Optimal Resource Allocation in Full-Duplex Ambient Backscatter Communication Networks for Wireless-Powered IoT

Optimal Resource Allocation in Full-Duplex Ambient Backscatter Communication Networks for Wireless-Powered IoT

Gang Yang, Member, IEEE, Dongdong Yuan, Ying-Chang Liang, Fellow, IEEE,
Rui Zhang, Fellow, IEEE, and Victor C. M. Leung, Fellow, IEEE
G. Yang and D. Yuan are with the National Key Laboratory of Science and Technology on Communications, and Center for Intelligent Networking and Communications (CINC), University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China (e-mails: yanggang@uestc.edu.cn, 201721260420@std.uestc.edu.cn).Y.-C. Liang is with Center for Intelligent Networking and Communications (CINC), University of Electronic Science and Technology of China (UESTC), Chengdu 611731, China (e-mail: liangyc@ieee.org). (Corresponding author: Y.-C. Liang.)R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore, 117583, Singapore (e-mail: elezhang@nus.edu.sg).V. C. M. Leung is with the Department of Electrical and Computer Engineering, the University of British Columbia, Vancouver, BC, V6T 1Z4, Canada (e-mail: vleung@ece.ubc.ca).
Abstract

This paper considers an ambient backscatter communication (AmBC) network in which a full-duplex access point (FAP) simultaneously transmits downlink orthogonal frequency division multiplexing (OFDM) signals to its legacy user (LU) and receives uplink signals backscattered from multiple BDs in a time-division-multiple-access manner. To maximize the system throughput and ensure fairness, we aim to maximize the minimum throughput among all BDs by jointly optimizing the backscatter time and reflection coefficients of the BDs, and the FAP’s subcarrier power allocation, subject to the LU’s throughput constraint, the BDs’ harvested-energy constraints, and other practical constraints. For the case with a single BD, we obtain closed-form solutions and propose an efficient algorithm by using the Lagrange duality method. For the general case with multiple BDs, we propose an iterative algorithm by leveraging the block coordinated decent and successive convex optimization techniques. We further show the convergence performances of the proposed algorithms and analyze their complexities. In addition, we study the throughput region which characterizes the Pareto-optimal throughput trade-offs among all BDs. Finally, extensive simulation results show that the proposed joint design achieves significant throughput gain as compared to the benchmark schemes.

I Introduction

Internet of Things (IoT) is a key application paradigm for the forthcoming fifth-generation (5G) and future wireless communication systems. IoT devices in practice have strict limitations on energy, cost, and complexity, thus it is highly desirable to design energy- and spectrum-efficient communication technologies [1, 2]. Recently, ambient backscatter communication (AmBC) has emerged as a promising candidate to fulfill such demand. On one hand, AmBC enables wireless-powered backscatter devices (BDs) to modulate their information symbols over ambient radio-frequency (RF) carriers (e.g., WiFi, TV, or cellular signals) without using any costly and power-hungry RF transmitter [3]. On the other hand, no dedicated spectrum is needed for AmBC due to the spectrum sharing between the backscatter transmission and the ambient transmission [4].

The existing AmBC systems can be divided into three categories, namely the traditional AmBC (TABC) system with separated backscatter receiver and ambient transmitter (and its legacy111Hereinafter, the term “legacy” refers to any existing wireless communication systems such as WiFi. receiver) [5, 6, 7, 8, 9, 10, 11, 4, 12, 13, 14], the cooperative AmBC (CABC) system with co-located backscatter receiver and legacy receiver [15, 16, 17], and the full-duplex AmBC (FABC) system with co-located backscatter receiver and ambient transmitter [18, 11].

The TABC systems are most studied in the literature [5, 6, 7, 8, 9, 10, 11, 4, 12, 13, 14]. One of the key challenges for TABC systems is the strong direct-link interference from the ambient transmitter received at the backscatter receiver. Frequency-shifting method is proposed in [8, 9] to avoid the direct-link interference, while in [10], the direct-link interference is cancelled out through using the specific feature of the ambient signals. There are also studies on TABC system performance and resource allocations [11, 4, 12, 13]. For example, in [4], a TABC system is modelled from a spectrum sharing perspective, and the ergodic capacity of the secondary backscatter system is maximized. In [11], the capacity bounds for backscatter communication are derived for a TABC system, under the assumption that the backscatter receiver knows legacy symbols.

In CABC systems, the signals from the ambient transmitter are recovered at the backscatter receiver instead of being treated as interference [15, 16, 17]. In particular, the optimal maximum-likelihood detector, suboptimal linear detectors, and the successive interference-cancellation based detectors are derived in [15]. In [16], the sum rate of the backscatter communication and the legacy communication is analyzed under both perfect and imperfect channel state information for a CABC system with multiple antennas at each node. In [17], the transmit beamforming is optimized to maximize the sum rate of a CABC system in which the ambient transmitter is equipped with multiple antennas.

In FABC systems, the backscatter receiver and ambient transmitter are collocated, thus the signals from the ambient transmitter can be cancelled out  [18, 11]. The authors in [11] analyze the capacity performances of both the backscatter communication and the legacy communication for an FABC system over OFDM carriers, and obtain the asymptotic capacity bounds in closed form when the number of subcarriers is sufficiently large. The authors in [18] build an FABC system prototype in which the WiFi access point (AP) decodes the received backscattered signal while simultaneously transmitting WiFi packages to its legacy client. However, only a single BD is considered in [11] and [18], which simplifies the analysis and implementation but limits the applicability in practice.

The aforementioned prior works mainly focus on the transceiver design and hardware prototyping for various single-BD AmBC systems. To our best knowledge, the existing literature still lacks fundamental analysis and performance optimization for a general FABC system with multiple BDs.

In this paper, we consider a full-duplex AmBC network (F-ABCN) over ambient OFDM carriers as shown in Fig. 1, consisting of a full-duplex access point (FAP) with two antennas for simultaneous signal transmission and reception, respectively, a legacy user (LU), and multiple BDs. The FAP transmits dowlink signal which not only carries information to the LU but also transfers energy to the BDs; while at the same time all BDs perform uplink information transmission via backscattering in a time-division-multiple-access (TDMA) manner. The backscattered signal in general interferes with the LU’s received information signal directly from the FAP. Thus, this proposed F-ABCN differs from the conventional full-duplex wireless-powered communication network (WPCN) in which the AP transmits solely downlink energy signal to all users in the first phase and each user uses its harvested energy to transmit uplink information signal via an additional RF transmitter in the second phase [19]. One typical application example of our considered F-ABCN is described as follows: a WiFi AP simultaneously transmits downlink information via OFDM modulation to its client(s) (e.g., smartphone, laptop) and receives uplink information from multiple domestic IoT devices (e.g., tags, sensors) in smart-home applications. We aim to optimize the throughput performance for a generic F-ABCN in this paper, where its main contributions are summarized as follows:

  • First, to ensure fairness, we formulate a problem to maximize the minimum throughput among all BDs by jointly optimizing the BDs’ backscatter time allocation, the BDs’ power reflection coefficients, and the FAP’s subcarrier power allocation, subject to the LU’s throughput requirement and the BDs’ harvested-energy constraints, together with other practical constraints. Such a joint optimization problem is practically appealing, since the system performance can benefit from adjusting design parameters in multiple dimensions. However, the formulated problem is non-trivial to solve in general, since the variables are mutually coupled and result in non-convex constraints.

  • Second, for the special case with a single BD, we obtain analytical solutions for the optimal resource allocation, and propose an efficient algorithm for obtaining it based on the Lagrange duality method. The optimal subcarrier power allocation is obtained in semi-closed form that provides useful insights to the optimal design. The convergence and complexity of the algorithm are also analyzed.

  • Third, for the general case with multiple BDs, we propose an iterative algorithm by leveraging the block coordinated decent (BCD) and successive convex optimization (SCO) techniques. The entire optimization variables are partitioned into three blocks for the BDs’ backscatter time allocation, the BDs’ power reflection coefficients, and the FAP’s subcarrier power allocation, respectively. The three blocks of variables are alternately optimized. However, for the non-convex subcarrier power allocation optimization problem with given backscatter time allocation and power reflection coefficients, we apply the SCO technique to solve it approximately. Also, we show the convergence of the proposed algorithm and analyze its complexity.

  • Fourth, we extend our study by characterizing the throughput region constituting all the Pareto-optimal throughput performance trade-offs among all BDs. Each boundary point of the throughput region is found by solving a sum-throughput maximization problem with a given throughput-profile vector.

  • Last, numerical results show that significant throughput gain is achieved by our proposed joint design, as compared to the benchmark scheme of F-ABCN with equal resource allocation and that of half-duplex AmBC network (H-ABCN) with optimal resource allocation. The BDs-LU throughput trade-off and the BDs’ throughput-energy trade-off are revealed as well. Also, the effect of system parameters like the peak power value on the throughput performance is numerically demonstrated.

The rest of this paper is organized as follows. Section II presents the system model for an F-ABCN over ambient OFDM carriers. Section III formulates the minimum-throughput maximization problem. Section IV analyzes the joint resource allocation for a single-BD F-ABCN and proposes an optimal algorithm by applying the Lagrange duality method. Section V proposes an efficient iterative algorithm by applying the BCD and SCO techniques to solve the joint resource allocation problem for a multiple-BD F-ABCN. Section VI studies the throughput region that characterizes the optimal throughput performance trade-offs among all BDs. Section VII presents the numerical results to verify the performance of the proposed joint design. Section VIII concludes this paper.

The main notations in this paper are listed as follows: The lowercase, boldface lowercase, and boldface uppercase letters, e.g., , , and , denote a scalar, vector, and matrix, respectively. means the operation of taking the absolute value of a scalar . denotes the statistical expectation of a random variable . denotes the transpose of a vector . The notation means the convolution operation. denotes the partial derivative operation. denotes the circularly symmetric complex Gaussian (CSCG) distribution with zero mean and variance . denotes the set of complex numbers. denotes the time complexity order of an algorithm.

Ii System Model

In this section, we present the system model for an F-ABCN over ambient OFDM carriers. As illustrated in Fig. 1, we consider two coexisting communication systems: the legacy communication system which consists of an FAP with two antennas for simultaneous information transmission and reception, respectively, together with its dedicated LU222We consider the case of a single LU, since the FAP typically transmits to an LU in a short period for practical OFDM systems like WiFi. The analyses and results can be extended to the case of multiple LUs., and the AmBC system which consists of the FAP and () BDs. The FAP transmits OFDM signals to the LU. We are interested in the AmBC system in which each BD transmits its modulated signal back to the FAP over its received ambient OFDM carrier from the FAP. Each BD contains a backscatter antenna, a switched load impedance, a micro-controller, an information receiver, an energy harvester, and other modules (e.g., battery, memory, sensing). To transmit information bits, the BD modulates its received ambient OFDM carrier by intentionally switching the load impedance to vary the amplitude and/or phase of its backscattered signal, and the backscattered signal is received and finally decoded by the FAP.

Fig. 1: System description for an F-ABCN.

The block fading channel model is considered, and the channel block length is assumed to be much longer than the OFDM symbol period. As shown in Fig. 1, let be the -path forward channel response from the FAP to the -th BD, for , be the -path backward channel response from the -th BD to the FAP, be the -path legacy channel response from the FAP to the LU, and be the -path interference channel response from the -th BD to the LU. Let be the number of subcarriers of the transmitted OFDM signals. For the downlink channel from the FAP to the -th BD, we define the frequency response at the -th subcarrier as , for . Similarly, for the backward channel from the -th BD to the FAP, we define its subcarrier response as ; for the interference channel from the -th BD to the LU, we define its subcarrier response as ; and for the legacy channel from the FAP to the LU, we define its subcarrier response as .

We consider a frame-based protocol as shown in Fig. 2. The frame duration of (seconds) is within the channel block length. In each frame consisting of slots, the FAP simultaneously transmits downlink OFDM signals to the LU, and receives uplink signals backscattered from all BDs in a TDMA manner. The -th slot of time duration (with time proportion ()) is assigned to the -th BD. Denote the backscatter time allocation vector . In the -th slot, BD reflects back a portion of its incident signal for transmitting information to the FAP and harvests energy from the remaining incident signal, and all other BDs only harvest energy from their received OFDM signals.

Fig. 2: Frame-based protocol for an F-ABCN.

Let be the FAP’s information symbol at the -th subcarrier, , in the -th OFDM symbol period of the -th slot. After inverse discrete Fourier transform (IDFT) at the FAP, a CP of length is added at the beginning of each OFDM symbol. The transmitted time-domain signal in each OFDM symbol period is

(1)

for the time index , where is the allocated power at the -th subcarrier in the -th slot. Denote the subcarrier power allocation matrix , where is the subcarrier power allocation vector in the -th slot.

In the -th slot, the incident signal at BD is . From [20], due to the impedance discontinuity of the antenna and the load, a proportion (, referred to as the power reflection coefficient) of the incident power is reflected backward, giving rise to the backscattered field, while the remaining power propagates to the energy-harvesting circuit. For convenience, denote the power reflection coefficient vector . Let (), , be the energy-harvesting efficiency constant [21, 22, 23] of BD . According to the aforementioned energy-harvesting scheme in the proposed protocol and from [21], the total energy harvested by BD in all slots is thus

(2)

where the first term in the square brackets relates to the harvested energy in the -th slot, and the second term relates to the harvested energy in all other slots.

From the antenna scatterer theorem [24], the electronic-magnetic (EM) field backscattered from the -th BD consists of the structural mode (load-independent) component and the antenna mode (load-dependent) component. The former is interpreted as the scattering from the antenna loaded with a reference impedance333The reference impedance can be arbitrary, which is typically taken as , and the antenna impedance for the short-circuit case, the open-circuit case, and the matched circuit case, respectively [24]., which depends on only the antenna’s geometry and material. The latter relates to the rest scattering of the antenna, which depends on the specific impedance of the load connected to the antenna. Let be the -th BD’s information symbol, whose duration is designed to be the same as the OFDM symbol period. We assume that each BD can align the transmission of its own symbol with its received OFDM symbol444BD can practically estimate the arrival time of OFDM signal by some methods like the scheme that utilizes the repeating structure of CP [10].. The signal backscattered by the -th BD, denoted by , can be written as [20]

(3)

where is the structural mode component, and the antenna mode component, denoted as , is defined as [11]. Since the structural mode component is fixed for each BD, it can be reconstructed and subtracted from the received signal at the FAP. Hence, for simplicity, we ignore the structural mode component and denote the backscattered signal as in the sequel.

Since the transmitted downlink signal is known by the FAP’s receiving chain, it can also be reconstructed and subtracted from the received signal. Therefore, the self-interference can be cancelled by using existing digital or analog cancellation techniques [18]. For this reason, we assume perfect self-interference cancellation (SIC) at the FAP in this paper. After performing SIC, the received time-domain signal backscattered from the -th BD is given by

(4)

where denotes the additive white Gaussian noise (AWGN) with power , i.e., .

After CP removal and discrete Fourier transform (DFT) at the FAP, the received frequency-domain signal is

(5)

where the frequency-domain noise .

The FAP performs maximum-ratio-combining (MRC) to recover the BD symbol as follows,

(6)

and the resulted decoding signal-to-noise-ratio (SNR) is

(7)

Hence, the -th BD’s throughput555This paper adopts normalized throughput with unit of bits-per-second-per-Hertz (bps/Hz). normalized to the frame duration is

(8)

Since the backscattered signal is transmitted at the same frequency as the downlink signal in the legacy system, the whole system in Fig. 1 is indeed a spectrum sharing system [4, 25, 26, 27]. The LU receives the superposition of the downlink legacy signal and the backscatter-link signal. Similar to (5), the received frequency-domain signal at the LU can be thus written as follows,

(9)

where the frequency-domain noise .

By treating backscatter-link signal as interference, the total throughput of the LU is given by

(10)

Iii Problem Formulation

Our objective is to maximize the minimum throughput among all BDs, by jointly optimizing the BDs’ backscatter time allocation (i.e., ), the BD’s power reflection coefficients (i.e., ), and the FAP’s subcarrier power allocation (i.e., ). Mathematically, the optimization problem is equivalently formulated as follows,

(11a)
(11b)
(11c)
(11d)
(11e)
(11f)
(11g)
(11h)
(11i)

Note that (11b) is the common-throughput constraint for each BD, (11c) is the LU’s requirement of a given minimum throughput ; (11d) is each BD’s requirement of a given minimum energy ; (11e) is the FAP’s maximum (total) transmission-power (i.e., a given value ) constraint; (11f) is the total backscatter-time constraint, and (11g) is the non-negative constraint for each backscatter time; (11h) is the non-negative and peak-power (i.e., a given value ) constraint for each subcarrier power; and (11i) is the constraint for each power reflection coefficient.

The above joint optimization problem is practically appealing. On one hand, by properly designing the power reflection coefficients of near BDs, more backscatter time can be allocated to far BDs to further enhance their throughput performance, alleviating the effect of double near-far problem for wireless-powered (backscatter) communication networks [19, 23]. On the other hand, by properly allocating subcarrier power at the FAP, better throughput trade-off can be achieved among the BDs and the LU. However, problem (11) is challenging to solve, due to the following two reasons. First, the backscatter time allocation variables ’s, the power reflection coefficient variables ’s and the subcarrier power variables ’s are all coupled in the constraints (11b), (11c), (11d), and (11e). Second, the logarithm function in the constraint (11c) is a non-convex function of the subcarrier power variables ’s. Therefore, problem (11) is non-convex, which is difficult to solve optimally in general.

Iv Joint Resource Allocation in a Single-BD F-ABCN

To obtain tractable analytical results, in this section, we consider the special case of , i.e., a single-BD F-ABCN. For brevity, the subscript for BD is omitted in the notations, as . The transmission power allocation matrix , the power reflection coefficient vector and the backscatter time allocation vector reduce to the vector , the scaler and the constant for the BD, respectively. Problem (11) is then simplified as follows,

(12a)
s.t. (12b)
(12c)
(12d)
(12e)
(12f)

Since the objective function in (12a) and the constraint functions in (12b) and (12c) are all monotonically increasing with respect to each individual , thus the constraint in (12d) should hold with equality at the optimal power allocation (otherwise, the objective function together with the left-hand-sides (LHSs) of the constraints in (12b) and (12c) can be further increased by increasing some ’s).

To obtain useful insights, we further assume that the interference from the BD to the LU is negligible, i.e., . This assumption is practical, since the interference signal goes through the FAP-to-BD channel fading, the power reflection loss at the BD, and the BD-to-LU channel fading, usually leading to much smaller interference power at the LU compared to the signal directly from the FAP. The general case of non-negligible interference will be studied in Section V. The optimal of problem (12) can be obtained by one-dimensional search, and we focus on optimizing the subcarrier power in the rest of this section. Since the logarithm function in (12a) is monotonically increasing with its argument, problem (12) for given can be rewritten as

(13a)
s.t. (13b)
(13c)
(13d)
(13e)

It can be easily checked that problem (13) is a convex optimization problem with respect to , thus can be solved by the Lagrange duality method, as shown as follows.

From (13a), (13b), (13c) and (13d), the Lagrangian of problem (13) is given by

(14)

where and denotes the dual variables associated with (13b), (13c), and (13d), respectively. The dual function of problem (13) is then given by

(15)

The dual problem of problem (13) is thus give by .

Theorem 1.

Given and , the maximizer of in (14) is given by

(16)

where .

Proof.

Please see Appendix A. ∎

We conclude that , since Theorem 1 implies , and the objective value is zero, if , which is in contradiction with the optimality of ’s.

From Theorem 1, the optimal solution of problem (13) can be obtained as follows. With obtained for each given pair of and , the optimal dual variables and that minimize can then be efficiently obtained by a sub-gradient based algorithm, with the sub-gradient of given by

(17a)
(17b)
(17c)

The overall steps for solving problem (13) are summarized in Algorithm 1. Since problem (13) is convex, the proposed Algorithm 1 is guaranteed to converge [28]. The computation time of Algorithm 1 is analyzed as follows. The time complexity of step 3 is , and those of step 4 and step 6 are . Since only three dual variables, , are updated by the sub-gradient method regardless of the number of BDs, . The time complexity of step 5 is thus . As a result, the total time complexity of Algorithm 1 is .

1:  Initialize dual variables , positive step-sizes , and small threshold constant . Let .
2:  repeat
3:     Given and , compute by using (16), and obtain the corresponding dual function value as in (15).
4:     Compute the sub-gradients and given in (17) by replacing by .
5:     Update dual variables
6:     Update iteration index .
7:  until 
8:  Obtain the optimal subcarrier power allocation .
Algorithm 1 Iterative algorithm for solving problem (13)

A numerical example is given here to demonstrate the optimal subcarrier power allocation. Fig. 3 depicts the optimal that maximizes the BD throughput in a single-BD F-ABCN with , and . We assume independent multi-path Rayleigh fading channels, and the power gains of multiple paths are exponentially distributed. The numbers of channel paths are set as , and . The FAP-to-BD distance, the BD-to-LU distance, and the FAP-to-LU distance are 4, 15, and 15 meters (m), respectively. Other parameters are set as the same as in Section VII. We consider two different peak-power values, and with . For the case of , we observe that 98.57% of the total power is allocated to subcarriers 3 to 6, among which subcarriers 5 and 6 are allocated with peak power of 0.3125, subcarriers 3 and 4 are allocated with power of 0.1751 and 0.1856, respectively, and any other subcarrier’s power is negligible. In contrast, for the case of , we observe that the power allocation is more concentrated, and 95% of the total power is allocated to subcarriers 5 and 6. Specifically, only subcarrier 6 is allocated with peak power of 0.6250, and the power at subcarrier 5 is 0.3245, while any other subcarrier’s power is much smaller and can be ignored. The allocation criterion can be explained as follows. Under the peak-power constraints, power is allocated with priority to the subcarriers with stronger backscatter-link channel , conditioned on that the LU’s throughput constraint and the BD’s harvested-energy constraint are satisfied.

Fig. 3: Optimal subcarrier power allocation for different peak-power constraints.

V Joint Resource Allocation in a Multiple-BD F-ABCN

In this section, we consider the joint resource allocation in an F-ABCN with multiple BDs. In general, there is no standard method for optimally solving the non-convex optimization problem (11) efficiently. Hence, we propose an efficient iterative algorithm to solve it sub-optimally by applying the block coordinate descent (BCD) [29] and successive convex optimization (SCO) [30] techniques. In each iteration, we optimize different blocks of variables alteratively. Specifically, for any given power reflection coefficient vector and subcarrier power allocation matrix , we optimize the backscatter time allocation vector by solving a linear programming (LP); for any given backscatter time allocation vector and subcarrier power allocation matrix , we optimize the power reflection coefficient vector by solving a convex problem; and for any given backscatter time allocation vector and power reflection coefficient vector , we optimize the subcarrier power allocation matrix by utilizing the SCO technique and solving an approximated convex problem. After presenting the overall algorithm, we show the convergence of the proposed algorithm and analyze its complexity.

V-a Backscatter Time Allocation Optimization

In iteration , for given power reflection coefficient vector and subcarrier power allocation matrix , the backscatter time allocation vector can be optimized by solving the following problem

(18a)
(18b)
(18c)
(18d)
(18e)
(18f)
(18g)

Since problem (18) is a standard LP, it can be solved efficiently by existing optimization tools such as CVX [31]. Moreover, it can be verified that either the constraint (18e) or (18f) is met with equality when the optimal is obtained for given and , since otherwise we can always increase ’s without decreasing the objective value.

V-B Reflection Power Allocation Optimization

For given backscatter time allocation vector and subcarrier power allocation matrix , the power reflection coefficient vector can be optimized by solving the following problem

(19a)
(19b)
(19c)
(19d)
(19e)

Given ’s and ’s, (19b) is a convex constraint, while (19d) and (19e) are linear constraints. Moreover, since the LHS of the constraint (19c) is a decreasing and convex function of , this constraint is convex. Hence, problem (19) is a convex optimization problem that can also be efficiently solved by CVX [31].

V-C Subcarrier Power Allocation Optimization

For given backscatter time allocation vector and power reflection coefficient vector , the subcarrier power allocation matrix can be optimized by solving the following problem

(20a)
(20b)
(20c)
(20d)
(20e)
(20f)

Since the constraint function in (20c) is non-convex with respect to , problem (20) is non-convex. Notice that the constraint function can be rewritten as

(21)

To handle the non-convex constraint (20c), we exploit the SCO technique [30] to approximate the second logarithm function in (V-C). Recall that any concave function can be globally upper-bounded by its first-order Taylor expansion at any point. Specifically, let denote the subcarrier power allocation matrix in the previous iteration. We have the following concave lower bound at the local point

(22)

With given local points and lower bound in (22), by introducing the lower-bound minimum-throughput , problem (20) is approximated as the following problem

(23a)
(23b)
(23c)
(23d)
(23e)
(23f)

Problem (23) is a convex optimization problem which can also be efficiently solved by CVX [31]. It is noticed that the lower bound adopted in (23c) implies that the feasible set of problem (23) is always a subset of that of problem (20). As a result, the optimal objective value obtained from problem (23) is in general a lower bound of that of problem (20).

V-D Overall Algorithm

We propose an overall iterative algorithm for problem (11) by applying the BCD technique [29]. Specifically, the entire variables in original problem (11) are partitioned into three blocks, i.e., the backscatter time allocation vector , power reflection coefficient vector , and subcarrier power allocation matrix , which are alternately optimized by solving problem (18), (19), and (23) correspondingly in each iteration, while keeping the other two blocks of variables fixed. Furthermore, the obtained solution in each iteration is used as the input of the next iteration. The details are summarized in Algorithm 2.

1:  Initialize with , and small threshold constant . Let .
2:  repeat
3:     Solve problem (18) for given and , and obtain the optimal solution as .
4:     Solve problem (19) for given and , and obtain the optimal solution as .
5:     Solve problem (23) for given , , and , and obtain the optimal solution as .
6:     Update iteration index .
7:  until The increase of the objective value is smaller than
8:  Return the optimal solution , , and .
Algorithm 2 Block coordinate descent algorithm for solving problem (11)

V-E Convergence and Complexity Analysis

From [29], for the classic BCD method, the subproblem for updating each block of variables is required to be solved exactly with optimality in each iteration so as to guarantee its convergence. However, in our proposed Algorithm 2, for subcarrier power allocation subproblem (20), we only solve its approximate problem (23) optimally. Thus, the convergence analysis for the classic BCD technique is not applicable to our case, and we prove the convergence of Algorithm 2 as follows.

Theorem 2.

Algorithm 2 is guaranteed to converge.

Proof.

First, in step 3 of Algorithm 2, since the optimal solution is obtained for given and , we have the following inequality on the minimum throughput

(24)

Second, in step 4 of Algorithm 2, since the optimal solution is obtained for given and , it holds that

(25)

Third, in step 5 of Algorithm 2, it follows that