Beamforming Design for BackCom Assisted NOMA Systems

Backscatter communication (BackCom) assisted non-orthogonal multiple access (NOMA) has received significant attention recently and can be applied to Internet of Things (IoT). The BackCom system aims to realize energy cooperation among the devices. NOMA is a well-known technique to enable spectrum cooperation among the devices, which motivates the combination of the two techniques. This letter focuses on the beamforming design in BackCom NOMA networks, where an uplink data rate maximization problem is formulated with a constraint to guarantee the downlink user’s quality of service requirements. As the formulated optimization problem is not concave, semidefinite relaxation (SDR) is applied to transform the considered problem into a concave form which can be solved efficiently. Simulation results are provided to demonstrate that the proposed algorithm can achieve superior performance over the considered benchmarking scheme.


I. INTRODUCTION
U LTRA-MASSIVE machine type communications (umMTC) is one of the important applications for the 6G communication network [1], [2]. On one hand, to improve spectral efficiency, the concepts of non-orthogonal multiple access (NOMA) have been proposed by encouraging spectrum cooperation [3]. NOMA has been recognized as the candidate technology for future communication networks for its better connectivity, higher throughput, and higher energy efficiency for multi-user environments [4]. On the other hand, the use of BackCom can support energy cooperation among the users, e.g., a device without a power source can modulate and reflect its received radio-frequency signals and hence achieve battery-less transmission [5]. With the aid of convex optimization concepts [6], and the algorithms for transforming the original problem to convex problem, the maximum sum uplink data rate will be achieved.
In order to enable spectrum and energy cooperation, it is natural to combine NOMA with BackCom, where initial Manuscript  work on reflection indices optimization has been considered in [7], [8]. The aim of this letter is to develop a power allocation scheme by designing beamforming vectors to maximize uplink throughput. The optimization problem is formulated and the power budget and downlink (DL) user data rates are constraints. While exploring the sum-rate of uplink (UL), the low complexity decoding method of QR decomposition is applied to obtain the desired signal. The sum of uplink data rates is selected as the objective function. The power constraint and the DL signal to interference noise ratios (SINR) form the expressions of constraint. Due to it is hard to determine the convexity or concavity of the objective function, it is difficult to find the optimal solution efficiently. After applying SDR [9], the objective function and the constraints are converted to affine form. Hence, the recast convex optimization problem can be solved. Due to the variable being replaced by the beamforming vector multiplied by its hermitian, the rank-one constraints should be added. The rank-one constraints are not convex, so for simplifying the problem, these constraints are relaxed. To decouple the vector from the solved semidefinite hermitian matrix, the Gaussian randomization technique is applied to generate sub-optimal beamforming vectors. Therefore, the sub-optimal UL throughput is obtained. Compared with BackCom-assisted orthogonal multiple access (OMA) networks, the advantage of BackCom NOMA systems, which is higher UL data rate, is demonstrated.

II. SYSTEM MODEL
Consider a NOMA assisted BackCom network with an N-antenna base station, K single-antenna user equipments (UEs) and M single-antenna BackCom devices (BDs). In this communication system, the channel coefficients between the BS and U k , BS and BD m , BD m and U k are denoted by g k ∈ C N ×1 , h m ∈ C N ×1 and g m,k ∈ C 1×1 , respectively. The symbol x = [x 1 , . . . , x K ] T transmitted by the BS activates the BDs. And the BS broadcasts the symbol of s 0 directly to the UEs at the same time. And the BD m reflects the normalized signal of s m which is unknown to the DL users. The signal received by UE k is given by: where the transmit power is denoted by P 0 , the reflection index for BD m is denoted by η m , the beamforming vector for UE k is denoted by w k and W = [w 1 , w 2 , . . . , w K ], n k is the additive noise for UE k . The term,

M m=1
√ P 0 η m g m,k h T m Wxs m , can be treated as interference, whose power is given by: where g m,k denotes the channel gain between the m th BD to the k th downlink user. Therefore, the data rate of DL users is: The received signal at the BS is given by: where

III. RESOURCE ALLOCATION STRATEGY
The aim of this letter is to maximize the sum UL data rate meanwhile guaranteeing the DL sum data rate is no lower than its requirement. Due to multi-antenna BS deployment, the part of self-interference should be added to the received signal as well as the thermal noise. Hence, the sum of thermal noise and self-interference is no longer considered as 'white' noise. The detection matrix for signal pre-whitening is (σ 2 I N + αP 0 C SI ) 1 2 . The signal is modified as: where [7], a method of QR decomposition for decoding the received is used in this letter. For the approach, QR composite is applied to decode the received signal. By assuming that the composite channel matrix H can be decomposed via QR decomposition as follows: H = QR, where Q ∈ C N ×N is a unitary matrix and R ∈ C N ×M is an upper triangular matrix. Therefore, the matrix Q H is selective as the detector, which means that the system model can be rewritten as follows: By using the upper triangular structure of R, successive interference cancellation can be implemented with lowcomplexity. In particular, during the (M − m + 1)-th step, the signal from BD m can be decoded with the following data rate: where R m,m is defined similar to D m,m . To further reduce system overhead, it is ideal to formulate the resource allocation problem based on the following average sum rate [7]: where which is a concave function [7]. It is noted that an alternative way to obtain (8) is to treat h T m Wx in (7) as a complex Gaussian variable with mean zero and variance |h T m W| 2 [7]. The condition that the data rate for user k in (3) is greater than its threshold is equivalent to the condition that the signal to interference and noise ratio (SINR) of user k is larger than NOMA k , 1 ≤ k ≤ K , which can be expressed by: The (9) can be further reformulated by: Considering the power constraint, the transmit power cannot be larger than power budget, the sum of beamforming vector w k multiplied by its hermitian w H k should be equal or less than 1. Therefore, the maximization problem is: By setting W k = w k w H k , the matrix of W is rank-one semidefinite hermitian matrix. The method of SDR can be applied to recast (10) as follows: By replacing (10) with (12) to finish the process of SDR, problem (11) is recast as: Due to f (x) is concave and the variable for f (x) is affine to W k , the second order derivative of the objective function must be negative. Therefore, the objective function (13a) is a concave function. Equation (13b) is the expression of linear form and (12) is a form of trace function. Equation (12) and (13b) are affine. Note that the ideal energy harvesting of BDs is considered in the system model, which can also be applied to more practical cases by adding the energy harvesting coefficient in (12) and (13a) [10].

A. Optimization Process in OMA Systems
While in OMA systems, the BS broadcasts its signal to the number of K DL users via the number of K orthogonal subcarriers. A signal reflected by one BD can potentially block all subcarriers. Similar to (2), the power of interference yields: where G m,k denotes the backward channel gain between U k and BD m , h m denotes to the channel gain between BD m and the downlink users. Suppose the SINR of the DL user k is larger than OMA k , which leads to the following constraint: Based on the principles of SDR, (15) can be reformulated as: Similarly, W k = w k w H k . The received signal by BD at the k th subcarrier is given by: where F m,k denotes to the backward channel of BD m at subcarrier k. After applying a pre-whitening course using , the received signal can be expressed as follow: where Assuming detecting matrix is Γ = diag{(αP 0 |h SI 1 | 2 + σ 2 ) 1 2 , . . . , (αP 0 |h SI k | 2 + σ 2 ) 1 2 }, the decoded signal for OMA system is: where H = ΓF, D = η QR decomposition is applied. The average uplink sum data rate is given by: The constraint of beamforming weights is the same as the one shown in (13b). With SDR, the optimization problem can be formulated as: SDR and Gaussian approximation will also be used to optimize beamforming vectors in BackCom assisted OMA systems, in a manner similar to that presented in Section III.

B. Gaussian Randomization
Due to the rank function being quasi-concave and subadditive, the rank-one constraint is not guaranteed in the optimization problem of (13). The rank-one constraint is relaxed in (13) and (21), hence the optimal solution W k cannot always be written as W k = w H k w k . Therefore, the approximation method of Gaussian randomization [9] can be implemented to determine the suboptimal solution. To obtain w k which follows w k ∼ CN(0, W k ), it is essential to set the number of randomizations L. The generated vectors must satisfy the constraints of (12) and (13b).

C. Successive Convex Approximation
Optimization problems with SDR are usually solved by using numerical solvers such as CVX. While CVX cannot support the format of {Constant}/{Affine}. Therefore, successive convex approximation (SCA) [11] is exploited to solve the maximization problem. The lemma of [7]. Therefore, the objective function for n th SCA iteration is expressed by: where <, > denotes to inner product. The constraints are similar to the form demonstrated in (13). Armijo rule [6] is applied to estimate the step size of its iteration until the convergence with required accuracy. The optimization problem for SCA is formulated as:

IV. SIMULATION ANALYSIS
In this section, performance of the proposed NOMA system is investigated by using OMA as a benchmarking scheme. To evaluate the performance, a BackCom assisted NOMA system is considered with a base station located either at the origin (0, 0) or on the border of the area. BDs and UEs are randomly  scattered in a square area with an original point as its center and a side length of 3. The pathloss exponent is set to 3. The power of thermal noise is −94 dBm and the transmit signal to noise ratio (SNR) is set to 30 dB. The lower bound on the downlink data rate is 1 bit per channel use (BPCU). The number of randomization is 5000. Fig. 2 depicts the performance of NOMA and OMA with various self-interference coefficient values when M = N = K = 2 and 8. The figure shows that the more severe self-interference causes the poorer performance in either NOMA or OMA system. While with the same number of BDs, the UL data rate of NOMA system is higher. Naturally, the beamforming weights optimization assisted NOMA system performs the best among the three strategies of optimal NOMA, random NOMA and OMA. Mathematically, when the value of α increases, the value of R m,m in (13a) decreases, leading to the decreasing of sum UL data rate. Fig. 3 shows the phenomenon that when the number of BDs varies and the self-interference index values are fixed, the data rate increases with the growth of the number of BDs cluster, regardless of whether NOMA or OMA is employed. However, as the values of M = N = K grow, the performance advantages of NOMA are becoming more and more significant compared with OMA. Even in the small number of BDs, the superior performance can be well stated in optimal algorithm with BackCom NOMA. If the number of BDs increases, the average uplink data rate will be improved. The reason is that more UL data rate is accumulated linearly in (13a). As the result, the UL throughput increases.   In this environment, the BS is located in the middle of the Cartesian coordinate system, and the self-interference coefficient is set to α = 0.0001. Fig. 4 shows that the average UL data rate decreases due to the increasing demand of DL users. As the minimum data rate increases, the constraint of the optimization problem becomes stricter, resulting in a reduction of the feasible area. Consequently, the average UL data rate decreases. Fig. 5 illustrates a scenario where the BS is not centrally located within the square, specifically at the coordinates (3, 0) when self-interference coefficient is α = 0.0001. The data rate is reduced when the distances between BS and BDs or UEs increase, resulting in increasingly severe pathloss. Even with different BS locations deployed, the performance of OMA is still inferior to that of optimal algorithm assisted NOMA. The longer distance weakens the channel gain, i.e., the value of h m becomes lower. Therefore, central BS deployment leads to better performance than border BS deployment.

V. CONCLUSION
In this letter, we have analyzed three power allocation schemes, namely optimization algorithm assisted allocation scheme with NOMA, random allocation with NOMA, and OMA, in BackCom-assisted wireless communication systems. The proposed sum-rate maximization problem with BackComassisted NOMA allocates power to optimize the UL data rate by applying SDR to transform the objective function to concave form so that the optimization problem can be efficiently solved by CVX solver. The results show that increasing the BD and reducing the SI coefficient result in increased UL throughput. Furthermore, the proposed optimization algorithm achieves the goal of updating the UL data rate as much as possible while maintaining a given power budget. The BackCom-assisted NOMA scheme with the proposed technique, SDR and Gaussian randomization is applicable for future communication platforms.