I Introduction
Internet of Things (IoT) is a key application scenario for the fifth generation (5G) and future mobile communication systems, and various IoT devices typically have strict limitations on energy, cost, and complexity. Recently, ambient backscatter communication (AmBC) which enables backscatter devices (BDs) to modulate their information symbols over the ambient RF carriers (e.g., WiFi, TV, or cellular signals) without using a complex and powerhungry RF transmitter [1], has emerged as a promising technology for energyefficient and costefficient IoT communications.
The existing AmBC systems can be divided into three categories, the standard AmBC system with separated backscatter receiver and ambient transmitter (its legacy receiver) [2, 3, 4, 5, 6], the cooperative AmBC system with colocated backscatter receiver and legacy receiver [7, 8], and the fullduplex AmBC system with colocated backscatter receiver and ambient transmitter [9]. Existing works on AmBC focus on the transceiver design and hardware prototype. To address the problem of strong directlink interference from ambient transmitter in standard AmBC systems, some studies shift the backscattered signal to a clean band that does not overlap with the directlink signal [5, 6]. In [3], the BD waveform and backscatter receiver detector are jointly designed to cancel out the directlink interference from the ambient orthogonal frequency division multiplexing (OFDM) signals. A fullduplex AmBC system is proposed in [9], in which the WiFi access point decodes the received backscattered signal while simultaneously transmitting WiFi packages to its legacy client.
Recently, there are a few literature on performance analysis and optimization for AmBC systems. In [10], the ergodic capacity for backscatter communication is maximized for a standard AmBC system. In [8], the transmit beamforming is optimized for a cooperative AmBC system with multiple antennas at the ambient transmitter. In [11], for an ambientbackscatter assisted cognitive radio network, the secondary transmitter’s rate is maximized by optimizing the time resource.
In this paper, we consider a fullduplex AmBC network (ABCN) over ambient OFDM carriers as shown in Fig. 1, consisting of a fullduplex access point (FAP), a legacy user (LU), and multiple BDs. We optimize the throughput performance for such an ABCN, which has not been studied in the literature to our best knowledge. The main contributions of this paper are summarized as follows:

First, by employing an FAP, we propose a new model to enable simultaneous downlink information transmission (energy transfer) to the LU (multiple BDs) and uplink information transmission from multiple BDs. We characterize the corresponding throughput and energy transfer performances of the BDs, as well as the throughput performance of the LU.

Second, to ensure fairness, we formulate a problem to maximize the minimum throughput among all BDs by jointly optimizing three blocks of variables including the BDs’ backscatter time portions, the BDs’ power reflection coefficients, and the FAP’s subcarrier power allocation. Through joint optimization, the system performance can benefit from multiple design dimensions.

Third, to solve the formulated nonconvex problem, we propose an iterative algorithm by leveraging the block coordinated decent (BCD) and successive convex optimization (SCO) techniques, in each iteration of which the three blocks of variables are alternately optimized. Also, we show the convergence of the proposed algorithm and analyze its complexity.

Finally, numerical results show that significant throughput gains are achieved by our proposed joint design, as compared to the benchmark scheme with equal resource allocation. Also, the BDLU throughput tradeoff and the BDs’ throughputenergy tradeoff are revealed.
The rest of this paper is organized as follows. Section II presents the system model for a fullduplex ABCN over ambient OFDM carriers. Section III formulates the minimumthroughput maximization problem. Section IV proposes an efficient iterative algorithm by applying the BCD and SCO techniques. Section V presents the numerical results. Finally, Section VI concludes this paper.
Ii System Model
In this section, we present the system model for a fullduplex ABCN. 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, and its dedicated LU^{1}^{1}1We 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 system. The analyses and results can be extended to the multipleLU case., 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. Each BD contains a backscatter antenna, a switched load impedance, a microcontroller, a signal processor, an energy harvester, and other modules (e.g., battery, memory, sensing). 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. The energy harvester collects energy from ambient OFDM signals and uses it to replenish the battery which provides power for all modules of the BD.
The block fading channel model is assumed. 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 directlink channel response from the FAP to the LU, and be the path interference channel response from the BD to the LU. Let be the number of subcarriers of the transmitted OFDM signals. For each channel, define the frequency response at the th subcarrier as , , , , for .
We consider framebased transmission, and the frame structure is shown in Fig. 2. In each frame of time duration (seconds) consisting of slots, the FAP simultaneously transmits downlink OFDM signals to the LU, and receives uplink signals backscattered from all BDs in a timedivisionmultipleaccess (TDMA) manner. The th slot of time duration (with time portion ()) is assigned to the
th BD. Denote the backscatter time portion 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.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 at the FAP, a cyclicprefix (CP) of length
is added at the beginning of each OFDM symbol. The transmitted timedomain 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. The subcarrier power values are subject to the average power constraint , where is the total transmission power in all slots. 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 , where means the convolution operation. Let () be the th BD’s power reflection coefficient, and denote the vector . Let () be the energyharvesting efficiency for all BDs. From the aforementioned system model and [12], the total harvested energy by BD in all slots is thus
(2) 
Let be the th BD’s information symbol, whose duration is designed to be the same as the OFDM symbol period. We assume each BD can align the transmission of its own symbol with its received OFDM symbol^{2}^{2}2
BD can practically estimate the arrival time of OFDM signal by some methods like the scheme that utilizes the repeating structure of CP
[3].. In the th slot, the backscattered signal from the th BD is thus .The transmitted downlink signal is known by the FAP’s receiving chain. Thus, this signal can be reconstructed and subtracted from the received signals. Therefore, the selfinterference can be cancelled by using existing digital or analog cancellation techniques. For this reason, we assume perfect selfinterference cancellation (SIC) at the FAP in this paper. After performing SIC, the received signal backscattered from the th BD is given by
(3)  
where the additive white Gaussian noise (AWGN) is assumed, i.e., .
After CP removal and discrete Fourier transform operation at the FAP, the received frequencydomain signal is
(4)  
where the frequencydomain noise .
The FAP performs maximumratiocombining (MRC) to recover the BD symbol as follows
(5) 
and the resulting decoding signaltonoiseratio (SNR) is
(6) 
Hence, the th BD’s throughput^{3}^{3}3This paper adopts normalized throughput with unit of bps/Hz. normalized to is
(7) 
Similar to (4), the received frequencydomain signal at the LU can be written as follows
(8)  
where the frequencydomain noise .
Similar to (II), treating backscatterlink signal as interference, the total throughput of the LU is given by
(9)  
Iii Problem Formulation
In this section, we formulate the optimization problem. The objective is to maximize the minimum throughput among all BDs, by jointly optimizing the BDs’ backscatter time portions (i.e., ), the BD’s power reflection coefficients (i.e., ), and the FAP’s subcarrier power allocation (i.e., ). We consider the following constraints: the total throughput of the LU needs to be larger than a given minimum throughput , i.e., ; each BD has a minimum energy requirement , i.e., ; the total power consumed by the FAP needs to be less than a given maximum power , i.e., ; the sum of backscatter time portions for all BDs should be no larger than 1, i.e., , with nonnegative time portion for each BD ; the peak power value for each subcarrier is , i.e., ; the power reflection coefficients are positive numbers and no larger than 1, i.e., . The optimization problem is thus formulated as follows
(10a)  
(10b)  
(10c)  
(10d)  
(10e)  
(10f)  
(10g)  
(10h)  
(10i) 
Notice that problem (10) is nonconvex and challenging to solve in general, since the variables are all coupled and the constraint function in (10c) is nonconvex over ’s.
Iv Proposed Algorithm
In this section, we propose an efficient iterative algorithm for the problem (10) by applying the block coordinate descent (BCD) [13] and successive convex optimization (SCO) techniques [14]. Then, we show the convergence of the proposed algorithm and analyze its complexity.
Iva Backscatter Time Allocation Optimization
In iteration , for given power reflection coefficients and subcarrier power allocation , the backscatter time portions can be optimized by solving the problem
(11a)  
(11b) 
where the variables ’s and ’s are replaced by given ’s and ’s, respectively, in all the constraints. Notice that problem (11
) is a standard linear programming (LP), it can be solved efficiently by existing optimization tools such as CVX
[15].IvB Reflection Power Allocation Optimization
For given backscatter time portions and subcarrier power allocation , the power reflection coefficients can be optimized by solving the following problem
(12a)  
(12b) 
where the variables ’s and ’s are replaced by given ’s and ’s, respectively. Since the lefthandside of the constraint (10c) with given and is a decreasing and convex function of , the constraint is convex. Hence, problem (12) is a convex optimization problem that can also be efficiently solved by CVX [15].
IvC Subcarrier Power Allocation Optimization
For given backscatter time portions and power reflection coefficients , the subcarrier power allocation can be optimized by solving the following problem
(13a)  
(13b)  
(13c) 
where the variables ’s and ’s are replaced by given values ’s and ’s, respectively. Since the constraint function in (13b) is nonconvex with respect to , problem (13) is nonconvex. Notice that the constraint function can be rewritten as
(14) 
To handle the nonconvex constraint (13b), we exploit the SCO technique [14] to approximate the second logarithm function in (IVC). Recall that any concave function can be globally upperbounded by its firstorder Taylor expansion at any point. Specifically, let denote the subcarrier power allocation in the last iteration. We have the following concave lower bound at the local point
(15)  
With given local points and lower bound in (15), by introducing the lowerbound minimumthroughput , problem (13) is approximated as the following problem
(16a)  
(16b)  
(16c) 
where the variables ’s and ’s are replaced by given ’s and ’s, respectively. Problem (16) is a convex optimization problem which can also be efficiently solved by CVX [15]. It is noticed that the lower bound adopted in (16b) implies that the feasible set of problem (16) is always a subset of that of problem (13). As a result, the optimal objective value obtained from problem (16) is in general a lower bound of that of problem (13).
IvD Overall Algorithm
We propose an overall iterative algorithm for problem (10) by applying the BCD technique [13]. Specifically, the entire variables in original problem (10) are partitioned into three blocks, i.e., , and , which are alternately optimized by solving problem (11), (12), and (16) 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 1.
IvE Convergence and Complexity Analysis
Notice that in our case, for subcarrier power allocation problem (13), we only solve its approximate problem (16) optimally. Thus, the convergence analysis for the classical BCD technique cannot be directly applied [13], and the convergence of Algorithm 1 needs to be proved, as follows.
Theorem 1.
Algorithm 1 is guaranteed to converge.
Proof.
First, in step 3 of Algorithm 1, since the optimal solution is obtained for given and , we have the following inequality on the minimum throughput
(17) 
Second, in step 4, since the optimal solution is obtained for given and , it holds that
(18) 
Third, in step 5, it follows that
(19) 
where (a) holds since the Taylor expansion in (15) is tight at given local point, which implies problem (16) at has the same objective function as that of problem (13); (b) is because is the optimal solution to problem (16); (c) holds since the objective value of problem (16) is a lower bound of that of its original problem (13).
From (17), (18), and (19), we have
(20) 
which implies that the objective value of problem (10) is nondecreasing after each iteration in Algorithm 1. Since the objective value of problem (10) is a finite positive value, the proposed Algorithm 1 is guaranteed to converge to the optimal objective value and solutions. This completes the convergence proof. ∎
V Numerical Results
In this section, we provide simulation results to evaluate the performance of the proposed joint design. We consider an ABCN with
BDs. Suppose that the FAPtoBD1 distance and FAPtoBD2 distance are 2.5 m and 4 m, respectively, the FAP (BD1, BD2)toLU distances are all 15 m. We assume independent Rayleigh fading channels, i.e., the channel coefficient of each path is a circularly symmetric complex Gaussian prandom variable, and the power gains of multiple paths are exponentially distributed. For each channel link, its firstpath channel power gain is assumed to be
, where is the distance with unit of meter. Let the number of pathes , , and . Other parameters are set as . Define the average receive SNR at the FAP as . Let . For performance comparison, we consider a benchmark scheme in which the backscatter time portion and subcarrier power are equally allocated, i.e., , and all BDs adopt a common power reflection coefficient that is optimized via CVX. The following results are obtainepd based on 100 random channel realizations.Fig. 3 plots the maxmin throughput of all BDs versus the LU’s throughput requirement for different SNRs ’s. We fix and J. As expected, the maxmin throughput decreases as increases, which reveals the throughput tradeoff between the BDs and the LU. We further observe that the maxmin throughput performance is significantly enhanced by using the proposed joint design, compared to the benchmark scheme. Also, higher maxmin throughput is achieved when the SNR at the FAP is higher.
Fig. 4 compares the maxmin throughput under different BDs’ energy requirements ’s and subcarrier peakpower values ’s, for both the proposed joint design and the benchmark scheme. We fix bps/Hz. In general, the maxmin throughput increases as the SNR increases. We have three further observations. First and foremost, the proposed joint design achieves significant maxmin throughput gains as compared to the benchmark scheme. Second, higher maxmin throughput is achieved for lower harvestedenergy requirement with given , which reveals the BDs’ throughputenergy tradeoff. This observation can be specifically obtained from the three red solid curves for our proposed joint design and the three blue dotted curves for the benchmark scheme, given . Third, higher maxmin throughput is obtained for higher peakpower value with given , which is demonstrated in the red and black solid curves with triangle marker for our proposed joint design.
Vi Conclusion
This paper has investigated a fullduplex AmBC network over ambient OFDM carriers. The minimum throughput among all BDs is maximized by jointly optimizing the BDs’ backscatter time portions, the BDs’ power reflection coefficients, and the FAP’s subcarrier power allocation. By utilizing the block coordinated decent and successive convex optimization techniques, an efficient iterative algorithm is proposed, which is guaranteed to converge. Numerical results show that significant throughput gains are achieved as compared to the benchmark scheme with equal resource allocation, benefitting from multiple design dimensions of the proposed joint optimization. The interesting BDs’ throughputenergy tradeoff and the throughput tradeoff between the BDs and the LU are also revealed.
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