Novel Spectrum Allocation Among Multiple Transmission Windows for Terahertz Communication Systems

The terahertz (THz) band communication (THzCom), operating in the electromagnetic spectrum ranging from 0.1 to 10 THz, is envisioned as an enabling technology to support emerging wireless applications that require an explosive amount of data in the sixth-generation (6G) and beyond systems. Specifically, the enormous available bandwidths in the order of a few tens to a hundred gigahertz (GHz) and the extremely short wavelengths offer enormous potentials for wireless transmission [1]. Despite the promise, THzCom encounters unique challenges that have never been addressed at lower frequencies, e.g., higher channel sparsity, severe spreading loss, and frequency-dependent molecular absorption loss [2]. These challenges need to be wisely tackled for developing ready-to-use THzCom systems.

To fully harness the potentials of the THz band, the huge available bandwidth that spans across multiple ultra-wide transmission windows (TWs) needs to be optimally useds [3, 4]. This necessitates the exploration of novel and efficient spectrum allocation schemes for THzCom system when the to-be-allocated spectrum of interest is composed of multiple TWs. In this work, we consider the entire spectrum regions that exist between neighboring molecular absorption coefficient peaks at the THz band as TWs [4]. Multi-band-based spectrum allocation scheme has recently been studied for micro- and macro-scale THzCom systems with multiple TWs in the to-be-allocated spectrum of interest. In this scheme, the spectrum of interest is divided into a set of non-overlapping sub-bands which have a relatively small bandwidth, and then the sub-bands are utilized to serve the users in the system. This scheme has the ability to efficiently allocate spectral resources while ensuring intra-band interference-free transmission. Triggered by this ability, multi-band-based spectrum allocation has been widely explored for micro- and macro-scale multiuser THzCom systems [4, 5, 6, 7, 8].

A novel sub-band assignment strategy for multi-band-based spectrum allocation was proposed in [4] to improve the throughput fairness among users in multiuser THzCom systems. Moreover, spectrum allocation problems in the THz band backhaul network and non-orthogonal multiple access (NOMA) assisted THzCom systems were addressed in [5], and [6], respectively. Furthermore, efficient transmit power allocation and sub-band assignment algorithms were developed in [7] and [8], respectively, for multiuser THzCom systems. It is noted that the studies [4, 5, 6, 7, 8] focused on multi-band-based spectrum allocation with equal sub-band bandwidth (ESB), where the spectrum of interest is divided into sub-bands with equal bandwidth. However, since the molecular absorption loss varies considerably within the THz band, the variation among the sub-bands would be very high when ESB is considered. This variation can be reduced by exploring spectrum allocation with adaptive sub-band bandwidth (ASB), which allows to change the sub-band bandwidth. This can eventually lead to an overall improvement in the data rate performance.

Motivated by the potentials of ASB, the multi-band-based spectrum allocation with ASB was first introduced in our previous study [9]. However, the design in [9] is only applicable when the spectrum of interest exists entirely within a specific region of a TW. When the to-be-allocated spectrum of interest spans across multiple TWs, on one hand, the number of sub-bands that exist within each TW needs to be optimally decided. On the other hand, the set of users that are served by each TW and the sub-band assignment policy need to be identified. These challenges have not been touched by [9], but need to be carefully addressed to exploit the potentials of the huge available bandwidth that spans across multiple TWs at the THz band. This motivates this work.

We note that since molecular absorption loss is high at the edges of TWs, it may be beneficial if some spectra at the edges of TWs are not used during spectrum allocation. Despite so, previous studies including ours [9, 4, 5, 6, 7, 8] considered that the entire spectrum existing within the to-be-allocated spectrum of interest is fully utilized. Several recent studies [10, 11] considered to avoid some spectra at the edges of TWs. However, the designs in [10, 11] considered the unused spectra at the edges of TWs to be pre-defined, i.e., the unused spectra are arbitrarily chosen and fixed, which may not guarantee the optimal performance.

In this work, we design a novel multi-band-based spectrum allocation strategy when the to-be-allocated spectrum of interest is composed of multiple TWs. Specifically, we explore the benefits of ASB and optimally avoiding some spectra that exist at the edges of TWs during spectrum allocation. With these considerations in mind and focusing on an indoor THzCom scenario where a single access point (AP) supports the downlink of multiple users, we formulate a spectrum allocation problem. To analytically solve this problem, we propose transformations, as well as modifications, and arrive at an approximate convex problem. Subsequently, we develop an iterative algorithm based on the successive convex approximation technique to solve the approximate convex problem. Using numerical results, we show that when the to-be-allocated spectrum of interest is composed of multiple TWs, enabling ASB achieves a significantly higher data rate (between $9\%$ and $15\%$) as compared to adopting ESB. We also show that an additional gain of up to $5\%$ can be obtained by optimally determining the unused spectra at the edges of TWs, as compared to avoiding using pre-defined spectra that exist at the edges of TWs, especially when the power budget constraint is stringent. We finally illustrate that the feasibility region of the spectrum allocation problem can be improved by considering ASB and optimally avoiding some spectra that exist at the edges of TWs.

We consider a three-dimensional (3D) indoor THzCom system where a single AP, which is located at the center of the ceiling of the indoor environment, supports the downlink of users which demand high data rates. We consider that the users are stationary and distributed uniformly on the floor in this environment. We denote $\mathcal{I}$ as the set for the users and further denote $l_{i}$ and $d_{i}{=}\sqrt{h_{\epsilon}^{2}{+}l_{i}^{2}}$ as the horizontal and 3D distances of the link between the AP and the $i$th user, respectively, where $h_{\epsilon}$ is the fixed difference between the heights of AP and the users [12].

In this section, we present the multi-band-based spectrum allocation problem with ASB when the to-be-allocated spectrum of interest is composed of multiple TWs. We aim to maximize the sum data rate among all users under given sub-band bandwidth, power, and rate constraints. Mathematically, this problem is formulated as

In this section, we present numerical results to examine the performance of the proposed multi-band-based spectrum allocation scheme. Specifically, we consider an indoor environment of size $20~{}\textrm{m}\times 20~{}\textrm{m}$, where an AP at the center of the indoor environment serves 30 users. We also consider that the spectrum that exists between $0.325~{}\textrm{THz}$ and $0.448~{}\textrm{THz}$ is used to serve the users. We clarify that the considered spectrum spans across two TWs and each of the TW is composed of a PACSR and an NACSR. We denote the two TWs and the four regions by $\{\mathrm{TW}_{2},\mathrm{TW}_{3}\}$ and $\{p_{2},n_{2},p_{3}\,n_{3}\}$, respectively, as depicted in Fig. 1. Unless otherwise stated, the values of the other parameters used for numerical results are as follows: $B_{\textrm{g}}{=}1~{}\textrm{GHz}$, $B_{\textrm{max}}{=}4.5~{}\textrm{GHz}$, $P_{\textrm{tot}}{=}{-}12.5~{}\textrm{dBm}$, $P^{\textrm{max}}{=}\frac{4P_{\textrm{tot}}}{3|\mathcal{I}|}$, $R_{\textrm{thr}}{=}2~{}\textrm{Gbps}$, $h_{\epsilon}{=}2~{}\textrm{m}$, $g_{\textrm{A}}{=}35~{}\textrm{dBi}$, $g_{\textrm{U}}{=}20~{}\textrm{dBi}$, $N_{0}{=}{-}174~{}\textrm{dBm/Hz}$, $\xi^{r}_{\phi,s}{=}10^{9.7},~{}\omega^{r}_{\phi,s}{=}10^{10.7}$, and $\varsigma^{r}_{\phi,s}{=}10^{-3}$.

In Fig. 3, we plot the performance gain achieved by (i) considering ASB when the to-be-allocated spectrum of interest is composed of multiple TWs, i.e., $0\leqslant B_{s}^{r}\leqslant B_{\textrm{max}},~{}\forall s\in\mathcal{S}^{r},r\in\mathcal{R}$, and (ii) optimally determining the unused spectra that exists at the edge of TWs, i.e., $B^{r}_{\Delta}\geq 0,~{}\forall r\in\mathcal{R}$. In Fig. 3(a), we first observe the aggregated multiuser data rate, $R_{\textrm{AG}}=\sum_{i\in\mathcal{I}}R_{i}$, achieved by the proposed scheme with ASB while considering the state-of-the-art ESB-based spectrum allocation scheme in [9] as the benchmark. We clarify that in the benchmark scheme, (i) ESB is considered, (ii) the sub-bands are optimally assigned to users while satisfying the constraints (4a), (4b), and (10g), and (iii) the transmit power is allowed to vary while satisfying the constraints (10b) and (10c). We clarify that for the two schemes, $B^{r}_{\Delta}$ is fixed at $B^{r}_{\ast}$, where $B^{r}_{\ast}$ denotes the bandwidth of the spectrum at the edges of each region where the molecular absorption coefficient is higher than 0.3 [10, 9]. We observe that for all power budget values, a significantly higher $R_{\textrm{AG}}$ (between $12\%$ and $15\%$) can be achieved by adopting ASB by varying the sub-band bandwidths. Next, we compare $R_{\textrm{AG}}$ achieved by the proposed spectrum allocation scheme with ASB when $B^{r}_{\Delta}\geq 0$ and $B^{r}_{\Delta}=B^{r}_{\ast}$, and then plot the data rate gain of considering $B^{r}_{\Delta}\geq 0$ relative to considering $B^{r}_{\Delta}=B^{r}_{\ast}$, $\Delta R_{\textrm{AG}}$. We observe that although $\Delta R_{\textrm{AG}}$ is marginal when the power budget is high, a noticeable data rate gain $\Delta R_{\textrm{AG}}$ ($\approx 5\%$) exists when the power budget constraint is stringent. These two observations show the significance of considering $0\leqslant B_{s}^{r}\leqslant B_{\textrm{max}}$ and $B^{r}_{\Delta}\geq 0$. We further observe in Fig. 3(b) that the average value of the unused bandwidth within regions, $E[B^{r}_{\Delta}]$, increases when the power budget decreases, which explains the increasing $\Delta R_{\textrm{AG}}$ when the power budget decreases.

In Fig. 4, we show how the spectral resources are allocated according to the proposed multi-band-based allocation scheme. We plot the allocation of spectral resources within the transmission window, $\mathrm{TW}_{2}$. We note that the gray bars represent the sub-bands and the width and the height of the bars represent the bandwidth of the sub-bands and the transmission distance of the users assigned to the sub-bands, respectively. We first observe that, according to the optimal sub-band assignment configuration, the users with longer distances are assigned to center sub-bands where the absorption loss is low while the users with shorter distances are assigned to edge sub-bands where the absorption loss is high. This sub-band assignment configuration is similar to that considered in [4, 5] for the scenario with ESB. This shows that, since the absorption loss increases exponentially with distance, the data rate gain achieved by the users with longer distances, through occupying the center sub-bands as compared to occupying the edge sub-bands, is higher than the data rate loss incurred by the users with shorter distances, through occupying the edge sub-bands as compared to occupying the center sub-bands. We also observe that although the edge sub-bands experience higher absorption losses, a wider bandwidth is allocated to the edge sub-bands compared to the center sub-bands, since the edge sub-bands are occupied by the users with shorter distances.

In order to examine the impact of different spectrum of interest on the proposed spectrum allocation scheme, we plot $R_{\textrm{AG}}$ versus the number of regions, $|\mathcal{R}|$, in Fig. 5. Here, the regions corresponding to $|\mathcal{R}|$ of 1, 2, 3, and 4 are $\{p_{2}\}$, $\{p_{2}{+}n_{2}\}$, $\{p_{2}{+}n_{2}{+}p_{3}\}$, and $\{p_{2}{+}n_{2}{+}p_{3}+n_{3}\}$, respectively. For the sake of fairness, we made the number of users in the system to be proportional to the total available bandwidth within the spectrum of interest. More precisely, the number of users corresponding to $|\mathcal{R}|$ of 1, 2, 3, and 4 are 10, 17, 27, and 30, respectively. We first observe that $R_{\textrm{AG}}$ achieved by our proposed scheme with ASB is equal to that achieved by the scheme in [9], which is only applicable when the spectrum of interest is entirely within a region, i.e., $|\mathcal{R}|{=}1$. This shows that the scheme investigated in [9] is a special case of our proposed multi-band based spectrum allocation scheme in this work. We also observe that the proposed spectrum allocation scheme with ASB outperforms the state-of-the-art spectrum allocation scheme with ESB by $9\%{-}13\%$ for all values of $|\mathcal{R}|$. Furthermore, we observe that an additional gain can be achieved by considering $B^{r}_{\Delta}\geq 0$ as compared to $B^{r}_{\Delta}=B^{r}_{\ast}$. Finally, we observe that $R_{\textrm{AG}}$ for the proposed spectrum allocation scheme improves when the upper bound on sub-band bandwidth, $B_{\textrm{max}}$, increases. This is expected since a larger $B_{\textrm{max}}$ gives more room to exploit the ASB capability, which improves $R_{\textrm{AG}}$.

Finally, we illustrate the improvement in the feasibility region of the problem $\mathbf{P^{o}}$ that occurs as a result of considering ASB and $B^{r}_{\Delta}\geq 0$. To this end, Fig. 6 plots the percentage of simulation trails for which the problem $\mathbf{P^{o}}$ is feasible, $\Psi$, for the benchmark scheme with ESB and the spectrum allocation scheme with ASB when $B^{r}_{\Delta}\geq 0$ and $B^{r}_{\Delta}=B^{r}_{\ast}$. We first observe that $\Psi$ increases when power budget, $P_{\textrm{tot}}$, and rate threshold, $R_{\textrm{thr}}$, increases and decreases, respectively. This is expected because the increase in $P_{\textrm{tot}}$ and the decrease in $R_{\textrm{thr}}$ will lead to the constraints (10b) and (10d) in $\mathbf{P^{o}}$ to be less stringent, thereby leading to more opportunities to attain a feasible solution to $\mathbf{P^{o}}$. Second, we observe that our consideration of ASB increases $\Psi$ as compared to considering ESB. Third, we observe that our consideration of $B^{r}_{\Delta}\geq 0$ increases $\Psi$ as compared to considering $B^{r}_{\Delta}=B^{r}_{\ast}$. The second and third observations are due to the fact that considering ASB and $B^{r}_{\Delta}\geq 0$ allows sub-band bandwidth to vary and some spectra that exist at the edges of TWs to be avoided from utilization during spectrum allocation, respectively, if that can yield a feasible solution point. This again shows the importance of our considerations of ASB and $B^{r}_{\Delta}\geq 0$ during spectrum allocation.

This paper investigated the benefits of ASB and optimally determining the unused spectra at the edges of TWs during multi-band-based spectrum allocation in THzCom systems where the to-be-allocated spectrum of interest is composed of multiple TWs. We first formulated an optimization problem, with the main focus on spectrum allocation. Thereafter, to analytically solve the formulated problem, we proposed transformations, as well as modifications, and developed iterative algorithms based on the SCA technique. Using numerical results, we showed that the proposed spectrum allocation scheme with ASB achieves a significantly higher data rate compared to the multi-band based spectrum allocation with ESB. We also showed that an additional gain can be obtained by optimally determining the unused spectra at the edges of TWs as compared to avoiding using pre-defined spectra that exist at the edges of TWs, especially when the power budget constraint is stringent.

We clarify that the solution proposed in this work is only applicable when the molecular absorption coefficient within the TWs can be modeled using exponential functions of frequency with minimal approximation errors. Nevertheless, to harness the potentials of the huge available bandwidths of THz band, it would be beneficial to develop a solution for the spectrum allocation with ASB when the molecular absorption coefficient varies in a highly non-linear manner within the TWs and cannot be modeled using exponential functions of frequencies, which is the case in many spectrum regions within the THz band.