You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Traditional simultaneous wireless information and power transfer (SWIPT) with power splitting assumes perfect channel state information (CSI), which is difficult to obtain especially in the massive multiple-input-multiple-output (MIMO) regime. In this letter, we consider a mixed-timescale joint beamforming and power splitting (MJBP) scheme to maximize general utility functions under a power constraint in the downlink of a massive MIMO SWIPT IoT network. In this scheme, the transmit digital beamformer is adapted to the imperfect CSI, while the receive power splitters are adapted to the long-term channel statistics only due to the consideration of hardware limit and signaling overhead. The formulated optimization problem is solved using a mixed-timescale online stochastic successive convex approximation (MO-SSCA) algorithm. Simulation results reveal significant gain over the baselines.'
author:
- '[Xihan Chen$^{1}$, Hei Victor Cheng$^{2}$, An Liu$^{1}$, Kaiming Shen$^{2}$]{}*,*[ and Min-Jian Zhao$^{1}$]{}[^1]'
bibliography:
- 'liu\_all.bib'
title: 'Mixed-Timescale Beamforming and Power Splitting for Massive MIMO Aided SWIPT IoT Network'
---

SWIPT, massive MIMO, mixed-timescale joint beamforming and power splitting, online stochastic successive convex approximation.

Introduction
============

The Internet of Things (IoT) [@Swan_survey12_IoT] is a revolutionary communication paradigm to provide massive connectivity for the next-generation 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 cost-effective and perpetual power source [@Ruizhang_2013twc_SpatialSWIPT]. 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, [@QJS_2014twc_JBPS] considers a multiuser joint beamforming and power splitting design problem under QoS constraints and proposes a semidefinite relaxation-based algorithm. To further reduce the computational complexity, an second-order cone programming relaxation method is proposed in [@QJS_2014tsp_SOCP]. Recently, [@Dong_2018globecom_ZFSWIPT] combines the SWIPT and massive multiple-input-multiple-output (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. In such scenarios, it is more reasonable to consider a mixed-timescale optimization of the long-term performance of the network, which only requires imperfect CSI plus the knowledge of channel statistics [@Anliu_tsp19_THCF]. To the best of our knowledge, this is first work on mixed-timescale optimization for massive MIMO aided SWIPT IoT network.

Contribution of this letter includes the algorithm design for mixed-timescale 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 long-term channel statistics due to the consideration of hardware limit and signaling overhead. We propose a *mixed-timescale online stochastic successive convex approximation* (MO-SSCA) algorithm to solve this joint optimization problem. Simulations verify the advantages of the proposed MJBP scheme over the baselines.

![\[fig:MBPS\]Architecture of the proposed MJBP scheme.](system_model.eps){width="1\columnwidth" height="2.8cm"}

System Model and Problem Formulation
====================================

Consider the downlink of a massive MIMO aided SWIPT IoT network, where the base station (BS) is equipped with $M$ antennas to simultaneously serve $K$ single-antenna IoT devices. As illustrated in Fig. \[fig:MBPS\], the BS employs digital beamformer $\mathbf{F}\triangleq[\bm{f}_{1},\cdots,\bm{f}_{K}]\in\mathbb{C}^{M\times K}$ to spatially multiplex devices and manage the multi-device interference, while device $k$ applies the power splitter $\rho_{k}$ ($0\leq\rho_{k}\leq1$) 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 long-term 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 [@Swan_survey12_IoT]; 2) such design can reduce the signaling overhead of sending $\rho_{k},\forall k,$ 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 $\mathbf{H}\triangleq[\bm{h}_{1},\cdots,\bm{h}_{K}]\in\mathbb{C}^{M\times K}$ is assumed to be constant within each block of length $T$. In this case, the received signal splitted to the information decoder (ID) of device $k$ is given by $y_{k}^{I}=\sqrt{\rho_{k}}(\bm{h}_{k}^{H}\sum_{m=1}^{K}\bm{f}_{m}s_{m}+n_{k})+z_{k},$ where $s_{m}\sim\mathcal{CN}(0,1)$ is the data symbol for device $m$, $n_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ is the additive noise (AN) at the PS of device $k$, and $z_{k}\sim\mathcal{CN}(0,\delta_{k}^{2})$ is the AN introduced by the ID at device $k$. Meanwhile, the received signal splitted to the energy harvester (EH) is given by $y_{k}^{E}=\sqrt{1-\rho_{k}}(\bm{h}_{k}^{H}\sum_{m=1}^{K}\bm{f}_{m}s_{m}+n_{k}).$

An achievable ergodic rate [@Caire_2018twc_MIMObound] at device $k$ is given by $$\begin{aligned}
\hat{r}_{k}^{\circ}(\rho_{k},\mathbf{F}) & =\mathbb{E}_{\mathbf{H}}[\log_{2}(1+\textrm{\ensuremath{\Gamma}}_{k}(\rho_{k},\mathbf{F},\mathbf{H}))]\nonumber \\
 & -\frac{1}{T}\sum_{m=1}^{K}\log_{2}(1+\frac{T}{\rho_{k}\sigma_{k}^{2}+\delta_{k}}\mathrm{Var}(\bm{h}_{k}^{H}\bm{f}_{m}),\label{eq:ergodiRate}\end{aligned}$$ where $\textrm{\ensuremath{\Gamma}}_{k}(\rho_{k},\mathbf{F},\mathbf{H})$ is the SINR of device $k$ with $$\textrm{\ensuremath{\Gamma}}_{k}(\rho_{k},\mathbf{F},\mathbf{H})=\frac{\rho_{k}|\bm{h}_{k}^{H}\bm{f}_{k}|^{2}}{\rho_{k}(\sum_{m\neq k}^{K}|\bm{h}_{k}^{H}\bm{f}_{m}|^{2}+\sigma_{k}^{2})+\delta_{k}^{2}}.$$

l error independent of $\hat{\bm{h}}_{k}$, and $\omega_{k}^{2}$ is the variance of the channel error. Consequently, the achievable rate is obtained by replacing $\bm{h}_{k}$ in (\[eq:ergodiRate\]) with $\hat{\bm{h}}_{k}+\bm{\phi}_{k}$.

*Proposition 1* : The ergodic rate at device is $k$ bounded as $\hat{r}_{k}^{\circ}(\rho_{k},\mathbf{\Theta})\leq\overline{r}_{k}(\rho_{k},\mathbf{\Theta})\triangleq\mathbb{E}_{\mathbf{\hat{H}},\bm{\phi}_{k}}[\log_{2}(1+\textrm{\ensuremath{\Gamma}}_{k}(\rho_{k},\mathbf{F},\hat{\bm{h}}_{k},\bm{\phi}_{k}))]$, and $\hat{r}_{k}^{\circ}(\rho_{k},\mathbf{\Theta})\geq$ $$\overline{r}_{k}(\rho_{k},\mathbf{\Theta})-\frac{1}{T}\sum_{m=1}^{K}\log_{2}(1+\frac{TP_{\textrm{max}}}{\delta_{k}^{2}}\mathbb{E}_{\mathbf{\hat{H}},\bm{\phi}_{k}}[||\hat{\bm{h}}_{k}+\bm{\phi}_{k}||^{2}]).$$

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.

![Cumulative distribution function (CDF) of the lower and upper bound of ergodic rate at each device. The detailed setup is given in Section \[sec:Simulation-Results-and\].](upper_final){height="3cm"}

where $\overline{\eta}_{k}(\rho_{k},\mathbf{\Theta})\triangleq\overline{r}_{k}(\rho_{k},\mathbf{\Theta})+\gamma_{k}\overline{e}_{k}(\rho_{k},\mathbf{\Theta})$ with the corresponding weight $\gamma_{k}$ is a weighted sum of the average rate and the harvested power, $\Phi\triangleq\{\bm{\rho}=[\rho_{1},\cdots,\rho_{K}]^{T}|\rho_{k}\in(0,1],\forall k\}$ is the feasible set of power splitters. The utility function $g\left(\overline{\eta}_{k}\right)$ is a continuously differentiable and concave function of $\overline{\eta}_{k}$. Moreover, $g\left(\overline{\eta}_{k}\right)$ is non-decreasing w.r.t. $\overline{\eta}_{k},$ and its derivative $\nabla_{\overline{\eta}_{k}}g\left(\overline{\eta}_{k}\right)$ is Lipschitz continuous.

Online Optimization Algorithm
=============================

In this section, we propose a MO-SSCA algorithm to solve the mixed-timescale stochastic non-convex optimization problem $\mathcal{P}$, and summarize it in Algorithm 1. In MO-SSCA, we focus on a coherence time of channel statistics, where the time is divided into $T_{f}$ frames and each frame consists of $T_{s}$ time slots. At beginning, the BS initializes the MO-SSCA algorithm with power splitter $\bm{\rho}^{0}$ and a weight vector $\bm{v}^{0}$. In subsequent, $\bm{\rho}$ and $\bm{v}$ are updated once at the end of each frame. Then we elaborate the implementation details of the iteration of the MO-SSCA algorithm at the $t$-th frame.

**Input:** [$\left\{ \alpha^{t}\right\} $, $\left\{ \beta^{t}\right\} $.]{}

**Initialize:**[ $\boldsymbol{\rho}^{0}\in\Phi$; $\boldsymbol{v}^{0}=\left[1,...,1\right]^{T}$, $t=0$.]{}

**Step 1** [(]{}**Short-term optimization at each time slot**[ $i\in\mathcal{T}_{t}$):]{}

****

****

****

****

****

****

****

Short-term FP-BCD Algorithm\[subsec:short-term algorithm\]
----------------------------------------------------------

At time slot $i\in\mathcal{T}_{t}\triangleq[tT_{s}+1,(t+1)T_{s}]$ within the $t$-th frame, BS obtains the estimated channel $\mathbf{\hat{H}}(i)$ by uplink channel training. Based upon the current $\bm{\rho}^{t}$, $\bm{v}^{t}$, and $\mathbf{\hat{H}}(i)$, 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 $$\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right):\max_{\mathbf{\mathbf{F}}\in\varLambda}\:\sum_{k=1}^{K}v_{k}\overline{\eta}_{k}^{\circ}(\rho_{k},\mathbf{F}|\bm{\hat{h}}_{k}),$$ where $\overline{\eta}_{k}^{\circ}(\rho_{k},\mathbf{F}|\bm{\hat{h}}_{k})\triangleq\overline{r}_{k}^{\circ}(\rho_{k},\mathbf{F}|\bm{\hat{h}}_{k})+\text{\ensuremath{\gamma_{k}}}\overline{e}_{k}^{\circ}(\rho_{k},\mathbf{F}|\bm{\hat{h}}_{k})$ , and $\mathcal{P}_{2}\left(\bm{\rho}^{t},\bm{v}^{t},\mathbf{\hat{H}}(i)\right)$ is solved at time slot $i\in\mathcal{T}_{t}$.

Since the objective function $\overline{\eta}_{k}^{\circ}(\rho_{k},\mathbf{F}|\bm{\hat{h}}_{k})$ contains expectation operators, it d

However, solving problem $\mathcal{P}_{3}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ is still challenging due to the nonlinear fractional term in $\hat{r}_{k}$ and coupling in the power constraint. To this end, we apply the Lagrangian dual transform method [@Kaiming_TSP18_FP] to recast problem $\mathcal{P}_{3}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ into a more tractable yet equivalent form, using the following lemma.

\[lem:transformation-F\] The optimal digital beamforming $\mathbf{\mathbf{F}}^{\ast}$ solves the problem in (\[lem:transformation-F\]) if and only if it solves $$\max_{\mathbf{\mathbf{\mathbf{F}}}\in\varLambda}\:\frac{1}{N}\sum_{n=1}^{N}\sum_{k=1}^{K}f\left(\mathbf{F},q_{k}^{n},\bm{\phi}_{k}^{n}\right)\label{eq:trans-short-sub}$$ where $\Gamma_{k}^{n}\triangleq\sum_{m=1}^{K}|\tilde{\bm{h}}_{k,n}^{H}\bm{f}_{m}|^{2}$ with $\tilde{\bm{h}}_{k,n}\triangleq\bm{\phi}_{k}^{n}+\bm{\hat{h}}_{k}$, and $q_{k}^{n}\triangleq\frac{\rho_{k}|\tilde{\bm{h}}_{k,n}^{H}\bm{f}_{k}|^{2}}{\rho_{k}\left(\sum_{m\neq k}^{K}|\tilde{\bm{h}}_{k,n}^{H}\bm{f}_{m}|^{2}+\sigma_{k}^{2}\right)+\delta_{k}^{2}}$ is the optimal auxiliary variable introduced for each ratio term.

In subsequent, we use the complex quadratic transformation [@Kaiming_TSP18_FP] to equivalently recast problem (\[eq:trans-short-sub\]) as where $$\begin{aligned}
 & \hat{r}_{k}\left(\mathbf{F},q_{k}^{n},\bm{\phi}_{k}^{n},w_{k}^{n}\right)\triangleq\sqrt{v_{k}\text{\ensuremath{\rho_{k}}}(1+q_{k}^{n})}\mathrm{Re}\left\{ \bm{f}_{k}^{H}\tilde{\bm{h}}_{k,n}\left(w_{k}^{n}\right)^{H}\right\} \\
 & +\left(w_{k}^{n}\right)^{H}w_{k}^{n}\left(\rho_{k}(\Gamma_{k}^{n}+\sigma_{k}^{2})+\delta_{k}^{2}\right)-q_{k}^{n}+\log_{2}(1+q_{k}^{n})\text{.}\end{aligned}$$ $\bm{q}=[\bm{q}_{1}^{T},\cdots,\bm{q}_{K}^{T}]^{T}$ with $\bm{q}_{k}=[q_{k}^{1},\cdots,q_{k}^{N}]^{T}$, and $\bm{w}=[\bm{w}_{1}^{T},\cdots,\bm{w}_{K}^{T}]^{T}$ with $\bm{w}_{k}=[w_{k}^{1},\cdots,w_{k}^{N}]^{T}$ is the auxiliary variable vector. Observing that the constraints are separable with respect to the three blocks of variables, i.e., $\bm{q},$ $\bm{w}$, and $\mathbf{F}$, we shall focus on designing a algorithm to find a stationary point of problem (\[eq:final-sub\]), and summarize it in Algorithm 2. For problem (\[eq:final-sub\]), this amounts to the following steps:

**Input:**[ $\boldsymbol{v},\boldsymbol{\rho}$, $\mathbf{\hat{H}}$, the sample number $N$.]{}

**Initialization:**[ Initialize $\mathbf{F}$ to feasible values.]{}

****

**Step 1 : Update**[ $\boldsymbol{q}$ according to (\[eq:update-q\]).]{}

**Step 2 : Update**[ $\boldsymbol{w}$ according to (\[eq:update-W\]).]{}

**Step 3 : Update**[ $\mathbf{F}$ by solvi]{}

****

### Optimization of $\boldsymbol{q}$

The optimal $\bm{q}^{\ast}$ is given by $$\left(q_{k}^{n}\right)^{\ast}=\frac{\rho_{k}|\tilde{\bm{h}}_{k,n}^{H}\bm{f}_{k}|^{2}}{\rho_{k}(\Gamma_{k}^{n}-|\tilde{\bm{h}}_{k,n}^{H}\bm{f}_{k}|^{2}+\sigma_{k}^{2})+\delta_{k}^{2}}.\label{eq:update-q}$$

### Optimization of $\boldsymbol{w}$

By applying the first-order optimal condition, the optimal $\bm{w}^{\ast}$ admits a closed-form solution as: $$\left(w_{k}^{n}\right)^{\ast}=\left(\rho_{k}(\Gamma_{k}^{n}+\sigma_{k}^{2})+\delta_{k}^{2}\right)^{-1}\!\!\!\!\!\sqrt{\rho_{k}v_{k}(1+q_{k}^{n})}\tilde{\bm{h}}_{k,n}^{H}\bm{f}_{k}.\label{eq:update-W}$$

### Optimization of $\mathbf{F}$

Long-term Optimization
----------------------

Before the end of $t$-th frame, device $k$ obtains a full channel sample $\bm{\hat{h}}_{k}^{t}\triangleq\bm{\hat{h}}_{k}(tT_{s}+1)$ and channel error sample $\bm{\phi}_{k}^{t}$. Based on $\bm{\hat{h}}_{k}^{t}$, $\bm{\phi}_{k}^{t}$ and $\mathbf{F}^{t}(i),\forall i\in\mathcal{T}_{t}$, we preserve the partial concavity of the original function and add the proximal regularization, to construct the concave surrogate function $\overline{g}^{t}(\rho_{k})$, resulting in the following

$$\overline{g}^{t}(\rho_{k})=g(\tilde{\eta}_{k}^{t})+\left(u_{k}^{t}\right)^{T}(\rho_{k}-\rho_{k}^{t})-\tau|\rho_{k}-\rho_{k}^{t}|^{2},\label{upsurrgate}$$

where $\tau>0$ is a postive constant; the recursive approximation of the weighted sum of the data rate and the harvested power $\tilde{\eta}_{k}$ is given by $$\tilde{\eta}_{k}^{t}=(1-\alpha_{t})\tilde{\eta}_{k}^{t-1}+\frac{\alpha_{t}}{N}\sum_{n=1}^{N}\sum_{i\in\mathcal{T}_{t}}\frac{\hat{\eta}_{k}^{\circ}(\rho_{k}^{t},\mathbf{F}^{t}(i)|\bm{\hat{h}}_{k}^{i},\bm{\phi}_{k}^{n}(i))}{|\mathcal{T}_{t}|},$$ with $\tilde{\eta}_{k}^{-1}=0$, and $\alpha_{t}\in\left(0,1\right]$ is a step-sizes sequence to be properly chosen; the recursive approximation of the partial derivative $\nabla_{\rho_{k}}g\left(\tilde{\eta}_{k}\right)$ is given by $$u_{k}^{t}=(1-\alpha_{t})u_{k}^{t-1}+\alpha_{t}J_{\rho_{k}}\left(\rho_{k}^{t},\mathbf{F}^{t}(i)|\bm{\hat{h}}_{k}^{t},\bm{\phi}_{k}^{t}\right)\nabla_{\overline{\eta}_{k}}g\left(\tilde{\eta}_{k}^{t}\right),$$ with $u_{k}^{-1}=0$, $J_{\rho_{k}}\left(\rho_{k}^{t},\mathbf{F}^{t}(i)|\bm{\hat{h}}_{k}^{t},\bm{\phi}_{k}^{t}\right)$ is the gradient of $\hat{\eta}_{k}^{\circ}\left(\rho_{k}^{t},\mathbf{F}^{t}(i)|\bm{\hat{h}}_{k}^{t},\bm{\phi}_{k}^{t}\right)$ w.r.t. $\rho_{k}$ at $\rho_{k}=\rho_{k}^{t}$ and $\mathbf{F}=\mathbf{F}^{t}(i)$. Moreover, the weight vector $\boldsymbol{v}$ is updated as

$$v_{k}^{t+1}=\left(1-\beta_{t}\right)v_{k}^{t}+\beta_{t}\bar{v}_{k}^{t},\label{eq:updatemu}$$

with $v_{k}^{t}=\nabla_{\overline{\eta}}g_{k}\left(\tilde{\eta}_{k}^{t}\right)$, where $\beta_{t}\in\left(0,1\right]$ is a step-sizes sequence satisfying $\sum_{t}\beta_{t}=\infty$, $\sum_{t}\left(\beta_{t}\right)^{2}<\infty$. Moreover, the optimal power splitting ratio for device $k$ can be obtained by solving the following quadratic optimization problem, i.e.,

$$\begin{aligned}
\underset{\rho_{k}\in\Phi}{\text{max}}\: & \bar{g}^{t}\left(\rho_{k}\right).\label{eq:Pitert}\end{aligned}$$

By applying the first-order optimality condition, it yields the *Remark 1* : Note that the stationary weight vector $v_{k}^{*}=\nabla_{\overline{\eta}_{k}}g_{k}\left(\tilde{\eta}_{k}^{\ast}\right)$ has captured the nature of the utility function. However, it is difficult to obtain $\boldsymbol{v}^{*}$, since it in turn depends on the stationary solution $\boldsymbol{\rho}^{*}$. Therefore, the basic idea of the proposed algorithm is to iteratively update the long-term variable $\boldsymbol{\rho}^{t}$ and the weight vector $\boldsymbol{v}^{t}$ such that $\boldsymbol{\rho}^{t}$ and $\boldsymbol{v}^{t}$ converge to a stationary solution $\boldsymbol{\rho}^{*}$ and the corresponding stationary weight vector $\boldsymbol{v}^{*}$, respectively.

The following theorem states that Algorithm 2 converges to a stationary point of $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ up to certain convergence error which vanishes to zero exponentially as $N\rightarrow\infty$.

\[Convergence of Algorithm 2\]\[thm:Convergence-of-Short-term\] Suppose problem $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ has a discrete set of stationary points, denoted by $\mathcal{F}^{*}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$. Let $\mathbf{F}^{N}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ denote the limiting point of the sequence generated by Algorithm 2 with input parameter $\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}$ and sample number $N$. Then for every small positive number $\epsilon>0$, there exist positive constants $\hat{a}(\epsilon)$ and $\hat{b}(\epsilon)$, independent of $N$, such that $$\mathrm{Pr}\left\{ \min_{\mathbf{F}\in\mathcal{F}^{*}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)}\!\!\!\|\mathbf{F}^{N}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)\!\!-\!\!\mathbf{F}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)\!\|\!\geq\!\epsilon\!\right\} \!\leq\!p(\epsilon,N),$$ for $N$ sufficiently large, where $p(\epsilon,N)\triangleq\hat{a}(\epsilon)e^{-N\hat{b}(\epsilon)}$.

Specifically, the proposed FP-BCD algorithm falls in the MM framework and similar proof is provided in [@Kaiming_arxiv19_D2D]. From Theorem 4.4 in [@Jacobson_TIP07_MM], every limiting point $\mathbf{F}^{N}$ of sequence generated by the short-term FP-BCD algorithm is a stationary point of problem $\mathcal{P}_{3}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$, where problem $\mathcal{P}_{3}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ is the sample average approximation of problem $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ with $N$ samples. As stated in [@Shapiro_SIAM09_SAA], problem $\mathcal{P}_{3}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ is equivalent to problem $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ w.p.1 when $N$ approaches to infinity, due to the classical law of large number for random functions. That is to say, as $N\rightarrow\infty$, any stationary point of $\mathcal{P}_{3}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ is also a stationary point of problem $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ w.r.1. When $N$ is finite, Algorithm 2 converges to approximate stationary points of problem $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ with the exponential convergence rate $\hat{a}(\epsilon)e^{-N\hat{b}(\epsilon)}$. This is consequence of [@SAA_convergenceRate], 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 $\mathcal{P}_{2}\left(\bm{\rho},\boldsymbol{v},\mathbf{\hat{H}}\right)$ satisfies the aforementioned assumption (a) and (b). This completes the proof.

Based on Theorem 2, the convergence of the proposed MO-SSCA algorithm is summarized in the following theorem.

\[Convergence of the Algorithm 1\]\[thm:Convergence-of-long\]Given problem (\[eq:originalP\]), suppose that $\tau>0$ in (\[upsurrgate\]) and the step-sizes $\{\alpha_{t}\}$ and $\{\beta_{t}\}$ are chosen so that

1.  $\alpha_{t}\rightarrow0$, $\frac{1}{\alpha_{t}}\leq O\left(t^{\kappa}\right)$ for some $\kappa\in\left(0,1\right)$, $\sum_{t}\left(\alpha_{t}\right)^{2}<\infty,$

2.  $\beta_{t}\rightarrow0$, $\sum_{t}\beta_{t}=\infty$, $\sum_{t}\left(\beta_{t}\right)^{2}<\infty$,

3.  $\lim_{t\rightarrow\infty}\beta_{t}/\alpha_{t}=0$.

Let $\{\bm{\rho}^{t},\boldsymbol{v}^{t},\mathbf{F}^{N}(i),\forall i\in\mathcal{T}_{t}\}_{t=1}^{\infty}$ denote the sequence of iterates generated by Algorithm 1, where $\mathbf{F}^{N}(i)\triangleq\mathbf{F}^{N}\left(\bm{\rho}^{t},\boldsymbol{v}^{t},\mathbf{\hat{H}}(i)\right),i\in\mathcal{T}_{t}$. Then every limit point $\bm{v}^{*},\bm{\rho}^{*}$ of $\left\{ \bm{v}^{t},\bm{\rho}^{t}\right\} _{t=1}^{\infty}$ almost surely satisfies

$$\bm{v}^{\ast}=\nabla_{\overline{\bm{\eta}}}g\left(\overline{\bm{\eta}}^{\ast}\right),\label{eq:stationary-weight}$$

$$(\boldsymbol{\rho}-\boldsymbol{\rho}^{*})^{T}\nabla_{\bm{\rho}}^{T}g\left(\bm{\overline{\eta}}\left(\bm{\rho}^{\ast},\bm{\Theta}^{N}\left(\bm{v}^{\ast},\bm{\rho}^{\ast}\right)\right)\right)\leq0,\forall\boldsymbol{\rho}\in\mathbf{\Phi},\label{eq:ooutconv}$$

where $\bm{\Theta}^{N}\left(\bm{v}^{\ast},\bm{\rho}^{\ast}\right)\triangleq\left\{ \mathbf{F}^{N}(\bm{v}^{\ast},\bm{\rho}^{\ast},\mathbf{\hat{H}}),\forall\mathbf{\hat{H}}\right\} $, and $\overline{\bm{\eta}}^{\ast}\triangleq\overline{\bm{\eta}}\left(\bm{\rho}^{\ast},\bm{\Theta}^{N}\left(\bm{v}^{\ast},\bm{\rho}^{\ast}\right)\right).$ Moreover, $\forall\mathbf{F}\in\varLambda\text{,}$ it satisfies $$\left(\mathbf{F}\!-\!\mathbf{F}^{N}(i)\right)^{T}\!\!\!\!J_{\mathbf{F}}\left(\bm{\rho}^{\ast},\mathbf{F}^{N}(i)|\hat{\bm{H}}(i)\right)\!\nabla_{\overline{\bm{\eta}}}g\left(\overline{\bm{\eta}}^{\ast}\right)\!\leq e\left(N\right),\label{eq:inerconv}$$ where $J_{\mathbf{F}}\left(\bm{\rho}^{\ast},\mathbf{F}^{N}(i)|\hat{\bm{H}}(i)\right)$ is the Jacobian matrix of the vector $\bm{\overline{\eta}}^{\circ}\triangleq[\overline{\eta}_{1}^{\circ},\cdots,\overline{\eta}_{K}^{\circ}]^{T}$ w.r.t. $\mathbf{F}$ at $\bm{\rho}=\bm{\rho}^{\ast}$ and $\mathbf{F}=\mathbf{F}^{N}(i)$, and $e\left(N\right)$ satisfies $\lim_{N\rightarrow\infty}e\left(N\right)=0$ almost surely.

Based on Theorem 2, Theorem \[thm:Convergence-of-long\] can be proven by a similar approach in [@Anliu_tsp19_THCF]. Thus, we omit the details due to the limited space.

According to equation (\[eq:inerconv\]) in Theorem 3, it implies that the short-term solution $\mathbf{F}^{N}(i)$ found by Algorithm 2 must satisfy the stationary condition approximately with certain error $e\left(N\right)$ that converges to zero exponentially as $N\rightarrow\infty$. Moreover, the limiting point $(\bm{v}^{\text{\ensuremath{\ast}}},\bm{\rho}^{\ast})$ generated by Algorithm 1 also satisfies the stationary conditions in (\[eq:stationary-weight\]) and (\[eq:ooutconv\]), respectively. Thus, Algorithm 1 converges to stationary solutions of the mixed-timescale optimization problem $\mathcal{P}$. Note that since $e\left(N\right)$ converges to zero exponentially, Algorithm 2 with a small $N$ can already achieve a good performance and avoids excessive computational complexity.

Simulation Results and Discussions\[sec:Simulation-Results-and\]
================================================================

We consider a single-cell of radius $100$ m, where BS is equipped with $64$ antennas. There are 12 devices randomly distributed in the cell. We adopt a geometric channel model with a *half-wavelength space* ULA for simulations [@Anliu_tsp19_THCF]. The channel between BS and device $k$ is given by $\boldsymbol{h}_{k}=\sum_{i=1}^{N_{p}}\varepsilon_{k,i}\boldsymbol{\textrm{a}}\left(\varphi_{k,i}\right)$, where $\boldsymbol{\textrm{a}}\left(\varphi\right)$ is the array response vector, $\varphi_{k,i}$’s are Laplacian distributed with an angular spread $\sigma_{\textrm{AS}}=10$, $\varepsilon_{k,i}\sim\mathcal{CN}\left(0,\sigma_{k,i}^{2}\right)$, $\sigma_{k,i}^{2}$ are randomly generated from an exponential distribution and normalized such that $\sum_{i=1}^{N_{p}}\sigma_{k,i}^{2}=G_{n,k}$, $G_{k}$ is the average channel gain determined by the pathloss model $30.6+36.7\log10\left(\textrm{ds}_{k}\right)$ [@3gpp_Rel9], and $\textrm{ds}_{k}$ is the distance between BS and device $k$ in meters. We consider $N_{p}=6$ channel paths for each device. The transmit power budget for BS is $P_{\mathrm{max}}=10$ dBm. We set $N=200$, $\omega_{k}^{2}=-40$ dB, $\sigma_{k}^{2}=-60$ dBm and $\delta_{k}^{2}=-50$ dBm. There are $T_{s}=10$ time slots in each frame and the slot size is 2 ms. The coherence time for the channel statistics is assumed to be 10 s. We use the average sum utility $g\left(\hat{\bm{\eta}}\right)=\sum_{k=1}^{K}\hat{\eta}_{k}$ 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 [@QJS_2014twc_JBPS]; 2) zero-forcing (ZF) scheme, which is obtained by fixing the ZF beamformer [@Dong_2018globecom_ZFSWIPT]. The power splitters of both MRT and ZF scheme are obtained by the long-term optimization.

In Fig \[fig:varSNR\], we plot the utility performance versus the signal-to-noise 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 multi-device interference to achieve better tradeoff between the average ergodic rate and the average harvested power,

In Fig \[fig:varK\], we plot the utility performance versus the number of devices $K$. We observe that the proposed MJBP scheme achieves significant gain over MRT scheme and ZF scheme, which demonstrates the importance of mixed-timescale joint optimization. Moreover, as the number of devices $K$ increases, the performance gap between the proposed MJBP scheme and other competing schemes becomes larger.

Conclusion
==========

In this letter, we considered mixed-timescale 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 MO-SSCA algorithm to find stationary solutions of the mixed-timescale non-convex stochastic optimization problem. Simulations verify that the proposed MJBP scheme achieves significant gain over existing schemes.

[^1]: This work was supported by the Science and Technology Program of Shenzhen, China, under Grant JCYJ20170818113908577, and the National Natural Science Foundation of China under Project No. 61571383. The work of An Liu was supported by the China Recruitment Program of Global Young Experts.

    Xihan Chen, An Liu and Min-Jian Zhao are with the College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou 310027, China (e-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>).

    Hei Victor Cheng, and Kaiming Shen are with the Electrical and Computer Engineering Department, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>, <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>).