towards power efficient 6G sub-THz transmission," in *2021 Joint European Conference on Networks and Communications & 6G Summit (EuCNC/6G Summit)*, pp. 25–30, 2021.

[6] N. T. Nguyen, J. Kokkoniemi, and M. Juntti, "Beam squint effects in THz communications with UPA and ULA: Comparison and hybrid beamforming design," in *2022 IEEE Globecom Workshops (GC Workshops)*, pp. 1754–1759, 2022.

[7] E. Ghaderi, A. Sivadhasan Ramani, A. A. Rahimi, D. Heo, S. Shekhar, and S. Gupta, "An integrated discrete-time delay-compensating technique for large-array beamformers," *IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 66, no. 9, pp. 3296–3306, 2019.

[8] Y. Wu, G. Song, H. Liu, L. Xiao, and T. Jiang, "3-d hybrid beamforming for terahertz broadband communication system with beam squint," *IEEE Transactions on Broadcasting*, vol. 69, no. 1, pp. 264–275, 2023.

[9] H. Ozen and G. M. Guvensen, "Interference and beam squint aware ttd-aided beamforming for dual wideband massive mimo," *IEEE Wireless Communications Letters*, vol. 13, no. 1, pp. 64–68, 2024.

[10] S. J. Orfanidis, "Electromagnetic waves and antennas." https://www.ece.rutgers.edu/~orfanidi/ewa/, 2016.

[11] G. Femenias and F. Riera-Palou, "Wideband cell-free mmwave massive mimo-ofdm: Beam squint-aware channel covariance-based hybrid beamforming," *IEEE Transactions on Wireless Communications*, vol. PP, pp. 1–1, 12 2021.

[12] S. Park, J. Park, A. Yazdan, and R. W. Heath, "Exploiting spatial channel covariance for hybrid precoding in massive mimo systems," *IEEE Transactions on Signal Processing*, vol. 65, pp. 3818–3832, July 2017.

[13] Q. Yuan, H. Liu, M. Xu, Y. Wu, L. Xiao, and T. Jiang, "Deep learning-based hybrid precoding for terahertz massive MIMO communication with beam squint," *IEEE Communications Letters*, vol. 27, no. 1, pp. 175–179, 2023.

[14] H. Van Trees, *Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory*. Detection, Estimation, and Modulation Theory, Wiley, 2004.

[15] H. Do, N. Lee, R. W. Heath, and A. Lozano, "Hybrid arrays: How many rf chains are required to prevent beam squint?," *IEEE Transactions on Wireless Communications*, vol. 23, no. 9, pp. 11708–11722, 2024.

[16] S. Boyd and L. Vandenberghe, "Convex optimization." https://web.stanford.edu/~boyd/cvxbook/, 2004.