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Energy-Efficient Backscatter Aided Uplink NOMA Roadside Sensor Communications under Channel Estimation Errors

This work presents non-orthogonal multiple access (NOMA) enabled energy-efficient alternating optimization framework for backscatter aided wireless powered uplink sensors communications for beyond 5G intelligent transportation system (ITS). Specifically, the transmit power of carrier emitter (CE) and reflection coefficients of backscatter aided roadside sensors are optimized with channel uncertainties for the maximization of the energy efficiency (EE) of the network. The formulated problem is tackled by the proposed two-stage alternating optimization algorithm named AOBWS (alternating optimization for backscatter aided wireless powered sensors). In the first stage, AOBWS employs an iterative algorithm to obtain optimal CE transmit power through simplified closed-form computed through Cardano's formulae. In the second stage, AOBWS uses a non-iterative algorithm that provides a closed-form expression for the computation of optimal reflection coefficient for roadside sensors under their quality of service (QoS) and a circuit power constraint. The global optimal exhaustive search (ES) algorithm is used as a benchmark. Simulation results demonstrate that the AOBWS algorithm can achieve near-optimal performance with very low complexity, which makes it suitable for practical implementations.

of sensing, computation, and communication components that consume power. Backscatter communication is an emerging technology that provides a single infrastructure for jointly sensing and transmitting data with microwatt levels of power consumption [2].
Backscatter communications provide batteryless connectivity through RF energy harvesting [3]. This technique is very different than the general energy harvesting techniques where devices harvest energy for the operation of active RF transmissions [4], [5]. On the contrary, backscatter communication is low complexity and low power technique that does not require any active RF transmission component. It allows transmitters to transfer their information to the backscatter reader (BR) installed on receivers by reflecting and modulating the RF carrier signals of CE. Such backscattering at transmitters is done through mismatching the impedance at the input of the antenna which results in varying reflection coefficients [6]. Its transmission consumes very little energy as compared to the conventional radio and its operation without active RF component results in much simpler and uncomplicated circuits. The implementation of the backscatter technique for the wirelessly powered sensors has been limited because of its limited coverage. Backscatter communication based on bistatic architecture has been proposed to mitigate this limitation [3], [7], [8]. In bistatic backscatter communication, the CE is dislocated from BR, which results in a more flexible network configuration and reduces the near-far effect. Therefore, it is much more suitable for future ITS to use bistatic backscatter aided roadside sensors.

A. Related Literature
Recently, backscatter communication has shown great potential for large-scale massive internet of things (IoT) networks [9]. To achieve low latency, high spectral, and energy efficiency, NOMA is the key technology to serve a large number of roadside sensors. The integration of NOMA with backscatter communication has proven great potential for collecting information from multiple sensors through the same sub-channel in a non-orthogonal manner [10], [11]. The information of multiple roadside sensors can be multiplexed on the same sub-channel by tuning the value of the reflection coefficient of each roadside sensor to a different value. After tuning their reflection coefficients, multiple roadside sensors in the same cluster can be separated through the power domain.
Roadside sensors collect large amounts of information and send it to central computing servers through roadside units (RSUs) or base stations (BSs) for analysis [23]. Besides, RSUs can also transfer sensed information to the vehicles that exist in their coverage area. For instance, sensors detect the pedestrians in the crosswalk and transmit it to RSUs through backscatter communication. Then, RSUs disseminate this information to surrounding vehicles in its coverage [20]. Hence, such an interconnected transport system results in better decision-making that leads to the improved safety of the environment. [2]. As the roadside sensors need to collect and transmit different useful information to the RSUs in an energyefficient manner. Therefore, backscatter communications can be employed by the roadside sensors to send their sensed information to the nearest RSU in the uplink scenario [17], [20]. Besides, Obtaining CSI is crucial for the performance analysis of backscatter communications. In practice, it is very challenging for the passive transmitter to guarantee the accuracy of CSI all the time [24]. Furthermore, the realization of backscatter enabled uplink sensor communications will demand advanced energy-efficient resource allocation (RA) frameworks. Their RA is incredibly challenging due to the diverse quality-of-service (QoS) requirements and its strong underlying dynamics. In literature, we believe that there is no energy-efficient RA optimization framework for NOMAenabled backscatter aided uplink roadside sensors communication with channel uncertainties. Therefore, in this article, we proposed a novel alternating optimization framework for EE maximization of wireless powered backscatter aided uplink sensors communications for future generation ITS under imperfect CSI. Our major contribution is summarized as follows, • A novel energy-efficient alternating optimization framework for NOMA-enabled wireless powered uplink sensor communication under imperfect CSI is proposed for ITS. The EE maximization problem is formulated under various QoS requirements of bistatic backscatter communications for roadside sensors. The EE is maximized under channel uncertainties by optimizing transmit power of CE and reflection coefficient of roadside sensors. The formulated problem is solved into two stages, which yields the proposed AOBWS algorithm. • The proposed two-stage AOBWS algorithm provides optimal EE performance in very low computational complexity, which makes it suitable for practical implementations. The detailed complexity analysis of the proposed 3 algorithm and benchmark algorithms is presented in the complexity analysis subsection. • The efficacy of the proposed AOBWS algorithm is analyzed through numerical simulations and is compared with the global optimal ES algorithm as a benchmark. From obtained results, it is observed that AOBWS can achieve desired EE performance with affordable computational complexity for practical implementations. The rest of the paper is organized as follows: The system model is provided in Section II. In Section III, we discussed the problem formulation with its solution for the energy efficient wireless powered uplink sensor communication under channel uncertainties for ITS while Section IV presents the numerical results to verify the efficacy of the proposed AOBWS algorithm. Finally, Section V provides concluding remarks of our analysis and discuses the future work. The definition of different acronyms and symbols used in this paper are respectively defined in Table I and II. II. SYSTEM MODEL This work considers NOMA-enabled backscatter communication for wireless powered passive sensors in ITS. The system model consists of a CE, multiple backscatters aided roadside sensors, and BR installed on RSU as depicted in Figure 1. Roadside sensors are multiplexed in various clusters, where each cluster consists of sensors. Practically, It is desireable to have two or three sensor per cluster for low decoding complexity and to guarantee timing constraints [25]. Roadside sensors harvest RF energy from the RF carrier signal emitted by CE [26] and backscatter their information to the BR. It works in two operational modes, namely the transmission mode ( ) and energy harvesting mode ( ℎ ). These two modes of roadside sensor constitute one slot such that + ℎ = 1 1 , where > 0 and ℎ > 0. In transmission mode, each cluster of roadside sensors backscatters the RF carrier signal of CE to transmit its sensed information to the BR. This backscattering is carried out with the help of an RF transistor. It reflect the incident carrier RF signal with altered phase and magnitude through mismatching the impedence at the input of the antenna which results in varying reflection coefficients. In the energy harvesting mode, roadside sensor harvests energy from incident RF signal instead of reflecting it for transferring information. This harvested energy is then reserved in the battery and is utilized to power its circuitry. For a more detailed description of backscatter enabled sensors, please refer to [25].
The uplink transmission scenario is considered in which one cluster of two roadside sensors is handled by a single CE and BR. Both roadside sensors and BR communicate through single antennas. The forward channel link between CE and ℎ roadside sensor is denoted by , , while , is used to denote the backscatter link between ℎ roadside sensor and BR, for ∈ {1, 2}. The channel coefficients of the forward 1 Optimizing time coefficients of transmission mode and energy harvesting mode ℎ of roadside sensor can further enhance the EE performance, however, it is left for our future work to focus here on optimizing CE transmit power and reflection coefficient of roadside sensors under channel uncertainties. and backscatter link are consist of the following components, respectively. and where ℎ , and ℎ , represent fast fading component of forward and backscatter link, respectively. , and , are the distance from CE to ℎ roadside sensor and ℎ roadside sensor to BR, respectively. is used as path-loss exponent. Let is the power of the carrier signal emitted from CE.
Then, the power of incident RF signal at ℎ roadside sensor is Roadside sensor can harvest energy from RF signals in both energy harvesting mode and transmission mode. Therefore, the energy harvested by ℎ roadside sensor during transmission mode ( ) and energy harvesting mode ( ℎ ) is given as follow where denotes the efficiency of energy harvester while Γ represents reflection coefficient of ℎ roadside sensor, where 0 < Γ ≤ 1. The reflected signal from ℎ roadside sensor toward BR installed on RSU is where is the transmitted signal form ℎ roadside sensor, satisfying E(| | 2 ) = 1. Then , the signal received at RSU is given as where = , , is the overall channel link between ℎ roadside sensor and BR, which can be estimated through LMMSE method [24].
can be written as = ×ℎ [27], where = − , × − , and ℎ = ℎ , × ℎ , . In practice, it is very challenging for backscatter aided communications to guarantee accurate CSI at all the time [24]. Therefore, it is necessary to investigate the backscatter aided sensors to RSU communication with imperfect CSI. Under imperfect CSI estimation, can be expressed by using the minimum mean square error (MMSE) channel estimation error model [28], [29] as follow whereˆ∼ is the estimated channel error, which is Gaussian distributed with zero mean and variance 2 . The relative channel error can be represented as = 2 . Then, 2 = and 2ˆ= (1− ) [31]. For convenience of analysis, the case of constant estimation error ( 2 = 2 ) for all NOMA users is considered in this paper [30], [32]. Thus, the signal received at RSU under imperfect CSI is given aŝ Without loss of generality, roadside sensors channel gains are arranged as: |ˆ2| 2 > |ˆ1| 2 . Unlink downlink scenario, In the uplink, all information received at RSU from a cluster of roadside sensors are desired signals, although they give rise to multiuser interference. In downlink NOMA, SIC is implemented in ascending order i.e. the roadside sensor with lower channel gain is first decoded and removed. While in the uplink, the RSU can decode the roadside sensors signals in an arbitrary order, as all information is desired at the RSU [33]. However, despite that, to implement SIC and decode the received information at the RSU, tuning the value of reflection coefficient of each roadside sensor to different values should be exploited in such a way that it maintained the distinctness among various signals. For the analysis, we assume that the SIC order that decodes the information of the 2nd roadside sensor (highest channel gain) first is employed at the RSU throughout this work [34]. According to the NOMA principles, the sum-rate of roadside sensors is given by The total power consumed by the cluster is the circuit power consumption of the roadside sensors in backscatter mode. The circuit operation is performed through the harvested power from the carrier signal during transmission mode as well as in energy harvesting mode. The power consumed by the cluster can be obtained by the sum of the harvested powers of roadside sensors paired in that cluster. Besides it, total energy consumption also consists of the energy consumed by the CE to transmit RF signals and RSU to receive the signals of roadside sensors. Therefore, the total power consumption of the system is as follow where is power amplifier efficiency that ranges between zero and one while and are the constant circuit power consumed by CE and RSU respectively. The energy efficiency of the proposed system model is defined as the ratio of the sum rate of roadside sensor to the consumed power of system in bits per Hertz per Joule (bits/Hz/J) [35], [36] as follows III. PROBLEM FORMULATION Energy-efficient NOMA-enabled wireless powered passive sensors to infrastructure communication under imperfect CSI is the primary objective of the optimization problem. The energy efficiency maximization problem is formulated as where represents the energy efficiency of the wireless powered sensor communications in NOMA-enabled ITS. = {Γ 1 , Γ 2 } is the vector of reflection coefficient of roadside sensors. C1 and C2 enforce roadside sensors to backscatter their information towards RSU with minimum QoS (minimum required rate) requirement. C3 ensures that the CE transmit power ( ) cannot exceed its maximum transmit power ( ). C4 limits the reflection coefficient of sensors between 0 and 1. C5 guarantees that the energy harvested by the ℎ sensor should exceed their minimum required circuit power ( ). The objective function in the problem defined in Eq. (16) has a non-linear fractional form, which is challenging to solve. The logarithmic approximation is implemented [37], which reduces complexity and transforms the optimization problem into tractable concave-convex fractional programming (CCFP) problem. The approximation is done as follow for any ≥ 0 , where Π = 0 1+ 0 and Φ = log 2 (1 + 0 ) − 0 1+ 0 log 2 ( 0 ). When = 0 , the bound becomes tight. Through lower bound of inequality in Eq. (17), the sum-rate of roadside sensors is presented as where and Hence the updated optimization problem can be formulated as The EE maximization problem is coupled on two kinds of optimization variables i.e., , and in problem defined in Eq. (21). Thus, it is very hard to find a globally optimal solution directly. It demands an approximate optimal algorithm through alternating optimization algorithm. Therefore, this problem can be solved through alternating optimization algorithm in two stages: ) In first stage, on the fixed value of reflection coefficient of roadside sensors, we compute the energy efficient transmit power of the carrier emitter ; ) 6 we substitute the optimal * obtained in stage 1 into the original problem and optimize the reflection coefficients of the roadside sensors in Stage 2.

A. Energy Efficent transmit power allocation for CE
Given the reflection coefficients of roadside sensors, the optimization problem in Eq. (21) can be simplified to CE transmit power optimization as follow By using the following lemma, we demonstrate that is a concave function with respect to .
Lemma 1 : is a concave function with respect to .
Proof : Please, refer to Appendix A.
As, is concave with respect to and the objective function denominator is an affine function of in Eq. (22). Thus, the problem in Eq. (22) is in the form of a concaveconvex fractional programming (CCFP) problem. Which can be solved efficiently through Dinkelbach's algorithm [38]. By using Dinkelbach method, problem in Eq. (22) can be transformed as where is the real parameter. Computing the roots of ( ) is analogous to solving the objective function in Eq. (22) [32]. The closed-form solution of optimal can be expressed as * = 3 √︂ + Proof : Please, refer to Appendix B Given the optimal CE transmit power allocation policy in Eq. (26), the primal problem's Lagrangian multipliers can be determined and updated iteratively by employing the subgradient method as presented in Eq. (27) -Eq. (31). Where is used for iteration index. 1 , 2 , 3 , 4 , and 5 present positive step sizes. The appropriate step sizes should be used for the convergence to an optimal solution.

B. Efficient selection of reflection coefficient for roadside sensor
Through simplified Cardano's formulae, a closed-form of optimal power for CE is obtained. Now the optimization problem for allocating efficient reflection coefficients to roadside sensors under their QoS and required circuit power constraint can be rewritten as, To solve optimization problem defined in Eq. (32), first we will present in Lemma 2 that is a concave function with respect to Γ 1 and Γ 2 .
Lemma3 : is a concave function with respect to Γ 1 and Γ 2 .
Proof : Please, refer to Appendix C As, is concave with respect to Γ 1 and Γ 2 . Therefore maximum value of optimization problem defined in Eq. (32) can be obtained by the upper bound of reflection coefficient of roadside sensor as follow, 7 L(P ce , , , ) = ,1 Π 1 log 2

Lemma 4 :
The optimal reflection coefficient of the optimization problem defined in Eq. (32) is presented as where, ℵ = Proof : For the given optimal * , the objective function of optimization problem in Eq. (32) increases with increase of . Therefore, the optimal reflection coefficient of roadside sensor can be computed by the upper bound of reflection coefficient. The range of Γ from the optimization problem in Eq. (32) can be determined by combining constraints C1, C3, and C5 for Γ 1 and constraints C2, C3, and C5 for Γ 2 . After some simple mathematical computations, the range of Γ can be presented as Thus, optimal Γ * can be calculated as 1 − + ℎ, , , 1 . Through Lemma 4, optimal reflection coefficients of roadside sensors can be determined. It is allocated to the roadside sensors in the following manner, • When the condition satisfied. Then, Lemma 4 will results in Γ * = 1 − + ℎ, , . This condition implies that the quality of service constraint and and circuit power constraint of sensors is guaranteed simulataneously. In such condition, Γ * = 1 − + ℎ, , presents the portion of backscattering signal in received RF signal at roadside sensor during transmission mode while remaining portion of received RF signal is used for harvesting.
• When the condition is satisfied. Then, it implies that the energy harvested during energy harvesting mode ℎ is enough for the circuit operation of roadside sensor and in transmission mode all the received RF signal at roadside sensor can be used for backscattering for EE maximization. In this condition Lemma 4 results in Γ * = 1. In this condition, if both sensors which are backscattering using NOMA results in reflection coefficient of 1, then the reflection coefficient of near sensor should be calculated as 1-. The value of can be selected in such a way to satisfy * Γ 2 |ˆ2| 2 − * Γ 1 |ˆ1| 2 ≤ (SIC constraint) [34].
• When the condition is satisfied. Then, it implies that QoS constraint and circuit power constraint of roadside sensor can not be guarnteed simulataneously. In such case, optimization problem defined in Eq. (32) is infeasible. The details of proposed algorithm AOBWS is summarized in Algorithm 1. 8 Once RSU received the information from roadside sensors, then that information can be provided to the vehicles through energy efficient schemes proposed in [28], [39]. Update the transmit power ( + 1) of CE by using equation (26) in Lemma: 2.

C. Complexity Analysis:
This subsection presents the complexity analysis of the proposed AOBWS algorithm and ES algorithm as benchmark algorithm. In literature, ES algorithm is employed for NOMA optimal power allocation [40]. Moreover, it is also used as a benchmark algorithm [41], [42], [43], because it shows the best performance as compare to all other algorithms but at high computational complexity. In this article, ES algorithm is used as a benchmark because it obtains global optimal EE performance for the considered system model. ) . It can be noticed that the ES algorithm is computationally expensive but is used as a benchmark because of its global optimal solution. For practical implementation AOBWS algorithm is proposed, which requires the complexity order of O ( 2 + ).  AOBWS algorithm employs a two-stage procedure, in which the first stage is consists of an iterative algorithm (OCETP) and 2nd stage is the non-iterative algorithm. OCETP algorithm during each iteration requires operations to calculate EE. Where is the total number of roadside sensors linked using NOMA with BR installed on RSU. Furthermore, operations are required to update dual variables. If I is the number of iterations that the OCETP algorithm needs to converge, then the total complexity of the OCETP algorithm is O ( 2 ). While in the 2nd Stage, AOBWS employs a non-iterative algorithm for optimal reflection coefficients, which requires K operations. After two-stage procedures, the complexity order of AOBWS becomes O ( 2 + ). So, the proposed AOBWS algorithm is always less computationally complex as compared to the ES algorithm. When the number of roadside sensors connected through NOMA with each RSU increases, the complexity of the ES algorithm increases exponentially while the proposed AOBWS algorithm still provides an optimal solution in polynomial time. The complexity analysis of analyzed algorithms is presented in Table III.

IV. SIMULATIONS
This section presents the simulation results to evaluate the efficacy of the proposed algorithm in terms of energy efficiency and complexity. For this purpose, the proposed AOBWS algorithm is compared with the global optimal ES algorithm (benchmark algorithm). ES algorithm achieves global optimal solution but at the cost of high computational complexity and is impractical. It is observed the proposed AOBWS algorithm achieves very close performance to the global optimal ES with very low complexity and is therefore suitable for practical implementation. During simulations, roadside sensors are deployed randomly around the CE on the road and their locations are modeled as binomial point process (BPP) unless otherwise specified. In our simulations, each subchannel is serving two roadside sensors through NOMA, which are selected randomly from the generated sesnors through BPP as shown in Fig. 2. The distance between far roadside sensor to RSU and near roadside sensor to RSU is set as 50 meters and 30 meters respectively unless otherwise specified. Time coefficient for roadside sensor transmission mode and energy harvesting mode is set to ,1 = ,2 = 0.5 and ℎ,1 = ℎ,2 = 0.5. Other main simulation parameters for our considered system model are described in Table IV. Fig. 3 presents the EE of roadside sensors versus the transmit power of CE in dBm. This figure provides the comparison of two proposed algorithms, where AOBWS refers to alternating optimization for backscatter aided wireless powered sensors, which provide optimal EE by optimizing both CE transmit power and reflection coefficient of roadside sensor through the proposed alternating optimization framework. AOBWS optimizes CE transmit power under given reflection coefficient of roadside sensor in the first stage while in the 2nd stage it optimizes reflection coefficient of roadside sensor under optimal CE transmit power obtained in the 1st stage. While OCETP refers to the first stage of AOBWS which obtains optimal CE transmit power under given reflection coefficient of roadside sensor. From the figure, it can be noticed that our proposed AOBWS has higher EE after performing its 2 stage operation as compare to OCETP. As our proposed alternating optimization framework (AOBWS) consists of two stages, where the first stage employs an iterative algorithm for optimal CE transmit power named OCETP while in 2nd stage it uses a non-iterative algorithm for optimal reflection coefficient of RS. Therefore, it is vital to analyze the convergence of the OCETP algorithm. In Fig. 4, EE convergence of OCETP algorithm versus iterations is demonstrated with different relative channel errors ( ). The obtained result presents that the OCETP algorithm usually converges in three iterations regardless of . It is observed  that affects EE, but their influence on the convergence of the OCETP algorithm is almost negligible. Fig. 5 depicts optimal EE obtained by the proposed AOBWS algorithm under different relative channel errors. It can be observed from the figure that EE as a function of first increases and then decreases with increasing . It is because when is increased beyond the optimal * , the RS sumrate increases very slowly relative to the power consumption. It can also be examined from the figure that higher relative channel errors result in higher EE degradation. Fig. 6 compares the EE of the proposed AOBWS algorithm with the global optimal ES algorithm and OFDMA. The comparison is done versus relative channel errors while considering different distances between RS and BSR installed on RSU. It can be analyzed from the figure that AOBWS obtains very close EE performance to the ES algorithm with very low complexity. Moreover, it can be observed that higher relative channel error has a higher degrading effect on the EE. From the figure, it can also be noticed that a greater distance between RSs and BSR installed on RSU results in higher EE degradation.
In Fig. 7, the EE of the proposed AOBWS algorithm is compared with the global optimal ES algorithm versus relative channel errors. This figure presents the effect of distance between dedicated CE and RS. From the results, It can be examined that a higher distance between CE and RS will result in lower EE. Moreover, the proposed AOBWS algorithm shows very close performance with the ES algorithm, which presents the efficacy of our proposed algorithm with low computational complexity. Fig. 8 shows the EE of RS versus QoS requirement of RS in term minimum data rate ( ) for proposed AOBWS algorithm. The analysis of this figure is done with ,1 = 10, ,2 = 5, ,1 = 50, and ,2 = 30. This figure depicts that higher relative channel error will result in lower guaranteed data rate of the network.   In this article, we proposed an alternating optimization framework for backscatter aided wireless powered roadside sensors to infrastructure communications with imperfect CSI. By considering various QoS requirements of bistatic backscatter communications for roadside sensors, we aim to maximize the EE of the network. The EE is maximized under channel uncertainties by optimizing transmit power of CE and reflection coefficient of RS, with very low computational complexity for practical implementations. The proposed problem for the solution is decoupled into two stages, which yields the proposed AOBWS algorithm. The complexity analysis of the proposed algorithm is analyzed in detail and is compared with the global optimal ES algorithm. It is observed that our proposed algorithm can obtain near-optimal EE performance with very low complexity. Moreover, numerical results present the efficacy of our proposed result in terms of computational 11 complexity and EE performance. In the future, we want to extend our work to a multi-cluster of RSs , full-duplex (FD) RSs, and multi-antenna RSs.

APPENDIX A CONCAVITY OF WITH RESPECT TO
Appendix A presents the concavity of with respect to . The first derivative of w.r.t. is given as .