title: Sum Rate Maximization for Intelligent Reflecting Surface Assisted Terahertz Communications
authors: Yijin Pan Kezhi Wang Cunhua Pan Huiling Zhu Jiangzhou Wang
text: 
  	
  	
  	
  	
  	
  	
  	
  	
  	Sum Rate Maximization for Intelligent Reflecting Surface Assisted Terahertz Communications
      Yijin Pan, Kezhi Wang, Cunhua Pan, Huiling Zhu and Jiangzhou Wang, Fellow, IEEE 
  
  	Y. Pan are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 211111, China. She is also with School of Engineering and Digital Arts, University of Kent, UK. Email: panyj@seu.edu.cn, y.pan@kent.ac.uk.
  	K. Wang is with the Department of Computer and Information Sciences, Northumbria University, UK. Email: kezhi.wang@northumbria.ac.uk.
  	C. Pan is with the School of Electronic Engineering and Computer Science, Queen Mary, University of London, UK. Email: c.pan@qmul.ac.uk.
  	J. Wang and H. Zhu are with the School of Engineering and Digital Arts, University of Kent, UK. Email: J.Z.Wang@kent.ac.uk, H.Zhu@kent.ac.uk.
      December 30, 2023
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  	In this paper, an intelligent reflecting surface (IRS) is deployed to assist the terahertz (THz) communications. The molecular absorption causes path loss peaks to appear in the THz frequency band, and the fading peak is greatly affected by the transmission distance. In this paper, we aim to maximize the sum rate with individual rate constraints, in which the IRS location, IRS phase shift, the allocation of sub-bands of the THz spectrum, and power control for UEs are jointly optimized. For the special case of a single user equipment (UE) with a single sub-band, the globally optimal solution is provided. For the general case with multiple UEs, the block coordinate searching (BCS) based algorithm is proposed to solve the non-convex problem. Simulation results show that the proposed scheme can significantly enhance system performance.
  	
  	
  	
  	Intelligent reflecting surface (IRS), Terahertz (THz) communication, Reconfigurable intelligent surface (RIS).
  	
  	
  	
  
  	
  
  § INTRODUCTION
  
  	
  	The terahertz (THz) band wireless transmission has been envisioned as a promising solution to meet the ultra-high data rate requirements of the emerging applications such as the virtual reality (VR) service.
  	However, due to its ultra-high frequency, the propagation at THz covers short-range area and is susceptible to blockages.
  	This issue becomes more pronounced in the indoor scenario with furniture and complex interior structure<cit.>.
  	Recently, intelligent reflecting surface (IRS), also known as reconfigurable intelligent surface (RIS), has been widely proposed to reconfigure wireless propagation environment  to enhance the system performance, such as the simultaneous wireless information and power transfer (SWIPT) system <cit.>, orthogonal frequency division multiple access (OFDMA) system<cit.> and multicell network <cit.> through careful design of the phase shifts of the IRS.
  	Due to its capability of constructing an alternative non-line-of-sight (NLoS) communication link, the IRS is very attractive for the applications in THz communications that are sensitive to blockages. 
  	However, the study on IRS-aided THz communications is still in its infancy  <cit.> and numerous practical design issues are not yet addressed. 
  		
  	
  
  	The sum rate performance of the IRS-aided THz communication system was maximized by optimizing the phase shift of the IRS<cit.>.
  	The IRS-assisted massive multiple input multiple output (MIMO) transmissions in THz band were investigated in <cit.>. 
  	Although the above-mentioned literature studied the IRS-aided THz communications, the special features associated with the THz band were not considered.
  	In fact, the bandwidth offered by the THz Band ranges from 0.1 THz to several THz.
  	Due to the propagation loss and high molecular absorption in THz band, severe path loss peaks appear in different frequencies.
  	Hence, the total bandwidth is divided into several sub-bands with different bandwidths<cit.>.
  	In addition, the path loss of the sub-band varies with the communication distance<cit.>.
  	Thus, the frequency and distance-dependent sub-bands in the THz communications should be taken into consideration in the system design.
  
  	
  	However, it is challenging to utilize multiple sub-bands in THz communications, as it needs to be intelligently selected to avoid the path loss peaks.
  	Meanwhile, the peaks of path loss in the THz band also depend on the transmission distance, which highly relies on the location of IRS in the NLoS links.
  	Fortunately, in indoor applications, the locations of UEs can be predicted by using the historical statistical information of users' movement and the layout of furniture, and this information can be exploited to decide the optimal deployment of the IRS during wireless network design.
  	Therefore, it is imperative to jointly optimize the deployment of IRS, the reflecting phase shift, along with the sub-band allocation to enhance the IRS-assisted THz transmission.
  	Unfortunately, this issue has not yet been addressed in the existing literature.
  	
  	
  	Against the above background, in this paper, the THz transmission in a blocked indoor scenario is studied with the assistance of an IRS.  
  	The THz wireless channel consists of a set of frequency and distance-dependent sub-bands, and the channel fading in each sub-band is affected by both spreading loss and absorbent molecules.
  	Our target is to maximize the summation of the achievable rates of UEs by optimizing the IRS location, the IRS phase shift, sub-band allocation, and power control.
  	The globally optimal solution is provided for the special case of single UE with a single sub-band, and a block coordinate searching (BCS) algorithm is proposed for the general case with multiple UEs.
  	Simulation results are also provided for performance evaluation.
  	
  	
  		
  	
  
  § SYSTEM MODEL
  
  	propositionProposition
  	Consider the downlink transmission of an access point (AP) operating in THz frequency to support the VR service in an indoor scenario as shown in Fig. <ref>.
  	The LoS link from the AP to a given area may be blocked by obstacles such as the pillar or wall, and this area is assumed to be rectangular with length L and width W. 
  	Then, the set 𝒰 of U UEs in this area are served by the AP with the aid of an IRS as shown in Fig. <ref>, in which the IRS has N passive reflecting units.
  	Suppose that the reflecting coefficients of all units share the same amplitude value of one and have different phase shifts.
  	Let ϕ_n denote the phase shift of the n-th reflecting unit of the IRS, which can be carefully adjusted by an IRS controller.
  	Then, the reflecting coefficient of the n-th reflecting unit is Γ_n =  e^jϕ_n.
  	
  		
  	
  	Assume that the IRS can be installed on the ceiling with height H and parallel to the Y-axis, and the location of the first reflecting unit is denoted by l_1 = [X, Y, H]^T.
  	The separation between adjacent IRS elements is denoted by Δ, so that the location of the n-th reflecting unit is l_n = [X,Y+(n-1)Δ,H]^T.
  	Furthermore, the location of AP is denoted by w_0 = [x_0, y_0,z_0]^T.
  	The location of UE u is denoted by w_u = [x_u, y_u,z_u]^T, which can be estimated by using the information of furniture layout and the statistics of users' historical activities.
  	
  	Assume that the IRS is a uniform linear array, so that the following steering vectors are introduced to help represent the channel gain. 
  	For the link from the AP to the IRS, we define steering vector e^t(f,l)= [1, e^-j θ_1(f,l), ⋯,e^-j θ_N(f,l) ], where l = [X,Y].
  	The phase θ_n(f,l) represents the phase difference of the incoming signal to the n-th reflecting unit relative to the first unit with transmission frequency f, and it can then be calculated as
  	
      θ_n(f,l) = 2π f /cr_0^T/|r_0|(l_n-l_1) 
      	= 2π f /c(Y-y_0)(n-1)Δ/|r_0|,
  
  	where |x| represents the Euclidean norm of vector x,  and r_0 = [X-x_0, Y-y_0,H-z_0]^T.
  	
  	For the link from the IRS to UE u, we define the steering vector e_u^r(f,l)= [1, e^-j ϑ_1(f,l), ⋯,e^-j ϑ_N(f,l)]. 
  	Then, the phase difference of the signal received at UE u from the n-th reflecting unit relative to the first unit with transmission frequency f is 
  	
      ϑ^u_n(f,l) = 2π f /cr_u^T/|r_u|(l_n-l_1) 
      	= 2π f /c(y_u-Y)(n-1)Δ/|r_u|,
  
  	where r_u = [x_u-X, y_u-Y,z_u-H]^T, u=1,⋯,U.
  
  	
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  	
  	In THz band, apart from the spreading loss affected by the frequency and distance, the absorbent molecules composited in the transmission medium cause several peaks of channel attenuation, as shown in Fig. <ref>.
  	As a consequence, the total bandwidth of THz band needs to be divided into several sub-bands, and the channel capacity is calculated by adding up the rates of all sub-bands. 
  	Let f_i denote the central frequency of the i-th sub-band, the set of total sub-bands is denoted by ℐ, and the total number of sub-bands is assumed to be I.
  	According to <cit.>, in the near field scenario, the cascaded NLoS channel gain of the AP-IRS-UE u link on the i-th sub-band is 
  	
      g_u,i(f_i, d_u)= ( c/4 π f_i d_u) e^-j 2π f_i d_u/c e^-1/2K(f_i)d_u,
  
  	where d_u = |r_0|+ |r_u|, K(f_i) is the overall absorption coefficient of the medium, and c is the  light speed.
  	Then, the reflecting channel of AP-IRS-UE u link can be expressed as 
  	
      h_u,i(f_i,Φ, l)
      	=  g_u,i(f_i, d_u)e^t(f_i,l)Φe^r_u(f_i,l)^T,
  
  	where Φ= diag (Γ_1, ⋯,Γ_N).
  	
  	To evaluate K(f_i), we adopt a simplified molecular absorption coefficient model for 200 – 400 GHz frequency band<cit.>, which has two major absorption peaks at about 325 GHz and 380 GHz. 
  	This simplified model only depends on the volume mixing ratio of water (humidity) μ_w and frequency f (Hz):
  	
      K(f) = A(μ_w)/B(μ_w) + (f/100c-10.835)
      	+ C(μ_w)/D(μ_w) + (f/100c-12.664)
      
      	+p_1 f^3 + p_2 f^2 + p_3 f+ p_4,
  
  	where A(μ_w)=0.2205μ_w(0.1303μ_w+0.0294), B(μ_w)=(0.4093μ_w+0.0925)^2, C(μ_w)=2.014μ_w(0.1702μ_w+0.0303), 
  	D(μ_w)= (0.537 μ_w+0.0956)^2, 
  	p_1 =5.54 × 10^-37Hz^-3, 
  	p_2 = -3.94 ×  10^-25Hz^-2,
  	p_3 = 9.06 × 10^-14Hz^-1, and
  	p_4 = -6.36 × 10^-3.
  	The volume mixing ratio of water vapour μ_w is evaluated as
  	
      μ_w = ϕ_H/100p_w(T,p_ss)/p_ss,
  
  	where ϕ_H and p_ss (measured in hectopascal) respectively represent the relative humidity and the pressure. 
  	The saturated water vapour partial pressure p_w(T,p_ss) also depends on temperature T (measured in ^∘C), according to Buck equation <cit.>, which is calculated as 
  	p_w(T, p_ss)= 6.1121(1.0007 +3.46× 10^-8p_ss)exp(17.502T/240.97+T).
  	
  	 
  	
  	Similar to <cit.>, we assume that each sub-band is less than coherence bandwidth, and the separation between adjacent sub-bands is assumed to be sufficiently large.
  	Therefore, the inter-symbol interference (ISI) and interband interference (IBI) can be eliminated and narrowband communication on each  sub-band can be achieved<cit.>. 
  	Let B_i denote the bandwidth of the i-th sub-band, and the power spectral density of the noise in the i-th sub-band is expressed as S_N(f_i).
  	The obtained transmission rate of UE u on the i-th sub-band is given by 
  	
      R_u,i(f_i, p^t_i,Φ,l) =B_i  log( 1 + p^t_i|h_u,i(f_i,Φ, l)|^2 /S_N(f_i) B_i),
  
  	where p^t_i  is the transmit power on the i-th sub-band.
  	
  	
  	
  	
  	
  
  § PROBLEM FORMULATION
  
  	
  	Let the binary variable α_u,i indicate that the signal is transmitted to the UE u on the i-th sub-band. Assume that each sub-band is only allocated to one UE. Then, we have the following constraint as
  	
      C1: α_u,i∈{0,1}, ∑_u =1^U α_u,i = 1, i ∈ℐ, u ∈𝒰.
  
  	The transmit power p^t_i is constrained by AP's maximum power p_max as
  	
      C2: ∑_i =1^I p^t_i≤ p_max.
  
  	In addition, as the phase shift of each reflecting element of the IRS varies within the range of [0, 2π], the following constraint is introduced on the IRS phase coefficient matrix Φ as
  	
      C3: |Γ_n|^2 =1, 1≤ n ≤ N.
  
  	Furthermore, we have the following restriction on the deployment of IRS location l = [X,Y] as
  	
      C4:   0 < X < W, 0 < Y < L.
  
  	Then, to serve multiple UEs, the rate requirement should be satisfied for each UE as
  	
      C5:  ∑_i =1^I α_u,iR_u,i(f_i, p^t_i,Φ,l) ≥ R_u^th, ∀ u ∈𝒰.
  
  	
  	Based on the above descriptions, we aim to maximize the sum rate of the IRS-aided THz system by jointly optimizing the IRS location l, sub-bands allocation {α_u,i}, the transmit power {p^t_i} and the phase shift of IRS {Φ}.
  	Then, we formulate the following optimization problem as
  	
  		
      {Φ},{α_u,i}, {p^t_i}, lmax      R =  ∑_u ∈𝒰∑_i  ∈ℐα_u,iR_u,i(f_i, p^t_i,Φ,l) 
      s.t.      C1-C5.
  
  	
  	However, Problem (<ref>) is challenging to solve due to the complex form of |h_u,i(f_i,Φ, l)|^2.
  	Therefore, in the following, we first present the optimal solution for the special case of single UE, and then we propose an iterative BCS algorithm to solve Problem (<ref>) for the multiple-UE case.
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  
  § SINGLE UE SINGLE SUB-BAND CASE
  
  	
  	Consider that only one UE is located in the service area.
  	It is assumed that the largest flat sub-band with central frequency f_0 is utilized, and all the power is allocated to this UE on the sub-band, i.e., p_0^t = p_max.
  
  	The resulting channel gain is expressed as |h_u,i(f_0,Φ, l)|^2, then the IRS phase shift is readily obtained as
  	
      ϕ_n^* = 2π f_0 (n-1)Δ/c( (y_u-Y)/|r_u| + (Y-y_0)/|r_0|) .
  
  	Then, the channel gain  is expressed as |h_u,i(f_0,Φ^*, l)|^2 = ( c/4 π f_0 d_u)^2 e^-K(f_0)d_uN^2.
  	It is observed that |h_u,i(f_0,Φ^*, l)|^2 decreases with d_u, so that the optimal IRS location (X,Y,H) is determined by solving the following problem:
  	
      0 < X < W, 0 < Y < Lmind_u = D_0(X,Y) +D_u(X,Y)
  
  	where D_k(X,Y) = √((X-x_k)^2+(Y-y_k)^2+ H_k) , k = 0,u, H_u = (H-z_u)^2, and H_0 = (H-z_0)^2.
  	Then, the Hessian matrix of D_k(X,Y) denoted by H_D_k is given by
  	
      H_D_k=D_k(X,Y)^-3/2[
      	(Y-y_k)^2+ H_k   -(X-x_k)(Y-y_k)   
      
      	-(X-x_k)(Y-y_k)     (X-x_k)^2+ H_k
      	]  .
  
  	It is verified that the determinant of H_D_k is non-negative, so that H_D_k is semi-positive definite.
  	As a result, Problem (<ref>) is a convex problem, and it can be solved by the standard algorithms, such as the interior point method.
  	
  	
  
  § MULTIPLE UES MULTIPLE SUB-BANDS CASE
  
  	In this section, we consider the more general case with multiple sub-bands and multiple UEs.
  	
  	Defining δ_i = 1/S_N(f_i) B_i and introducing the auxiliary variables {t_u,i}, we reformulate Problem (<ref>) as:
  	
  		
      {α_u,i}, {p^t_i}, {Φ},l, {t_u,i}max    R_s(t_u,i)=∑_u ∈𝒰∑_i  ∈ℐα_u,i B_i  log( 1 + δ_i t_u,i) 
      s.t.    p^t_i|h_u,i(f_i,Φ, l)|^2≥α_u,i t_u,i,∀ u ∈𝒰, i ∈ℐ, 
          ∑_i ∈ℐB_i α_u,ilog( 1 + δ_i t_u,i) ≥ R_u^th, ∀ u ∈𝒰, 
          t_u,i≥ 0,∀ u ∈𝒰, i ∈ℐ,  C1-C4.
  
  	
  	
  	
  	To solve Problem (<ref>),  we first fix the location l of the IRS, and optimize the IRS phase shift and the sub-bands/power allocation. 
  	As the expression of |h_u,i(f_i,Φ, l)|^2 is intractable, the two-dimensional search is then adopted to find the optimal location of IRS.
  	
  	
  
   §.§ IRS phase Shift Optimization
  
  	
  	With given l, {t_u,i}, {α_u,i} and {p_i^t}, (<ref>) can be represented as
  	
      p^t_i|g_u,i(f_i, d_u) e^t(f_i,l)Φe^r_u(f_i,l)^T|^2 ≥  t_u,i,∀ u ∈𝒰, i ∈ℐ_u,
  
  	where ℐ_u represents the set of sub-bands that are allocated to UE u.
  	Then, we define
  	
      e_u (f_i) =√(p^t_i) g_u,i(f_i, d_u)[1,  e^-j(θ_1(f_i) + ϑ_1(f_i)), ⋯, 
       e^-j (θ_N-1(f_i) + ϑ_N-1(f_i))], ϕ = [Γ_1, ⋯, Γ_N]^T.
  
  	To optimize the IRS phase shift vector ϕ, Problem (<ref>)  is simplified to
  	
  		
      ϕmax      R_s(t_u,i) 
      s.t.      |Γ_n|^2 =1, 1≤ n ≤ N, 
            |e_u (f_i)ϕ|^2 ≥ t_u,i , u ∈𝒰, i ∈ℐ_u.
  
  	
  	As the feasible set of Problem (<ref>) is non-convex, Problem (<ref>) is difficult to solve in this original formulation.
  	Define w_u,i = e_u (f_i)ϕ. Then, T_u,i(w_u,i) = w_u,i ^2  is convex with respect to w_u,i.
  	Thus, its lower bound surrogate function at ŵ_u,i could be obtained by the first-order Taylor approximation as
  	
      T_u,i(w_u,i)  ≥ T_u,i(ŵ_u,i) 
      	+ ▿_w_u,i T_u,i|_w_u,i = ŵ_u,i(w_u,i-ŵ_u,i)
      
      	+ ▿_w_u,i^* T_u,i|_w_u,i=ŵ_u,i(w_u,i^*-ŵ_u,i^*).
  
  	Substituting w_u,i = e_u (f_i)ϕ and ŵ_u,i = e_u (f_i)ϕ̂ into the right hand side of (<ref>), we have
  	
      T_u,i(e_u (f_i)ϕ) ≥  2 {e_u (f_i)ϕ̂^H e_u (f_i)^Hϕ} - |e_u (f_i)ϕ̂|^2.
  
  	Then, Problem (<ref>) can be addressed by solving a sequence of simpler problems.
  	Based on (<ref>), for given ϕ̂, the convex approximation problem of Problem (<ref>) can be constructed as
  	
  		
      ϕmax    R_s(t_u,i) 
      s.t.    |Γ_n|^2 =1, 1≤ n ≤ N, 
          2 {Θ_u,i(ϕ̂)ϕ}≥Ψ_u,i(ϕ̂) + t_u,i , u ∈𝒰, i ∈ℐ_u,
  
  	
  	where 
  	Ψ_u,i(ϕ̂) =|e_u (f_i)ϕ̂|^2, and
  	Θ_u,i(ϕ̂) = e_u (f_i)ϕ̂^H e_u (f_i)^H.
  	However, as the constraint (<ref>) is non-convex, the Lagrangian dual method cannot be applied to solve Problem (<ref>) due to the non-zero dual gap.
  	In the following, we adopt a pricing mechanism to solve Problem (<ref>), where a series of non-negative prices {ρ_u,i} are introduced in  constraints (<ref>).
  	Then, a penalty term is introduced to the objective function, and the problem is transformed to
  	
      |Γ_n|^2 =1max R_s(t_u,i) +∑_u ∈𝒰∑_i  ∈ℐ_uρ_u,i(2 {Θ_u,i(ϕ̂)ϕ}.
       - . Ψ_u,i(ϕ̂)-t_u,i).
  
  	With given {ρ_u,i}, the optimal solution ϕ to Problem (<ref>) is given by 
  	
      ϕ^*_n= Θ̂_n(ρ_u,i), and Θ̂(ρ_u,i) = (∑_u ∈𝒰∑_i  ∈ℐ_u 2ρ_u,iΘ_u,i(ϕ̂)) ,
  
  	where Θ̂_n(ρ_u,i) is the n-th element of Θ̂(ρ_u,i).
  	
  	If the obtained solution ϕ^* is not feasible, the introduced penalty term will decrease the objective value.
  	Consequently, the pricing factors {ρ_u,i} should be optimized so that the introduced penalty term  ρ_u,i(2 {Θ_u,i(ϕ̂)ϕ}  - Ψ_u,i(ϕ̂) - t_u,i) is minimized.
  	Then, to obtain pricing factor ρ_u,i^*, we employ the sub-gradient descent based method.
  	To be specific, {ρ_u,i^(t)} in the t-th iteration is updated as
  	
      ρ_u,i^(t)=[ρ_u,i^(t-1) - τ_u,i^(t)(2 {Θ_u,i(ϕ̂)ϕ} -Ψ_u,i(ϕ̂) - t_u,i)]^+ ,
  
  	where [a]^+ = max{0,a}, τ_u,i^(t) is the positive step-size in the t-th iteration. 
  	
  	
  	
  	
  	
  		The SGD algorithm can find the globally optimal solution to Problem (<ref>). 
  	
  	Proof: see Appendix A. 	
  
  	
  
   §.§ Sub-band Allocation and Power Control Optimization
  
  	
  	With the given IRS location l and the IRS coefficient Φ, the auxiliary variables, the sub-bands and the power allocation can be optimized by solving the following problem:
  	
  		
      {t_u,i}, {p_i^t},{α_u}max    R_s(t_u,i) =∑_u =1^U∑_i =1^Iα_u,i B_i  log( 1 +  δ_u,it_u,i) 	 
      s.t.    ∑_i =1^I α_u,i B_i  log( 1 +  δ_u,it_u,i) ≥ R_u^th, u ∈𝒰, 
            p^t_i h_u,i^2 ≥α_u,i t_u,i,∀ u ∈𝒰, i ∈ℐ, 
         (<ref>), C1-C2.
  
  	
  	By introducing the transformation x_u,i = α_u,i t_u,i, the above Problem (<ref>) can be solved by the dual-based method given in <cit.> and <cit.>.
  
  	In summary, based on the above analysis, we propose the following BCS algorithm to solve the original problem in (<ref>), and the detailed algorithm is presented in Algorithm <ref>.
  	
  	
  	
  	
  	
  	In the proposed algorithm, according to <cit.> and <cit.>, the obtained solution p^t_i and α_u,i satisfies the Karush–Kuhn–Tucker (KKT) conditions of Problem (<ref>). 
  	Furthermore, according to <cit.>, by successively constructing the approximated Problem (<ref>), the converged result obtained in step 8 - step 11 is a local minimum to Problem (<ref>).
  	Then, it can be verified that the objective R^(n) is non-decreasing over each iteration, and the convergence of the procedure from  step 4 - step 14 is guaranteed.
  	
  	The complexity of the proposed algorithms consists of three parts: 1) The first part is the dual algorithm to find α_u,i and p^t_i in step 6, of which the complexity can be concluded as 𝒪(U^4 + U^3I)<cit.>; 2) The second part is the successively approximation to find ϕ. Let S denote the iteration number in Step 8, and the number of iterations in SGD algorithm is T. Then the complexity to find ϕ is 𝒪(ST); 3) The last part is the two-dimensional search to find IRS location l. Let W denote the iteration number in Step 5. Then, the total complexity can be concluded as 𝒪(⌊L/δ_x⌋⌊W-(N-1)δ/δ_y⌋ W(ST +U^4 + U^3)).
  	
  	
  	
  
  	
  
  § SIMULATION RESULTS
  
  	
  	In this section, simulation results are presented to show the performance of the proposed scheme.
  	UEs are uniformly distributed in a 5m × 8m rectangular area, the height of ceiling is 3m, and the AP is located at (0,0,2).
  	The UE's rate requirement is 1Gbps, the bandwidth of the sub-band is 50GHz, the transmission frequency is 200-400GHz, and p_max is 1W.
  	The number of IRS reflecting elements is 20, and the distance between each element is 5mm.
  	All the results are obtained by averaging over 100 random realizations of UE locations.
  	For comparison, we consider three different algorithms: 1) The location of the IRS is randomly chosen; 2) The phase shift of IRS is randomly generated; 3) The location of the IRS is selected to minimize the sum of  transmission distances of all UEs, i.e., (X,Y) = 0 < X < W, 0 < Y < L min∑_u  ∈𝒰 d_u.
  	The above three algorithms are labelled as “RanLoc”,  “RanPhi” and “MiniDis”, respectively.
  	
  	
  	
  	Fig. <ref> illustrates the sum rate obtained by different algorithms.
  	It is observed that the proposed “BCSAlg" algorithm always achieves the best performance.
  	The sum rates decrease with the number of UEs for all the algorithms, which is due to the rate requirement constraint.
  	The “MiniDis” approach is better than the “RanLoc” and “RanPhi” schemes due to the optimized IRS location and phase shifts.
  	Moreover, it is observed that the achieved rate of “RanLoc” is slightly larger than that of “RanPhi”.
  	 This implies that IRS phase shift has a slightly more significant impact on the sum rate performance compared with the IRS location.
  	
  	
  
  § CONCLUSION
  
  	
  	In this paper, the sum rate of the UEs in an IRS-assisted THz transmission system has been maximized by optimizing the IRS location, the IRS phase shift, sub-band allocation, and power control. 
  	Although the formulated problem is nonconvex, the proposed algorithm has been shown to improve the sum rate performance significantly.
  	Through simulations, it is observed that our proposed solution has considerable performance gain over the benchmark schemes.
  	
  	
  	
  	
  	
  
  § PROOF OF PROPOSITION <REF> 
    
  	   
  	
  	
  	The proposition is proved by using the contradiction method.
  	
  	Let {ρ_u,i^*} and ϕ^*(ρ_u,i^*) denote the converged results obtained by SGD algorithm.
  	We define function f_u,i(ϕ^*) = 2 {Θ_u,i(ϕ̂)ϕ^*}  - Ψ_u,i(ϕ̂) - t_u,i.
  	
  	Assume that ϕ^*(ρ_u,i^*) is not the globally optimal solution to Problem (<ref>), so that the constraint (<ref>) cannot be satisfied for all u ∈𝒰.
  	Let 𝒰_1 denotes the set of UEs that satisfy the constraint (<ref>), and the set of the left unsatisfied UEs are denoted by 𝒰_2, i.e., 𝒰 =𝒰_1 + 𝒰_2.
  	Then, the globally optimal solution to Problem  (<ref>) is denoted by ϕ̃, and the following inequalities hold:
  	
      ∑_u ∈𝒰_1∑_i  ∈ℐ_uρ^*_u,i   f_u,i(ϕ^*) 
      	< 0 < ∑_u ∈𝒰_1∑_i  ∈ℐ_uρ^*_u,i  f_u,i(ϕ̃) .
  
  	Adding the same term to both sides of  (<ref>), we have
  	
      ∑_u ∈𝒰∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ^*)
      	< ∑_u ∈𝒰_1∑_i  ∈ℐ_uρ_u,i^*f_u,i(ϕ̃)
       + ∑_u ∈𝒰_2∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ^*).
  
  	Meanwhile, as the phase vector ϕ^* obtained in Algorithm <ref> achieves the globally optimal solution to Problem (<ref>), we have
  	
      ∑_u ∈𝒰∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ^*) 
      	>∑_u ∈𝒰∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ̃) .
  
  	
  	
  	Then, combining the left hand side of (<ref>) and the left hand side of (<ref>),  as well as removing the common terms in 𝒰_1, we have
  	
      ∑_u ∈𝒰_2∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ^*)> ∑_u ∈𝒰_2∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ̃) .
  
  	
  	Then, we consider two cases for ρ^*_u,i: 1) ρ^*_u,i=0, ∀ u ∈𝒰_2; 2) ρ^*_u',i > 0, u' ∈𝒰', 𝒰' ⊆𝒰_2 and  i ∈ℐ_u'.    
  	
  	In the first case, the left hand side and the right hand side of (<ref>) both equal zero, which contradicts the assumption.
  	
  	In the second case, as ρ^*_u',i > 0 and SGD algorithm is based on the sub-gradient method, then with a sufficient small step size, the converged result ϕ^* obtained by SGD algorithm satisfies the condition of f_u,i(ϕ^*)=0, ∀ u' ∈𝒰', i ∈ℐ_u'.
  	Then, combining with the left hand side of (<ref>), we have
  	
      0 > ∑_u ∈𝒰_2∑_i  ∈ℐ_uρ^*_u,if_u,i(ϕ̃) .
  
  	
  	As ρ_u',i ^*>0, it is inferred that f_u,i(ϕ̃)  <0, ∀ u' ∈𝒰', i ∈ℐ_u'.
  	This contradicts the constraints in (<ref>).
  	However, as the ϕ̃ is the globally optimal solution to Problem  (<ref>), so that ϕ̃ should satisfy all the constraints of Problem  (<ref>).    
  	As a result, the assumption does not hold, and ϕ^*(ρ_u,i^*) is the globally optimal solution to Problem  (<ref>).
  	Hence, the proof is completed.
  	
  	
  	ieeetran
  	
  	
  