I Introduction
Full-duplex (FD) transceivers can transmit and receive signals at the same time and frequency, and hence, provide improved spectral efficiency. However, the self-interference (SI), which suppresses the weak received signal of interest, limits their performance. With current SI cancellation (SIC) techniques, small power transceivers are identified as being suitable for FD deployment [1].
Recently, radio-frequency (RF) signals have been investigated for simultaneous wireless information and power transfer (SWIPT) [2]. Commonly, the RF signals’ energy is harvested either by power- or time-splitting receivers. For the former (latter), the received signal power (transmission time) is divided into two parts: one used for information gathering and another one used for energy harvesting. Very recently, FD has been combined with SWIPT to boost both the spectral efficency and energy efficiency (EE) of the system [3, 4]. Furthermore, the idea of self-energy (SEg) recycling from the SI is conceptualized in [5, 6, 7]. Both in [5] and [6], FD is used at the relay terminal and the time-splitting protocol is applied for the SEg harvesting. The system throughput is maximized in [5], whereas the signal-to-noise ratio (SNR) is maximized in [6]. In [7], the authors introduce a three-port circuit for recycling the SI and show significant improvement in the EE.
Since the SI carries high energy, it could potentially be harvested for some fraction of a total transmission time. Inspired from this idea, in this paper, we consider the SEg harvesting by the SI at a small cell base station (SBS). Particularly, we propose a time-splitting based two-phase protocol for SEg harvesting at the FD SBS. The SBS is equipped with an SIC switch: when it is turned-on, the SIC is activated; otherwise, SIC is disabled. In the first phase, the SIC switch is off and the SBS sends the information-bearing signal to its downlink (DL) users. The energy harvesting device at the SBS harvests the SI energy and also receives energy-bearing signals from its uplink (UL) UEs. In the second phase, the SIC switch is on, and no energy harvesting is possible from the residual SI signal. In this phase, the SBS continues to transmit the information-bearing signal to DL UEs and starts receiving the information-bearing signals from the UL UEs. We explore the optimal time-splitting factor that maximizes the EE of the system along with the optimal beamforming and power allocation design for the DL and UL UEs, respectively. Simulation results show the significant EE improvement offered by the proposed SEg harvesting scheme.
Notation:
Bold uppercase and lowercase letters denote matrices and vectors,
and represent the Hermitian and transpose operations, and denote the absolute value and -norm, and and are used as the trace and expectation operators, respectively. represents the identity matrix. represents a set where each element has value greater than .Ii System Model
We consider an FD SBS equipped with transmit and receive antennas, which serves and single antenna DL and UL UEs, respectively. The sets of DL and UL UEs are denoted by and , respectively. The SBS is powered by a regular grid source, and is also equipped with an RF power harvesting device and a rechargeable battery for energy storage. We assume a flat fading channel model, in which all channels remain unchanged for a time block of duration and change independently to new values in the next block.
The transmission time is divided into phases of duration and , where is the time-splitting factor. In the first phase, the SBS transmits the information-bearing and receives the energy-bearing signals to and from the DL and UL UEs, respectively. The SIC switch in this phase is turned-off for SEg harvesting. Then, the received signal at the DL UE is given by
(1) |
where and with are the beamforming vector and data of DL UE , respectively; is the power coefficient allocated to the UL UE during phase; the vector and complex scalar denote the channel from the SBS to DL UE and from the UL UE to the DL UE , respectively; and is complex additive white Gaussian noise (AWGN) at DL UE
, with variance
. The signal vector at the receive antennas of the SBS is given as(2) |
where and are the SI channel matrix of the SBS and channel vector of UL UE to the SBS, respectively; and is a complex AWGN vector at the receiver of the SBS. Consequently, the total SEg harvested at the SBS in the first phase is given by
(3) |
where represents the energy conversion efficiency of the harvester. In (3), the noise term is ignored as its contribution is negligible. In the second phase, the SBS turns the SIC switch on, which brings the SI power to the noise level. In this phase, the signals received at the DL UE and SBS are, respectively, given as
(4) | |||
(5) |
where and are the power coefficient allocated to UL UE and the complex AWGN vector; and is the SI channel matrix, which captures the effect of SIC.
Using (1) and (4), the received signal-to-interference plus noise ratios at DL UE in the first and second phases can be written as
(6) |
where represents the phase. At the end of transmission time , the achievable rate for DL UE is given as .
Next, for the UL transmission, using (5), the received SINR of UL UE at the SBS, which applies the minimum-mean-squared error with successive interference cancellation receiver, is given by
(7) |
where . Then, the achievable rate for UL UE , at the end of transmission time , is .
Energy usage model: The combined energies consumed in the circuit and decoding operations are comparable or even dominate the actual transmit energy [8]. Consequently, these energies play a significant role in representing the total power consumption. Thus, the total power consumed at the SBS can be expressed as
(8) |
where is the amplifier efficiency of the SBS; is the total circuit power consumption, in which and correspond to the active RF blocks, and to the cooling and power supply, respectively; and is the power consumption for data decoding of the UL UE , which is a function of the achievable rate of the UE, i.e., for the UL UE , where models the decoder efficiency, being decoder specific [8, 9].
Energy efficiency function: In the context of 5G networks, EE maximization is of paramount importance for both operators and users [10]. Thus, here as well our interest is to jointly optimize the DL UE beamformers, UL UE power coefficients and such that the system EE is maximized. Unlike [10], we introduce a novel EE function that measures the efficiency of the aggregated energies draw from the grid source at both SBS and UL UEs as , where and capture the throughput and total grid energy consumed by the system, respectively. The sets w and p collect the optimization variables and , and and , respectively. In particular, the throughput is given as , and total grid power consumption is given as . denotes the consumption during the period, with denoting the grid power consumed during the phase. denotes the energy drawn from the battery source at the UL UEs during both the and phases.
Iii Energy Efficiency Maximization
Using the notation introduced above, the constrained EE maximization problem is formulated as
s.t. | ||||
where , and and denote the maximum transmit powers of the SBS and UL UE, respectively. Constraint (III) ensures that each UL UE achieves the minimum specified data-rate of . Constraints (III) and (III) represent the restrictions on the maximum transmit powers of the SBS and the UL UEs, respectively. Constraint (III) restricts the SBS to use the harvested SEg in the second phase, if it is sufficient; otherwise, the SBS draws the energy from the grid source to sustain the transmissions. Evidently, (III) is a nonconvex problem and obtaining an optimal solution is challenging and converges slowly. Hence, we seek a rapidly converging suboptimal solution in the following section.
Iv Proposed Solution Method
There are two main steps involved to arrive at the rapidly converging solution. In the first step, we perform a few equivalent transformations on (III), similar to [9] and [11], to expose the hidden convexity and gain tractability. Accordingly, the resulting problem is expressed equivalently as
s.t. | ||||
where and is a rank-1 positive semi-definite (PSD) matrix. The introduction of helps convexify (IV) [11], which is otherwise a difficult constraint to handle. For notational compactness, a set is introduced, which collects all the optimization variables, where collect , , and , respectively. , , , , , , , , , , , , , are slack variables. It is easy to see that a solution to (IV) is also feasible for (III). Moreover, all the constraints (IV)-(IV) are active at optimality, and hence, (IV) is an equivalent formulation of (III). Note that, to write (IV) as a second-order cone (SOC) constraint, we introduce in the objective function; however, its maximization is a nonconvex problem. Hence, we equivalently replace the objective function with , which also maximizes . Next, to achieve further tractability, we relax the nonconvex rank-1 constraint (IV) by dropping it. Now, (IV) can be equivalently expressed as
(12) |
In the second step, we identify the nonconvex parts of (12) and linearize them with a first-order Taylor approximation around the point of operation [12]. This step leads to an iterative procedure and a local solution to (12). In (IV), the constraints (IV)-(IV), (IV), (IV)-(IV) are nonconvex. Particularly, the nonconvexity in (IV), (IV), (IV) and (IV) is due to the convex function of form , on the greater side of the inequalities. Functions of this form can be approximated, around a point at the th iteration, as . The constraints (IV), (IV), (IV), (IV), (IV) and (IV) also have nonconvexity due the presence of functions of the forms and , which we linearize around a point , as [13]. The constraints (IV) and (IV) have nonconvexity on the lesser side of the inequalities of the form . Using the result from [14], we replace with its convex upper bound around a point as . The approximations employed above satisfy the following three conditions [12]: i) ; ii) ; iii) , and hence, the convergence of the iterative procedure is ensured. Now, by replacing the nonconvex parts of the constraints with the approximations discussed above, (12) can be formulated as a convex problem at the th iteration as
s.t. | ||||
where . Pseudocode for solving (IV) is outlined in Algorithm 1. Note that the constraints in (IV) are exponential cones, which we approximate as a set of SOC constraints [15, 16] in Algorithm 1. Since the problem is upper bounded due to power constraints, the algorithm generates a monotonic non-decreasing sequence of objective function values and converges to a Karush-Kuhn-Tucker (KKT) point of (12) [12]. The detailed proof follows similar lines as the one discussed in [17], and hence, is omitted here for brevity.
The feasible initial point to Algorithm 1 is generated by solving the following problem; subject to , where are the newly introduced variables. The feasible initial point is obtained when and requires three iterations at most.
V Numerical Results
In this section, a performance evaluation of the proposed SEg harvesting scheme is presented. The parameters used in simulations with their values are listed in Table I. The algorithm is implemented using the CVX parser [20] and mosek as an internal solver. The minimum data-rate required for each UL UE is set to Mbit/s. A small cell of radius m is considered with uniformly distributed UEs within the cell area. The channels are Rayleigh faded with each coefficient following the distribution. The SI channel is modeled as Rician, with Rician factor , i.e., , where is a deterministic matrix and denotes the Kronecker product. denotes the ratio of the average SI powers before and after the SIC. Results are obtained based on runs.
In Fig. 1, we show the convergence behavior of the iterative Algorithm 1. The objective function values are plotted versus the number of iterations for two independent random channels states and dBm. We observe that Algorithm 1 converges in less than fifty iterations for all three channel realizations. For benchmarking purpose, the objective function values are also compared with the global ones, which are obtained using the branch-reduce-and-bound algorithm [21].
In Fig. 2, the average EE (AEE) with and without the SEg harvesting scheme versus the transmit power of the SBS is plotted. It can be seen that the average gain of the scheme that harvests SEg is significantly higher than the one that does not harvest. The AEE of both schemes saturates in the high regime; however, the former saturates later than the latter. In low-power regime, for the proposed scheme, the SBS harvests less SEg and draws more energy from the grid source for decoding the UL users data, and hence, the AEE drops. Note that the AEE of the latter is obtained by using Algorithm 1 with fixed . The time-splitting factor is also shown on the right-hand side y-axis of the figure. Its values increase with but saturates in the high regime.
Lastly, we have observed that in the first phase Algorithm 1 allocates zero power to each UL UE for all values of , and accordingly, the SBS harvests only the SEg.
Vi Conclusion
We have considered a FD SBS, in which, by installing an additional on and off SIC switch, the FD SBS can harvest SEg from the SI. The SEg is harvested when the SIC switch is off; otherwise, there is no harvesting. The fraction of the transmission time for which the SIC switch is on has been investigated jointly with the beamformer and power allocations that maximize the EE of the SBS. Numerical results have shown that significant AEE gain is attained by the system that allows SEg harvesting.
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