An Unsupervised Learning Approach for Spectrum Allocation in Terahertz Communication Systems

Terahertz (THz) communication (THzCom) has been envisioned as a key wireless technology in the sixth-generation (6G) and beyond era [1]. The THz band has enormous potential to support tera-bits-per-second (Tbps) data rates and massive connections in 6G networks, due to the tens up to a hundred GHz bandwidth [2]. Thanks to the recent advancements in manufacturing THz transceivers and antennas, as well as the experimental licenses (95 GHz–3 THz) opened by the US Federal Communications Commission, increasing endeavors have been devoted to the design and development of practical THzCom systems over the past few years [3].

Despite the promise, THzCom encounters unique challenges that are different from those experienced at lower frequencies, e.g., severe spreading loss, higher channel sparsity, and the unique molecular absorption loss [4]. The molecular absorption loss is frequency-dependent, divides the whole THz band into several ultra-wide transmission windows, and introduces substantially varying distance-dependent path loss within transmission windows. These challenges need to be wisely tackled for developing practical THzCom systems.

There have recently been increasing endeavors to exploring novel and efficient THz spectrum allocation strategies, e.g., [4, 5, 2], due to its benefits of exploiting the potentials of the THz band. Specifically, multi-band-based spectrum allocation is envisioned as one promising solution to supporting micro- and macro-scale THzCom systems, since it can effectively assign spectral resources when the variation in molecular absorption loss among the users in multiuser systems is very high [6]. In this allocation, the spectrum of interest is divided into non-overlapping sub-bands, and the sub-bands are utilized to support the users in the system.

Although different designs have been proposed to optimize the performance of multi-band-based spectrum allocation, e.g., [6, 7, 8, 9], they all considered the spectrum of interest is divided into sub-bands with equal bandwidth. It is noted that the consideration of equal sub-band bandwidth (ESB) can lead to high variation in the molecular absorption loss among the sub-bands. Different from ESB, adaptive sub-band bandwidth (ASB), where the spectrum of interest is divided into sub-bands with unequal bandwidths, can effectively reduce the variation in molecular absorption loss among the users, thus leading to an overall improvement in the data rate performance. Motivated by this, the multi-band-based spectrum allocation with ASB was first proposed in our previous studies [10, 11]. We note that the design in [10] is only applicable when the molecular absorption coefficient within the to-be-allocated spectrum is simple such that it can be modeled as a piecewise exponential function of frequency with minimal approximation errors. In many spectrum regions within the THz band, the molecular absorption coefficient varies in a highly non-linear manner; thus, it cannot be modeled as a piecewise exponential function of frequencies in such spectrum regions. It follows that the designs in [10, 11] cannot be adopted in such spectrum regions. This motivates the current work.

In the current work, we study multi-band-based spectrum allocation with ASB to improve the spectral efficiency of THzCom systems. We formulate an optimization problem to determine the optimal sub-band bandwidth and optimal transmit power. We then propose an unsupervised learning-based approach to obtaining a near-optimal solution to the formulated optimization problem. In the proposed approach, we first train a deep neural network (DNN) where a loss function inspired by the Lagrangian of the formulated optimization problem is used. Thereafter, using the trained DNN, we approximate the near-optimal solution to the formulated optimization problem. Using numerical results, we demonstrate that when the values of the molecular absorption coefficient within the spectrum of interest can be modeled as an exponential function of frequency, the data rate obtained by our proposed unsupervised learning-based approach converges to the optimal data rate achieved by the existing spectrum allocation approach with ASB. We also demonstrate that when the values of the molecular absorption coefficient within the spectrum of interest change rapidly and thus cannot be modeled as an exponential function of frequency, our proposed unsupervised learning-based approach outperforms the existing approach.

Notations: Matrices and column vectors are denoted by uppercase and lowercase boldface symbols, respectively. Scalar variables are denoted by italic symbols. Given a matrix $\mathbf{A}$, $\mathbf{A}_{\mathrm{r}}$ denotes the $r$th row vector of $\mathbf{A}$. Given a column vector $\mathbf{a}$, $\mathbf{a}^{\mathrm{T}}$ and $a_{\mathrm{r}}$ denote the transpose and the $r$th element of $\mathbf{a}$, respectively. Moreover, $\mathbf{I}_{m}$ and $\mathbf{L}_{m}$ denote $m\times m$ identity matrix and $m\times m$ lower triangular matrix with all non-zero values as 1, respectively. Furthermore, $\mathbf{1}_{m}$ denotes the $m\times 1$ vector with all entries as 1. The curled inequality symbols, e.g., $\preceq$, denote generalized inequalities, i.e., componentwise inequality between vectors.

We consider a three-dimensional (3D) indoor THzCom system where the high data rate demands of $n_{\mathrm{I}}$ users are supported by a single AP. We consider that the users are stationary and distributed uniformly on the floor in the indoor environment. We denote $\mathbf{d}$ as the $n_{\mathrm{I}}\times 1$ communication distance vector of the AP-user channels. For notational convenience, we consider that the elements in $\mathbf{d}$ are ordered such that $d_{1}<d_{2}<\cdots<d_{n_{\mathrm{I}}}$.

In this section, we present an unsupervised learning-based approach to solving problem $\mathbf{P^{o}}$ in (4). In this approach, we first employ an offline training phase to train a DNN, utilizing (i) a loss function which is inspired by the objective function of the dual problem of $\mathbf{P^{o}}$ and (ii) a batch of realizations of distance vectors. Thereafter, during the implementation phase, we use the trained DNN to approximate the optimal solution to $\mathbf{P^{o}}$ for the given distance vector. Next, we will discuss the architecture of the adopted DNN in Section III-A, and the unsupervised learning model used in the training phase in Section III-B.

As mentioned in Section III, it is impossible to analytically solve the spectrum allocation problem $\mathbf{P^{o}}$ using traditional optimization techniques for the generalized system model considered in this work. Despite so, for a special case of the considered system model, the solution to $\mathbf{P^{o}}$ can be obtained using the convex optimization theory and standard convex problem solvers [10, 11]. We next present this special case system and its solution using convex optimization, the performance of which will be used as a benchmark in Section V for demonstrating the effectiveness and benefits of our proposed approach in Section III.

In the considered special case system, the molecular absorption coefficient within the spectrum of interest, $k(f)$, can be modeled as an exponential function of frequency with minimal approximation errors. We note that there indeed exist PACSRs and NACSRs within the THz band where $k(f)$ can be modeled as an exponential function of frequency, e.g., between $0.771$ and $0.821~{}\textrm{THz}$, where the maximum modeling approximation error is $5\%$.

For the considered special case system, the data rate, $r_{s}$, can be approximated as a tractable expression in terms of b. Specifically, by modeling ${k}(f)$ within NACSRs as an exponential function of frequency, we can approximate $k({f})$ as

where $\boldsymbol{\eta}=[\eta_{1},\eta_{2},\eta_{3}]^{\mathrm{T}}$ is the model parameter vector. This enables us to obtain $r_{s}$ as a tractable expression of b as

With the simplified expression $r_{s}$ in (23), although $\mathbf{P^{o}}$ becomes tractable, it is still non-convex. To overcome the non-convexity, we consider the following substitution for $\mathbf{b}$, given by

where $\mathbf{z}$ is the $n_{\mathrm{S}}\times 1$ new design variable vector that would replace $\mathbf{b}$ in the optimization problem and $\boldsymbol{\xi}=[\xi_{1},\xi_{2},\xi_{3}]^{\mathrm{T}}$ is a vector with real constants as its elements. Using (24), the spectrum allocation problem for the special case system can be transformed into an equivalent standard convex problem given by

where $z_{\textrm{tot}}=({\xi_{3}}^{-\xi_{2}}e^{{b}_{\textrm{tot}}-\xi_{1}n_{\mathrm{S}}})^{n_{\mathrm{S}}}$, $z_{\textrm{min}}=\xi_{3}^{-1}e^{\frac{-\xi_{1}}{\xi_{2}}}$, and $z_{\textrm{max}}=\xi_{3}^{-1}e^{\frac{b_{\textrm{max}}-\xi_{1}}{\xi_{2}}}$. The transformed problem for the special case system, $\mathbf{\hat{P}^{o}_{SC}}$, can be solved efficiently by using standard convex problem solvers [15]. The details of the convex optimization-based approach can be found in our previous work [10, 11].

The numerical results are obtained by considering a rectangular indoor environment of size $25~{}\textrm{m}\times 25~{}\textrm{m}$ where an AP, which is located at the center of the ceiling of the indoor environment, serves 15 users. We consider the spectrum with a bandwidth of 50 GHz that exist in either (i) the NACSR $sr_{n1}$ between $0.557$ and $0.671~{}\textrm{THz}$ or (ii) the NACSR $sr_{n2}$ between $0.752$ and $0.868~{}\textrm{THz}$ is used to serve the users. We clarify that on one hand, the molecular absorption coefficient, $k(f)$, in $sr_{n2}$ can be modeled as an exponential function of frequency with minimal approximation errors. Hence, the spectrum allocation problem in $sr_{n2}$ is a special case of the general spectrum allocation problem considered in this work. On the other hand, $k(f)$ in $sr_{n1}$ cannot be modeled as an exponential function of frequency.

We implement the unsupervised learning model presented in Section III in Python. We utilize a five-layer DNN for the unsupervised learning model, which has 100, 100, 50, 25, and 30 neurons in the first, second, third, fourth, and fifth layers, respectively. The initial weights of the DNN are Gaussian random variables with zero mean and unit variance, and the initial biases are set to 0. The initial values of $\boldsymbol{\lambda}$ are set to a small constant of $0.1$. The values of other system parameters, hyper-parameters of the unsupervised learning model, and the parameters of the convex optimization-based approach used for numerical results are summarized in Table I.

In Fig.4, we show the convergence of the proposed unsupervised learning-based approach when $k(f)$ within the spectrum of interest can be modeled as an exponential function of frequency, i.e., considering the special case system investigated in Section IV. To this end, we plot the aggregated multiuser data rate, $R_{\textrm{AG}}=\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{r}$, the value of $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{p}-p_{\textrm{tot}}$, and the value of $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{b}-b_{\textrm{tot}}$ obtained from the unsupervised learning-based approach in Figs. 4(a), 4(b), and 4(c), respectively, when the spectrum allocation problem is implemented in $sr_{n2}$. Also, the value of $R_{\textrm{AG}}$ obtained from the convex optimization-based approach is plotted in Fig.4(a). We first observe from Fig.4(a) that for several initial iterations, the value of $R_{\textrm{AG}}$ obtained using the unsupervised learning-based approach is higher than that obtained using the convex optimization-based approach. This is due to the occasional violation of the constraints (5b) and (5d) that occur at corresponding initial iterations, which can be validated by observing $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{p}-p_{\textrm{tot}}$ and $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{b}-b_{\textrm{tot}}$ in Figs. 4(b) and 4(c), respectively. Moreover, after 200 iterations, we observe that the value of $R_{\textrm{AG}}$ obtained using the unsupervised learning-based approach converges to the optimal value obtained using the convex optimization-based approach, which shows the correctness of our proposed unsupervised learning-based approach. Finally, we observe from Figs. 4(b) and 4(c) that after 200 iterations, the values of $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{p}-p_{\textrm{tot}}$ and $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{b}-b_{\textrm{tot}}$ of the proposed unsupervised learning-based approach are very close to zero, which reflects that constraints (5b) and (5d) are satisfied when the proposed unsupervised learning-based approach is used.

We next show the convergence of the proposed unsupervised learning-based approach when $k(f)$ within the spectrum of interest cannot be modeled as an exponential function of frequency, i.e., considering the generalized system investigated in Sections II and III. To this end, we plot $R_{\textrm{AG}}$, $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{p}-p_{\textrm{tot}}$, and $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{b}-b_{\textrm{tot}}$ in Figs. 5(a), 5(b), and 6(c), respectively, when the spectrum allocation problem is implemented in $sr_{n1}$. We clarify that in $sr_{n1}$, $k(f)$ cannot be modeled as an exponential function of frequency with minimal approximation errors. However, for the sake of comparison, we obtain approximate solutions to the spectrum allocation problem in $sr_{n1}$ using the convex optimization approach while utilizing an inaccurate approximation for $k(f)$ in $sr_{n1}$, and plot the resulting $R_{\textrm{AG}}$ in Fig.5(a). Once again, we observe the overshoot in $R_{\textrm{AG}}$, $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{p}-p_{\textrm{tot}}$, and $\mathbf{1}_{n_{\mathrm{S}}}^{\textrm{T}}\mathbf{b}-b_{\textrm{tot}}$ for several initial iterations and the satisfaction of the constraints (5b) and (5d) after 200 iterations in Figs. 5(a)-5(c), which are similar to Figs. 4(a)-4(c). Apart from them, we further observe that the value of $R_{\textrm{AG}}$ obtained using the unsupervised learning-based approach converges to a value higher than that obtained using the convex optimization-based approach. This is because when the convex optimization-based approach is used, where an exponential function with high approximation errors is used to model $k(f)$, only the sub-optimal $R_{\textrm{AG}}$ is obtained. This shows the significance of our proposed unsupervised learning-based approach, i.e., it gives much higher $R_{\textrm{AG}}$ for the spectrum allocation problem when $k(f)$ within the spectrum of interest cannot be modeled as an exponential function of frequency.

Finally, we plot $R_{\textrm{AG}}$ achieved by the spectrum allocation strategy with ESB [6, 7, 8, 9] and the spectrum allocation strategy with ASB that is obtained from both convex optimization and unsupervised learning-based approaches, versus the upper bound on sub-band bandwidth, $b_{\textrm{max}}$, in Fig.6. We first observe that the proposed spectrum strategy with ASB achieves a significantly higher $R_{\textrm{AG}}$ compared to the strategy with ESB for different $b_{\textrm{max}}$, which demonstrates the benefits of our proposed strategy with ASB. Second, we observe that in $sr_{n1}$, the value of $R_{\textrm{AG}}$ obtained using the unsupervised learning-based approach converges to values higher than that obtained using the convex optimization-based approach for all $b_{\textrm{max}}$. This again shows the significance of our proposed unsupervised learning-based approach to obtain higher $R_{\textrm{AG}}$, especially when $k(f)$ within the spectrum of interest cannot be modeled as an exponential function of frequency.

We investigated multi-band-based spectrum allocation with ASB for multiuser THzCom systems. We first formulated an optimization problem and then proposed an unsupervised learning-based approach to obtaining near-optimal sub-band bandwidth and transmit power. In the proposed unsupervised learning-based approach, we first employed an offline training phase to train a DNN while utilizing a loss function inspired by the Lagrangian of the formulated problem. Then using the trained DNN, we approximated the near-optimal solutions to the optimization problem for the given distance vector. Numerical results showed that when the values of the molecular absorption coefficient within the spectrum cannot be modeled as an exponential function of frequency, the data rate obtained by our proposed unsupervised learning-based approach outperforms that obtained by the existing convex optimization-based approach.