I Introduction
The Internet of Things (IoT) [1] is a revolutionary communication paradigm to provide massive connectivity for the nextgeneration wireless cellular networks. The limited battery life of devices poses a significant challenge for designing green and sustainable IoT. One promising solution is to leverage the simultaneous wireless information and power transfer (SWIPT) with radio frequency to prolong the IoT network, due to its ability to provide costeffective and perpetual power source [2]. This requires receiver circuits to decode information and harvest energy from the same received signal independently and simultaneously, which renders SWIPT impractical.
To overcome these limitations, the telecommunication industry is increasingly turning towards power splitting (PS), a receiver architecture that divides the received signal into two streams of different power for decoding information and harvesting energy. Based on the PS architecture, [3] considers a multiuser joint beamforming and power splitting design problem under QoS constraints and proposes a semidefinite relaxationbased algorithm. To further reduce the computational complexity, an secondorder cone programming relaxation method is proposed in [4]. Recently, [5] combines the SWIPT and massive multipleinputmultipleoutput (MIMO) to further improve the spectral and energy efficiency of IoT networks. The aforementioned works focus on optimizing the weighted sum of objective function under perfect CSI, which is difficult to obtain in the massive MIMO regime due to the large number of antennas and the limited pilot sequences [6]. In such scenarios, it is more reasonable to consider a mixedtimescale optimization of the longterm performance of the network, which only requires imperfect CSI plus the knowledge of channel statistics [7]. To the best of our knowledge, this is first work on mixedtimescale optimization for massive MIMO aided SWIPT IoT network.
Contribution of this letter includes the algorithm design for mixedtimescale joint beamforming and power splitting (MJBP) scheme for the downlink transmission of massive MIMO aided SWIPT IoT network, to maximize a general network utility. Specifically, the digital beamformer is adapted to the imperfect CSI, while the power spiltters are adapted to the longterm channel statistics due to the consideration of hardware limit and signaling overhead. We propose a mixedtimescale online stochastic successive convex approximation (MOSSCA) algorithm to solve this joint optimization problem. Simulations verify the advantages of the proposed MJBP scheme over the baselines.
Ii System Model and Problem Formulation
Consider the downlink of a massive MIMO aided SWIPT IoT network, where the base station (BS) is equipped with antennas to simultaneously serve singleantenna IoT devices. As illustrated in Fig. 1, the BS employs digital beamformer to spatially multiplex devices and manage the multidevice interference, while device applies the power splitter () to coordinate information decoding and energy harvesting from the received signal. With MJBP, both the digital beamformer and the power splitters are optimized at the BS. Furthermore, the digital beamformer is adapted to instantaneous CSI. For the power splitter implemented at each device, it is adapted to longterm channel statistics due to following reasons: 1) the hardware capability of the IoT device is limited, and thus the power splitter cannot be changed frequently due to hardware limitations [1]; 2) such design can reduce the signaling overhead of sending to the corresponding device, especially when the number of devices is large.
We consider flat fading channels with block fading model, but the proposed algorithm can be easily modified to cover the frequency selective channels. The channel is assumed to be constant within each block of length . In this case, the received signal splitted to the information decoder (ID) of device is given by where is the data symbol for device , is the additive noise (AN) at the PS of device , and is the AN introduced by the ID at device . Meanwhile, the received signal splitted to the energy harvester (EH) is given by
In practice, perfect CSI is challenging to obtain due to device mobility, processing latency and other limitations. Thus, we model the channel imperfection as where
is the estimated channel from from BS to device
, is the channel error independent of , andis the variance of the channel error. Consequently, the achievable rate is obtained by replacing
in (1) with . For convenience, we let , and . Further, we define as the collection of shortterm optimization variables for all possible estimated channel states , where is the feasible set of .Proposition 1 : The ergodic rate at device is bounded as , and
Here the lower bound follows from the properties of variance and the Cauchy–Schwarz inequality. From Proposition 1, optimizing the lower and upper bound provide the same optimal solution. Moreover, as verified in Fig. 2, we find that both bounds are tight. Therefore, we optimize the lower (upper) bound of the ergodic rate at each device as it is more tractable for optimization.
The average harvested power conditioned on imperfect CSI of device follows a nonlinear function [8] and can be expressed as where
where is a constant denoting the maximum harvested power at the th device, , , and parameter and are constants related to the circuit specifications, and is the input RF power for the th device. Then, the average harvested power of user is defined as .
We are interested in a mixedtimescale joint optimization of digital beamformer and power splitter to balance the average ergodic rate and the average harvested power. This can be formulated as the following network utility maximization problem:
(2) 
where with the corresponding weight is a weighted sum of the average ergodic rate and the harvested power, is the feasible set of power splitters. The utility function is a continuously differentiable and concave function of . Moreover, is nondecreasing w.r.t. and its derivative is Lipschitz continuous.
Iii Online Optimization Algorithm
In this section, we propose a MOSSCA algorithm to solve the mixedtimescale stochastic nonconvex optimization problem , and summarize it in Algorithm 1. In MOSSCA, we focus on a coherence time of channel statistics, where the time is divided into frames and each frame consists of time slots. At beginning, the BS initializes the MOSSCA algorithm with power splitter
and a weight vector
. In subsequent, and are updated once at the end of each frame. Then we elaborate the implementation details of the iteration of the MOSSCA algorithm at the th frame.Iiia Shortterm FPBCD Algorithm
At time slot within the th frame, BS obtains the estimated channel by uplink channel training. Based upon the current , , and , we can obtain digital beamforming by maximizing the a weighted sum of the average data rate and the average harvested power conditioned on imperfect CSI, which can be formulated as
where with and , and is solved at time slot .
Since the objective function contains expectation operators, it does not have a closedform expression. To address the challenge, we resort to the Sample Average Approximation (SAA) method [9]. Specifically, a total of samples are generated for independently drawn from the distribution , and the th sample of is defined as In this case, the SAA version of is formulated as where .
However, solving problem is still challenging due to the nonlinear fractional term in and coupling in the power constraint. To this end, we apply the Lagrangian dual transform method [10] to recast problem into a more tractable yet equivalent form, using the following lemma.
Lemma 1.
The optimal digital beamforming solves the problem in (1) if and only if it solves
(3) 
where with , and is the optimal auxiliary variable introduced for each ratio term.
In subsequent, we use the complex quadratic transformation [10] to equivalently recast problem (3) as
(4) 
where
with , and with is the auxiliary variable vector. Observing that the constraints are separable with respect to the three blocks of variables, i.e., , and , we shall focus on designing a fractional programming block coordinate descent (FPBCD) algorithm to find a stationary point of problem (4), and summarize it in Algorithm 2. For problem (4), this amounts to the following steps:
IiiA1 Optimization of
The optimal is given by
(5) 
IiiA2 Optimization of
By applying the firstorder optimal condition, the optimal admits a closedform solution as:
(6) 
IiiA3 Optimization of
The subproblem w.r.t. is nonconvex due to the involvement of the nonlinear energy harvesting model. To overcome this difficulty, we first transform it into a more tractable yet equivalent form by the introduction of new auxiliary variables and some manipulations, which can be expressed as
(7)  
where , and . Note that the constraint in problem is nonconvex. Thus, we apply the the majorization minimization (MM) method [11] to approximate this nonconvex constraint using its firstorder Taylor expansion as
(8)  
where and represents the last iteration of and , and . Note that problem (8) is convex, which can be efficiently solved by the CVX toolbox [12].
IiiB Longterm Optimization
Before the end of th frame, device obtains a full channel sample and channel error sample . Based on , and , we preserve the partial concavity of the original function and add the proximal regularization, to construct the concave surrogate function , resulting in the following
(9) 
where is a postive constant; the recursive approximation of the weighted sum of the data rate and the harvested power is given by
with , and is a stepsizes sequence to be properly chosen; the recursive approximation of the partial derivative is given by
with , is the gradient of w.r.t. at and . Moreover, the weight vector is updated as
(10) 
with , where is a stepsizes sequence satisfying , . Moreover, the optimal power splitting ratio for device can be obtained by solving the following quadratic optimization problem, i.e.,
(11) 
By applying the firstorder optimality condition, it yields the closedform solution where denotes the projection onto the feasible region . Consequently, the longterm variable is updated as
(12) 
Remark 1 : Note that the stationary weight vector has captured the nature of the utility function. However, it is difficult to obtain , since it in turn depends on the stationary solution . Therefore, the basic idea of the proposed algorithm is to iteratively update the longterm variable and the weight vector such that and converge to a stationary solution and the corresponding stationary weight vector , respectively.
IiiC Convergence Analysis
The following theorem states that Algorithm 2 converges to a stationary point of up to certain convergence error which vanishes to zero exponentially as .
Theorem 2 (Convergence of Algorithm 2).
Suppose problem has a discrete set of stationary points, denoted by . Let denote the limiting point of the sequence generated by Algorithm 2 with input parameter and sample number . Then for every small positive number , there exist positive constants and , independent of , such that
for sufficiently large, where .
Proof:
Specifically, the proposed FPBCD algorithm falls in the MM framework and similar proof is provided in [13]. From Theorem 4.4 in [11], every limiting point of sequence generated by the shortterm FPBCD algorithm is a stationary point of problem , where problem is the sample average approximation of problem with samples. As stated in [9], problem is equivalent to problem w.p.1 when
approaches to infinity, due to the classical law of large number for random functions. That is to say, as
, any stationary point of is also a stationary point of problem w.r.1. When is finite, Algorithm 2 converges to approximate stationary points of problem with the exponential convergence rate . This is consequence of [14], Theorem 3.1, which provides a general convergence result for the original problem that satisfies the following assumptions: (a) The feasible set of optimization variables is a nonempty closed convex set; (b) The objective function of the original problem is continuously differentiable on the feasible set for any given random system states, and its gradient is Lipchitz continuous. Clearly, problem satisfies the aforementioned assumption (a) and (b). This completes the proof. ∎Based on Theorem 2, the convergence of the proposed MOSSCA algorithm is summarized in the following theorem.
Theorem 3 (Convergence of the Algorithm 1).

, for some ,

, , ,

.
Let denote the sequence of iterates generated by Algorithm 1, where . Then every limit point of almost surely satisfies
(13) 
(14) 
where , and Moreover, it satisfies
(15) 
where is the Jacobian matrix of the vector w.r.t. at and , and satisfies almost surely.
Proof:
According to equation (15) in Theorem 3, it implies that the shortterm solution found by Algorithm 2 must satisfy the stationary condition approximately with certain error that converges to zero exponentially as . Moreover, the limiting point generated by Algorithm 1 also satisfies the stationary conditions in (13) and (14), respectively. Thus, Algorithm 1 converges to stationary solutions of the mixedtimescale optimization problem . Note that since converges to zero exponentially, Algorithm 2 with a small can already achieve a good performance and avoids excessive computational complexity.
Iv Simulation Results and Discussions
We consider a singlecell of radius m, where BS is equipped with antennas. There are 12 devices randomly distributed in the cell. We adopt a geometric channel model with a halfwavelength space ULA for simulations [7]. The channel between BS and device is given by , where is the array response vector, ’s are Laplacian distributed with an angular spread , ,
are randomly generated from an exponential distribution and normalized such that
, is the average channel gain determined by the pathloss model [15], and is the distance between BS and device in meters. We consider channel paths for each device. The transmit power budget for BS is dBm. We set , mW, , , , dB, dBm and dBm. There are time slots in each frame and the slot size is 2 ms. The coherence interval , which corresponds to a coherence time of 2 ms and a coherence bandwidth of 200 kHz [16]. The coherence time for the channel statistics is assumed to be 10 s. We use the average sum utility as an example to illustrate the advantages of the proposed scheme. Two schemes are included as baselines: 1) maximum ratio transmission (MRT) scheme, which is obtained by fixing the MRT beamformer [3]; 2) zeroforcing (ZF) scheme, which is obtained by fixing the ZF beamformer [5]. The power splitters of both MRT and ZF scheme are obtained by the longterm optimization.In Fig 3, we plot the utility performance versus the signaltonoise ratio (SNR). We can see that as the SNR increases, the average sum utility of all schemes increases gradually. It is observed that the average sum utility achieved by the proposed MJBP scheme is higher than that achieved by the other schemes for moderate and large SNR. This indicates that the proposed MJBP scheme can better mitigate the multidevice interference to achieve better tradeoff between the average ergodic rate and the average harvested power, which is further validated in Fig 3.
In Fig 4, we plot the utility performance versus the number of devices . We observe that the proposed MJBP scheme achieves significant gain over MRT scheme and ZF scheme, which demonstrates the importance of mixedtimescale joint optimization. Moreover, as the number of devices increases, the performance gap between the proposed MJBP scheme and other competing schemes becomes larger.
In Fig 4, we plot the utility performance versus the number of antennas at BS. It shows that the performance of all these schemes is monotonically increasing with the number of antennas. Again, it is seen that the proposed MJBP scheme outperforms all the other schemes for all regime.
V Conclusion
In this letter, we considered mixedtimescale joint beamforming and power splitting (MJBP) scheme in the downlink transmission of massive MIMO aided SWIPT IoT network to maximize the network utility under the power budget constraint. We proposed a MOSSCA algorithm to find stationary solutions of the mixedtimescale nonconvex stochastic optimization problem. Simulations verify that the proposed MJBP scheme achieves significant gain over existing schemes.
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