Cooperative Fair Throughput Maximization in a Multi-Cluster Wireless Powered Network

The operational lifetime of a communication network employing battery powered wireless devices can be enhanced by employing recent advances in Wireless Power Transfer (WPT) and harvesting technologies in which the wireless devices receive power/energy from dedicated wireless power transmitters [1]. One important application of WPT is in Wireless Powered Communication Network (WPCN), where the wireless devices harvest the wireless energy during the downlink communication in order to power the uplink phase. In a WPCN, the energy transmitter and the information receiver are placed either at the same location as a Hybrid Access Point (HAP) or at separate locations [2]. The authors of [3] have proposed a Harvest-Then-Transmit (HTT) protocol for a WPCN consisting of ${K}$ single-antenna users and a single-antenna HAP. They have shown that in a WPCN which employs a HAP, an inherent unfair “doubly near-far” phenomenon causes the users which are far from the HAP to harvest less energy in the downlink where they require more energy to transmit their information in the uplink. Several methods have been proposed in the literature to overcome the doubly near-far phenomenon [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

In [4, 5, 6, 7, 8, 9, 10], the user cooperation is proposed for a three-node WPCN assuming that all nodes are equipped with single-antenna. In the proposed methods in [4] and [5], the closer user to the HAP transmits the signal of the more distant user along with its own signal to the HAP and the sum throughput of the network is maximized to find optimal time and power allocations. The method in [4] is extended in [6] for a system employing an Intelligent Reflecting Surface (IRS). In [7], the max-min throughput fairness was considered for a single-pair WPCN, where both users harvest the transmitted energy from an energy source and then help each other to send their information to a destination node. The outage performance is studied for a single-user WPCN in [8] and [9], where the user sent its information to the HAP via a dedicated relay. A single-pair is considered in [10] with cooperation in both downlink and uplink phases.

Several references employ multiple antennas energy transmitter and design the Energy Beamforming (EB) to focus the transmitted energy toward the specific receivers [11, 12, 13, 14, 15]. The throughput maximization is investigated for a WPCN in [11, 12, 13] consisting of two cooperating single-antenna users and a multiple antennas HAP. The throughput of a single-user WPCN is calculated in [14] employing a dedicated single-antenna relay, where a multiple antennas energy source first powers the single-antenna user and the relay. Then, the source transmits its information to a single-antenna destination node with the help of the relay. In [15] a three-node cooperative wireless powered scenario is considered consisting of a source, destination and a dedicated relay which all have multiple antennas. The system in [15] is designed to maximize the throughput.

The above works have considered the WPCN only in the context of a single-pair of users. This paradigm is extended in [16, 17, 18] to consider multiple users in a single-cluster. In [16], all single-antenna users transmit their signals to the single-antenna cluster head (CH) and then the CH transmits the user signals as well as its own signal to the multiple antennas HAP via Time-Division-Multiple-Access (TDMA) protocol. The transmit time allocations and EB matrix were designed to improve the throughput fairness among the users. The model in [16] is extended in [17] to a cognitive WPCN by considering a secondary single-cluster WPCN and a primary point-to-point communication. In [18], a fairness-aware transmission is employed for a single-cluster WPCN where each user sends its information to all the other users and cooperatively send their signals to the HAP using TDMA.

Grouping of users into clusters may practically lead to significant improvement in rate and energy [21]. Similar order of enhancement is obtained in the throughput fairness for a two-cluster cooperating WPCN employing two multiple antennas HAPs and multiple single-antenna users in [19] where all the information transmissions in [19] are divided in time domain to avoid inter-cluster interference which increases the user latency. The sum throughput maximization is studied in [20] for a multi-cluster non-cooperating WPCN where HAP and all users have single-antenna. To the best of our knowledge, the design of multi-cluster cooperating WPCN with inter-cluster interference has not been addressed in the literature. In this paper, we evaluate the minimum and sum throughput enhancement for a multi-cluster Multiple-Input Multiple-Output (MIMO) WPCN with user cooperation. We consider the general MIMO case and manage the inter-cluster interference where the main advantages of our proposed scheme are: (i) lower user latency in comparison with [19]; (ii) lower path loss attenuation contrary to [20]. Precisely, the main contributions of this manuscript are summarized as follows:

• •

  We propose a new multi-cluster WPCN under HTT protocol with user cooperation, where the HAP as well as users have multiple antennas all operating in the same frequency band. As shown in Fig. 1, the CH assists its cluster members to transmit their informations to the HAP more efficiently [22]. Moreover, in the downlink phase, the HAP employs beamforming technique to transfer energy to the users. In the uplink phase, the users in each cluster transmit their signals to the HAP and to their CH. Finally, the CHs first relay the signals of their cluster members and then transmit their own information signals to the HAP. Note that we must deal with the inter-cluster interference in this model.

• •

  We design the EB matrix, transmit covariance matrices of the users and time allocations among energy transfer and cooperation phases in order to maximize max-min and sum throughputs of the network. These problems are non-convex and consequently are hard to solve. We propose a proper minorizer and develop algorithms (using Alternating Optimization (AO) and minorization-maximization (MM) techniques) to solve these problems. We recast the main sub-problems for max-min and sum throughput cases as Second Order Cone Programming (SOCP) and Quadratic Constraint Quadratic Programming (QCQP).

• •

  We also investigate the impact of imperfect Channel State Information (CSI) as well as the non-linearity in the Energy Harvesting (EH) circuits in our design problem and extend the proposed design to the case of deterministic energy signal.

• •

  Via numerical examples, we explore the impact of cooperation and non-linear EH circuits as well as optimal location of the CHs.

Organization: We present the multi-cluster cooperative model for WPCN in Section II. We analyze the max-min throughput optimization problem and present our proposed design method in Section III. We consider the special case of deterministic energy signal in Section IV and investigate the sum throughput maximization problem. Numerical results in Section V reveal the efficiency of the proposed method. We finally draw conclusions in Section VI.

Notation: Bold lowercase (uppercase) letters are used for vectors (matrices). The notations $\mathbb{E}[\cdot]$, $\Re\{\cdot\}$, $\|\cdot\|_{2}$, ${(\cdot)^{{T}}}$, $(\cdot)^{{H}}$, $\mbox{tr}\{\cdot\}$, $\mathbf{\lambda}_{\textrm{max}}(\cdot)$, $\mathbf{\lambda}_{\textrm{min}}(\cdot)$, $\textrm{vec}(\cdot)$ and ${\nabla}^{2}_{{\mathbf{x}}}f(\cdot)$ indicate statistical expectation, real-part, $l_{2}$-norm of a vector, vector/matrix transpose, vector/matrix Hermitian, trace of a square matrix, principal eigenvalue of a Hermitian matrix, minimum eigenvalue of a Hermitian matrix, column-wise stacking of the elements of a matrix and the Hessian of the twice-differentiable function with respect to (w.r.t.) $\mathbf{x}$, respectively. The symbol $\otimes$ stands for the Kronecker product of two matrices/vectors. We denote $\mathcal{CN}(\mathbf{\omega},\mathbf{\Sigma})$ as the circularly symmetric complex Gaussian (CSCG) distribution with mean $\mathbf{\omega}$ and covariance $\mathbf{\Sigma}$. The symbol $\mathbb{R}_{+}$ represents non-negative real numbers, ${\mathbb{C}}^{N\times N}$ (${\mathbb{C}}^{N}$) is the set of ${N\times N}$ (${N\times 1}$) complex matrices (vectors), $\mathbb{S}_{++}^{N}\subseteq{\mathbb{C}}^{N\times N}$ denotes the set of positive definite matrices, and $\mathbf{I}_{N}$ is the $N\times N$ identity matrix. Finally, the notation $\mathbf{A}\succ\mathbf{B}$ ($\mathbf{A}\succeq\mathbf{B}$) means that $\mathbf{A}-\mathbf{B}$ is positive (semi)definite.

We consider a multi-cluster MIMO WPCN consisting of a HAP and ${KN}$ cooperating users denoted by $U_{k}^{n},1\leq k\leq{K},1\leq n\leq{N}$, in ${K}$ clusters as shown in Fig. 1. The HAP and user $U_{k}^{n}$ have ${M}_{h}$ and ${M}_{k,n}$ antennas, respectively, and all of them operate in the same frequency band. We assume that the HAP has a sustained power supply, whereas the users have no fixed batteries [4]. Therefore, the HAP shall first broadcast radio frequency (RF) energy to enable users to transmit their information to the HAP by using the energy collected during the downlink phase. We assume that the CH of $k$th cluster denoted by $U_{k}^{N}$ (which can be one of users) is tasked to relay the information of its cluster members $U_{k}^{n},1\leq n\leq{N-1}$. We further assume that the channels are reciprocal and follow a flat block fading model. We denote the channel response matrix from HAP to the user $U_{k}^{n}$ and the channel matrix between $U_{i}^{n}$ and the CH, i.e., $U_{k}^{N}$, respectively by $\mathbf{H}_{k,n}\in{\mathbb{C}}^{{M}_{k,n}\times{M}_{h}},1\leq k\leq K,1\leq n\leq N$ and $\mathbf{G}_{k,n,i}\in{\mathbb{C}}^{{M}_{k,N}\times{M}_{i,n}},1\leq k,i\leq K,1\leq n\leq N-1$. We assume that the entries of $\mathbf{H}_{k,n}$ and $\mathbf{G}_{k,n,i}$ are independent random variables with zero mean and variances $\sigma^{2}_{{h}_{k,n}}$ and $\sigma^{2}_{g_{k,n,i}}$, respectively.

Fig. 2 shows the timing protocol of the proposed WPCN which is an enhancement of the HTT protocol in [3]. In this scheme, one complete operation of the system takes $T$ seconds and is divided into four time slots. The first phase/time slot has a fixed duration of $\tau_{1}$ and is allocated to the channel estimation and user clustering [16]. In this phase, users (except CHs) transmit pilot signals to the HAP and CHs. We model the Linear Minimum Mean Square Error (LMMSE) estimation of $\mathbf{H}_{k,n}$ (at HAP) and $\mathbf{G}_{k,n,i}$ (at CHs) by [23]

where $\widehat{\mathbf{H}}_{k,n}$ and $\widehat{{\mathbf{G}}}_{k,n,i}$ are the estimates of the channel matrices $\mathbf{H}_{k,n}$ and ${\mathbf{G}}_{k,n,i}$, respectively. Moreover, the channel estimation errors $\Delta\mathbf{H}_{k,n}$ and $\Delta{\mathbf{G}}_{k,n,i}$ are uncorrelated with $\widehat{\mathbf{H}}_{k,n}$ and $\widehat{{\mathbf{G}}}_{k,n,i}$. We also assume that the elements of $\widehat{\mathbf{H}}_{k,n}$, $\widehat{{\mathbf{G}}}_{k,n,i}$, $\Delta\mathbf{H}_{k,n}$ and $\Delta{\mathbf{G}}_{k,n,i}$ are independent random variables with variances $\sigma^{2}_{{\widehat{h}}_{k,n}}$, $\sigma^{2}_{{\widehat{g}}_{k,n,i}}$, $\sigma^{2}_{{h,\Delta}_{k,n}}$ and $\sigma^{2}_{{g,\Delta}_{k,n,i}}$, respectively. The properties of the LMMSE estimator allows to write the following expressions

where the parameters $0\leq{\rho}_{{h}_{k,n}}\leq 1$ and $0\leq{\rho}_{g_{k,n,i}}\leq 1$ indicate the estimation accuracy. The CHs forward the estimates to the HAP and therefore, we assume that the HAP has access to these CSI estimates [16], using which users are clustered by the HAP as a network coordinator (see [20, 24, 25] for more details).

During the next downlink phase $\tau_{2}$, the HAP emits power to the all users. In the next two uplink phases, the users transmit their information signals to the HAP using their harvested energy. In the third phase, the users of each cluster transmit the information signals to their CH, i.e., $U_{k}^{N}$, and the HAP via a time division scheme where $\tau_{3}=\sum_{n=1}^{N-1}\tau_{3,n}$. More precisely, in time slot $\tau_{3,n}$, the $n$th users from all clusters simultaneously transmit to the HAP and their CH. During the $\tau_{4,n}$ sub-interval of the fourth phase $\tau_{4}=\sum_{n=1}^{N}\tau_{4,n}$, the CHs transmit the decoded signals of $n$th cluster members to the HAP in a similar time division scheme for all $n\neq N$. During $\tau_{4,N}$, the CHs transmit their own signals to the HAP. All users of each cluster, i.e., $U^{n}_{k},1\leq n\leq N$ could benefit from this cooperation [22]. This is obvious for $U^{n}_{k},1\leq n\leq N-1$ but for the CH, i.e., $U^{N}_{k}$, this claim can be verified using the fact that this cooperation allows the HAP to allocate more time for information transmission (i.e., $\tau_{3}+\tau_{4}$) instead of energy transmission (i.e., $\tau_{2}$) and therefore the CH’s throughput loss (due to cooperation) could be compensated by longer uplink time [4]. By dedicating $T$ as the total upper bound operation time of one block, we obtain the following constraint for different phases [16]

We now model the signal and system in more details and obtain the link throughputs. The HAP transfers its energy signal $\mathbf{x}_{0}\in{\mathbb{C}}^{{M}_{h}}$ with the EB matrix $\mathbf{Q}=\mathbb{E}[\mathbf{x}_{0}{\mathbf{x}_{0}}^{{H}}]\succeq\mathbf{0}$ during the downlink phase $\tau_{2}$ to the users. We consider $\mathbf{x}_{0}$ as a random vector and design its EB matrix. We discuss the special case of deterministic $\mathbf{x}_{0}$ severalty in Subsection IV-A.. The transmit power of the HAP is constrained by $p_{0}$, i.e.,

The received signal at $U_{k}^{n}$ during $\tau_{2}$ for all $k,n$ is expressed as

where $\mathbf{z}^{(\textrm{HAP-U})}_{k,n}\sim\mathcal{CN}\left(\mathbf{0},{\mathbf{L}}^{(\textrm{HAP-U})}_{k,n}\right)$ is the received noise with ${\mathbf{L}}^{(\textrm{HAP-U})}_{k,n}\in\mathbb{S}_{++}^{{M}_{k,n}}$. Following [26], we consider a non-linear circuit model for the EH circuit in which the curve fitting procedure is performed in logarithmic scale to obtain more accurate results in low power regimes [27]. Therefore, the energy harvested by $U_{k}^{n}$ (by ignoring the harvested energy due to the receiver noise [16]) for the non-linear EH model (eq. (21) of [27] in linear scale) for all $k,n$ becomes

where $a$, $b$ and $c$ are the curve fitting parameters and

During the uplink sub-interval $\tau_{3,n}$, the $n$th users $U_{k}^{n}$ of all clusters transmit their independent message signals to the HAP and their CH by using the energy harvested in the previous phase. Let $\mathbf{V}_{k,n}\in{\mathbb{C}}^{{M}_{k,n}\times{{d}_{k,n}}}$ denote the linear precoder matrix used at $U_{k}^{n}$ to convert the message stream $\mathbf{m}_{k,n}\in{\mathbb{C}}^{{d}_{k,n}}$, viz. a Gaussian random vector with zero mean and covariance matrix $\mathbf{I}_{{d}_{k,n}}$, to the transmitted vector ${\mathbf{x}}_{k,n}\in{\mathbb{C}}^{{M}_{k,n}}$ with ${\mathbf{x}}_{k,n}=\mathbf{V}_{k,n}\mathbf{m}_{k,n}$. The total consumed energy at $U_{k}^{n}$ in time slot $\tau_{3,n}$ shall be less than the harvested energy. Thus, we must satisfy the following constraint for all $k$ and $n\neq N$

where $P_{c_{k,n}}$ is the constant circuit power consumption, $\eta_{k,n}$ accounts for power amplifier efficiency and ${\mathbf{{S}}}_{k,n}=\mathbb{E}[{\mathbf{x}}_{k,n}{\mathbf{x}}^{{H}}_{k,n}]=\mathbf{V}_{k,n}{\mathbf{V}^{{H}}_{k,n}}\in{\mathbb{C}}^{{M}_{k,n}\times{M}_{k,n}}$ is the information transmit covariance matrix at $U_{k}^{n}$. The received signals at the CH $U_{k}^{N}$ and the HAP in time slot $\tau_{3,n}$ for all $k$ and $n\neq N$ are respectively given by

where ${\mathbf{{z}}}^{(\textrm{U-CH})}_{k,n}\sim\mathcal{CN}\left(\mathbf{0},{\mathbf{L}}^{(\textrm{U-CH})}_{k,n}\right)$ and $\mathbf{\widetilde{z}}^{(\textrm{U-HAP})}_{n}\sim\mathcal{CN}\left(\mathbf{0},{\mathbf{L}}^{(\textrm{U-HAP})}_{n}\right)$ are the additive noises at the $k$th CH and the HAP, respectively, with covariance matrices ${{\mathbf{L}}}^{(\textrm{U-CH})}_{k,n}\in\mathbb{S}_{++}^{{M}_{k,N}}$ and ${\mathbf{L}}^{(\textrm{U-HAP})}_{n}\in\mathbb{S}_{++}^{{M}_{h}}$. The linear decoder matrix $\mathbf{W}_{k,n}\in{\mathbb{C}}^{{d}_{k,n}\times{M}_{k,N}}$ used at $k$th CH estimates the message $\mathbf{m}_{k,n}$ as:

We express the achievable throughput from $U_{k}^{n}$ to $k$th CH during $\tau_{3,n}$ $\forall k$ and $n\neq N$ as [28]:

It is shown in [28] the LMMSE is the optimal decoder for interference suppression. Thus, we propose to employ the following LMMSE decoder for all $k$ and $n\neq N$ (see Appendix A):

We now use the matrix inversion lemma, Sylvester’s determinant property i.e., $\textrm{det}(\mathbf{I}+\mathbf{A}\mathbf{B})=\textrm{det}(\mathbf{I}+\mathbf{B}\mathbf{A})$ and substitute (15) in (14) in order to rewrite (14) as follows for all $k$ and $n\neq N$

The signal transmitted by $U_{k}^{n}$ is also received by the HAP as in (12). For similar arguments, we propose to employ the LMMSE decoder and Successive Interference Cancellation (SIC) technique to decode the messages of the users at the HAP. In this case, we can write the achievable transmission throughput from $U_{k}^{n}$ to the HAP for all $n\neq N$ as [29]:

During $\tau_{4,n}$, $U_{k}^{N}$ relays the signal of its $n$th cluster member $U_{k}^{n}$ along with its own signal (for $n=N$) to the HAP. We express the transmitted vectors of $U_{k}^{N}$ during $\tau_{4,n}$ as follows for all $k,n$:

where $\widetilde{{\mathbf{m}}}_{k,n}\in{\mathbb{C}}^{{{d}_{k,n}}}$ is the symbol stream and $\widetilde{\mathbf{V}}_{k,n}\in{\mathbb{C}}^{{M}_{k,N}\times{{d}_{k,n}}}$ is the precoder employed by $U_{k}^{N}$ for relaying the signals of its $n$th cluster member (for all $n\neq N$) and its own signal (for $n=N$). We further assume that the elements of messages are independent Gaussian random variables with zero mean and unit variance. The total energy consumed at the $k$th CH $U_{k}^{N}$ during $\tau_{4}=\sum_{n=1}^{N}\tau_{4,n}$ shall be upper bounded by its harvested energy in the downlink phase. Thus, the following constraint must be satisfied for all $k$

where $\widetilde{{\mathbf{{S}}}}_{k,n}=\mathbb{E}[\widetilde{{\mathbf{x}}}_{k,n}\widetilde{{\mathbf{x}}}^{{H}}_{k,n}]=\widetilde{\mathbf{V}}_{k,n}\widetilde{{\mathbf{V}}}^{{H}}_{k,n}\in{\mathbb{C}}^{{M}_{k,N}\times{M}_{k,N}}$ is the transmit covariance matrix at $U_{k}^{N}$. Therefore, we can express the received signal at the HAP during $\tau_{4,n}$ for all $n$ as

where $\mathbf{\widetilde{z}}^{(\textrm{CH-HAP})}_{n}\sim\mathcal{CN}\left(\mathbf{0},{{{\mathbf{L}}}}^{(\textrm{CH-HAP})}_{n}\right)$ denotes the receiver additive noise with ${{\mathbf{L}}}^{(\textrm{CH-HAP})}_{n}\in\mathbb{S}_{++}^{{M}_{h}}$. Similar to the third phase, we propose to use LMMSE decoder along with SIC technique at the HAP to decode the corresponding symbol streams, during the last phase. In this case, we can express the achievable throughput for decoding messages of $U_{k}^{n}$ for $1\leq n\leq N-1$ and $U_{k}^{N}$ in time slot $\tau_{4,n}$ at HAP as

By combining (16), (17) and (21), the achievable throughput of $U_{k}^{n}$ for all $n\neq N$ becomes [4]:

Moreover, the achievable throughput for $k$th CH is given by (21), i.e.,

In this section, we cast the max-min fairness throughput problem in which we maximize the throughput of all cooperating users by allocating optimal time durations and optimizing energy and information transmit covariance matrices of the users and HAP in different phases. We formulate this max-min throughput problem by using (5), (6), (10), (19), (22) and (23) as

where $\mathbf{S}=[\mathbf{S}_{k,n},\hskip 2.0pt\forall k,n\neq N]$, $\widetilde{\mathbf{S}}=[\widetilde{\mathbf{S}}_{k,n},\hskip 2.0pt\forall k,n]$, $\bm{\tau}=[\tau_{2},\tau_{3,1},...,\tau_{3,N-1},\tau_{4,1},...,\tau_{4,N}]^{{T}}$ and without loss of generality, we normalize the total time to $T=1$. We note that the problem in (24) is not convex due to the coupled design variables in the objective function and in the constraints $\textrm{C}_{3}$ and $\textrm{C}_{4}$. To deal with this problem, we devise a method based on AO with partitioning $[\bm{\tau}\hskip 5.0pt\vdots\hskip 5.0pt\mathbf{{Q}},\mathbf{{S}},\widetilde{\mathbf{{S}}}]$. We first solve the problem for $[\mathbf{{Q}},\mathbf{{S}},\widetilde{\mathbf{{S}}}]$ assuming a fixed solution for $\bm{\tau}$. Then, we solve the problem for $\bm{\tau}$ assuming a fixed solution for $[\mathbf{{Q}},\mathbf{{S}},\widetilde{\mathbf{{S}}}]$. These two steps will continue till a predefined stop criterion is satisfied for the convergence.