The initializing procedure for algorithm 1.

## Abstract

This chapter describes transceiver design methods for simultaneous wireless power transmission (WPT) and information transmission in two typical multiuser MIMO networks, that is, the MIMO broadcasting channel (BC) and interference channel (IC) networks. The design problems are formulated to minimize the transmit power consumption at the transmitter(s) while satisfying the quality of service (QoS) requirements of both the information decoding (ID) and WPT of all users. The mean-square error (MSE) and the signal-to-interference-noise ratio (SINR) criteria are adopted to characterize the ID performance of the BC network and the IC network, respectively. The designs are cast as nonconvex optimization problems due to the coupling of multiple variables with respect to transmit precoders, ID receivers, and power splitting factors, which are difficult to solve directly. The feasibility conditions of these deign problems are discussed, and effective solving algorithms are developed through alternative optimization (AO) framework and semidefinite programming relaxation (SDR) techniques. Low-complexity algorithms are also developed to alleviate the computation burden in solving the semidefinite programming (SDP) problems. Finally, simulation results validating those proposed algorithms are included.

### Keywords

- wireless power transfer (WPT)
- energy harvesting
- multiuser MIMO
- transceiver design
- alternating optimization
- semidefinite programming relaxation (SDR)

## 1. Introduction

Wireless power transfer (WPT) through radio frequency (RF) signals has been redeemed as one of the promising techniques to provide perpetual and cost-effective power supplies for mobile devices [1, 2, 3, 4]. Compared with traditional energy harvesting (EH) methods depending on external sources, such as solar power and wind energy, the RF WPT is able to power the wireless devices at any time. Moreover, since RF signals carry energy as well as information, wireless devices can be charged while communicating. These merits of WPT bring great convenience and provide quality of service (QoS) guarantee for wireless devices.

On the other hand, multiple-input, multiple-output (MIMO) techniques are widely used in many wireless communication systems such as WiFi and the fifth generation mobile (5G) systems, due to their potential in providing increased link capacity and spectral efficiency combined with improved link reliability. Moreover, the evolution of MIMO techniques to the massive MIMO systems, where tens or hundreds of antennas are equipped at transmitters or/and receivers, accompanied by shrinking coverage of base stations (BSs) in the future wireless systems, makes it possible to transmit wireless power with higher efficiency. It is envisaged that the power line connected to the mobile devices would be eliminated completely in future wireless communications by combing WPT with MIMO wireless information transmission (WIT) system [5, 6, 7, 8, 9].

Transceiver design plays a very important role in achieving this vision. The objective of the transceiver design is to improve the energy and spectral efficiency of the transmitter by optimizing the beam patterns of the transmitting antennas and the filters at the information decoding (ID) receivers. However, it is not a trivial task to design transceivers for multiuser MIMO systems operating in simultaneous wireless information and power transfer (SWIPT) mode due to the presence of inter-user interference. The interference makes the whole design complicated since it is harmful to WIT but beneficial to WPT, and it is very challenging to balance the role of interference in ID and EH. And what makes things worse is that the interference and the PS factors are coupled together, which makes the joint transceiver design and power splitting (JTPS) problems nonconvex. These problems are NP-hard in general, so effective algorithms should be found to get feasible solutions.

In this chapter, we will discuss transceiver design methods for SWIPT in two typical multi-user MIMO scenarios, that is, the broadcasting channel (BC) network and interference channel (IC) network. We focus on the QoS-constrained problems that are formulated as minimizing the transmit power consumption subject to both the minimum ID and EH requirements. The mean-square error (MSE) and the signal-to-interference-noise ratio (SINR) criteria are adopted to characterize the ID performance of the two kinds of network, respectively. The formulated optimization problems are nonconvex with respect to the optimization variables, that is, the parameters of transmit precoders, ID receivers, and PS factors. In order to develop effective solutions, the feasibility is first investigated and found to be independent with EH constraints and PS factors. Based on this, we develop an effective initializing procedure for the design problems. Then, effective iterative solving algorithms are developed based on alternative optimization (AO) framework and semidefinite programming relaxation (SDR) techniques. Specifically, we find that the original problems can be equivalently reformulated as convex semidefinite programming (SDP) with respect to the transceivers and PS ratios when the receivers are fixed. On the other hand, when the transmitters and PS factors are fixed, the original problems degenerate to the classical linear MSE minimization receiver design problem for the BC network and the SINR maximization receiver design problem for the IC network, respectively. Since the SDP problems can be solved exactly in polynomial time, feasible solutions can be obtained for the proposed algorithms effectively.

However, the SDP solving is not computationally efficient for large number of variables case [10], and the computational complexity of the SDP-based algorithms is prohibitively high for large number of antenna and user. This greatly restricts its application. To break this, low-complexity schemes should be developed. In this chapter, closed-form power splitting factors with given the transceivers designed from traditional transceiver design algorithms are developed.

Notations: _{,} and

## 2. Joint transceiver design and power splitting optimization based on MSE criterion for MIMO BC channel

### 2.1 System model

A downlink MIMO BC channel network shown in Figure 1, where one base station (BS) serves

The BS transmits the signal

where

As shown in Figure 1, each user divides the received signal into two parts through power splitters, one part is for information decoder (ID) and the other is for EH. For easy analysis, solution, and implementation, we adopt the uniform PS model [11] in this chapter, that is, the power splitters of all the antennas of a user have the same PS factor. Denote

where

At the ID receiver, a filter

It is noted that a scaling factor

Consequently, the MSE at the

where

where

### 2.2 Problem formulation and feasibility analysis

#### 2.2.1 Problem formulation

A case that each MS has its dedicated ID and EH QoS requirements and the BS has to satisfy all the users with minimum transmit power consumed is considered. This scenario can be modeled by the following QoS constrained power minimization problem

where

Obviously, (7) is nonconvex with respect to the precoders, receivers, and power splitters and thus difficult to be solved directly. Before developing an effective algorithm for it, its feasibility should be analyzed first.

#### 2.2.2 Feasibility analysis

Some sufficient and necessary conditions for the feasibility of the original problem (7) can be given by the following propositions.

* Problem*(8)

is feasible if and only if the following problem is feasible:

For the sake of brevity, we omit the proof to these propositions. Interested readers are suggested to refer to [12, 13, 14, 15, 16]. Proposition 1 reveals that the feasibility of the original problem (7) is irrelevant to the EH constraints, while Proposition 2 further shows that the EH constraints are irrelevant to the feasibility. Since the feasibility of the formulated problem depends on neither the EH constraints nor the PS factors, checking its feasibility can be simplified to checking the feasibility of (9), which is a traditional MSE-based multiuser MIMO transceiver design problem [17, 18, 19]. So, in the following, we assume that problem (9) is feasible under the given MSE QoS requirements and focus on how to solve it.

### 2.3 Alternative optimization solution based on semidefinite programming relaxation

#### 2.3.1 Alternative optimization framework

By reviewing the MSE expression (5), we know that it is convex with respect to either

Specifically, when the receiver

It is noted that problem (10) is not convex in its current form. We will further process it based on SDR techniques in the next subsection.

When the precoders and PS factors are fixed, the transmit power at the BS and the EH power are fixed. Considering that only MSE at the ID receiver of user

Problem (11) is the traditional unconstrained MSE minimization problem and its closed-form solution can be given by

Therefore, by alternatively optimizing the transmitter together with PS factors according to (10) and the receivers according to (12), an iterative optimization framework is established and is summarized in Algorithm 1.

** Algorithm 1**Alternating optimization framework for JTDPS.

1: Initialize the receivers

2: Optimize the transmitter

3: Optimize the receive filters

4: Repeat 2 and 3 until convergence or the maximum number of iterations is reached.

#### 2.3.2 Convergence analysis

For the AO framework, it is vital to analyze its convergence property. The following proposition reveals this property.

** Proposition 3**For the initial receivers

The proof can be found in [12]. According to Proposition 3, two critical prerequisites should be satisfied in order to guarantee finding a feasible solution for problem (7) through Algorithm 1. 1) the subproblem (10) should be feasible and 2) the initialization of the receivers

#### 2.3.3 Feasibility of the transmitter design subproblem

By checking the MSE constraints of (9), a necessary condition for the feasibility of problem (9) is established as

This condition shows that the Frobenius norm of the receiver should be small enough to make the problem feasible. In order to satisfy this condition, we introduce a positive scaling parameter

After replacing the MSE constraints with (14), problem (10) can then be reformulated as

It can be proved that a sufficient and necessary condition for the feasibility of problem (15) is given by the following proposition [12].

** Proposition 4**Problem (15) is feasible if and only if the following problem is feasible:

Proposition 4 reveals that the feasibility of the transmitter and PS problem (15) does not depend on the EH constraints either. Therefore, the feasibility of the joint transmitter design and PS problem (15) can be simply verified by checking whether problem (16) is feasible or not. To guarantee the feasibility of problem (16), the following proposition is proposed.

** Proposition 5**Fix

where

** Proof.**The MSE constraints in problem (16) can be recast as

According to the Schur complement lemma, the inequality (18) is equivalent to

The proposition is obtained.

Problem (17) is convex, and its optimal solution can be obtained. Therefore, problem (16) is feasible. With the solution of problem (16), an effective initialization procedure for problem (16) can be constructed. Specifically, the receiver

1: Given the MSE requirements 2: Generate random matrix 3: By fixing 4: Return |

Through Proposition 4 and Proposition 5, it is known that the joint transmitter design and power splitting subproblem is feasible. However, Algorithm 1 cannot be carried out in its current form, since problem (10) is still nonconvex. In the following, the SDP relaxation is adopted to reform it in to convex form.

#### 2.3.4 Algorithm description

By introducing two variables

Adopting the Schur complement lemma, the constraints

Problem (21) is a nonconvex inhomogeneous quadratically constrained quadratic program (QCQP) [20, 21], which are NP-hard [10, 22, 23, 24, 25]. In order to solve it effectively, SDR is utilized. Specifically, a new variable

Problem (22) is a convex SDP with respect to

It is noted that the optimal objective of (22) is a lower bound of that of the nonconvex QCQP problem (21), since the same objective function is minimized over a larger set [27]. Let

The complexity of Algorithm 1 is mainly introduced by the SDP (22). Given a solution accuracy

### 2.4 Low-complexity design scheme

In this section, a low-complexity algorithm is derived by first designing ID transceivers to satisfy the MSE constraints and then optimizing the transmit power together with PS factors with the designed transceivers. The scheme is of quite low computational complexity.

It is noted that the MSE-constrained transceiver design problem (9) can be solved efficiently by existing methods proposed in [17]. So, let _{,} and decrease the receiver

where the scaling factor

** Proposition 6**The optimal solution of problem (23) is given in closed form by

where _{,} and

** Proof.**Problem(23) can be transformed to

By summing (27) and (28) for user

where

When

The optimal solution of problem (32) is given by

For the optimal

Given

** Algorithm 2**Low-complexity closed-form PS (CF-PS) algorithm.

1: Solve problem (9) to obtain

2: Optimize the optimal scaling factor

3: Return the feasible solution

The main computational complexity of Algorithm 2 comes from solving problem (9), which is of

### 2.5 Simulation results and analysis

The performance of the described algorithms is validated through simulations. The user number is set to be

The convergence property of the described scheme is shown in Figure 2. It can be seen that the optimized transmit power and the PS factors decrease monotonically along with the increase of the number of iterations. It needs special explanation that

Figure 3 shows the per-user MSE and harvested energy of the proposed scheme along with iterations. Similar to the MSE-QoS-TRAD scheme, the SDP-JTDPS scheme satisfies the MSE QoS requirements in each iteration. Moreover, the EH requirements can also be satisfied.

The performance of SDP-JTDPS and CF-PS algorithms are compared in Figure 4. It can be observed that all of them achieve the MSE QoS requirements exactly. This is consistent with the analysis that the MSE constraints can be satisfied with equality for both schemes. It is also shown that the SDP-JTDPS can exactly reach the EH targets of all users, which implies that the EH constraints are satisfied with equality. Different from the SDP-JTDPS, the CF-PS harvests more energy than the predefined threshold, but at the expense of more transmit power, which is shown in Figure 5.

The optimized transmit power achieved by the algorithms are compared in Figure 5. During the simulations, all algorithms run at the same independently generated initial receivers in each trail. It can be observed that SDP-JTDPS and CF-PS consume higher transmit power than the traditional MSE-QoS-TRAD at most of the trials, and CF-PS consumes more power than SDP-JTDPS. This is obvious because more power is needed to satisfy the EH requirements. CF-PS consumes more transmit power than SDP-JTDPS does, which has been mentioned in the previous section that the low complexity is achieved at the cost of high transit power.

As shown in Figure 6, CF-PS performs the best with respect to the computational complexity. For the SDP-JTDPS scheme, its execution time is well fitted as a power function on the number of transmit antennas _{,} and

## 3. Joint transceiver design and power splitting optimization based on SINR criterion for MIMO IC channel

In this section, we further consider the joint transceiver design and wireless power transfer for MIMO IC networks.

### 3.1 System model

A

The received baseband signal at the

where

Similar to Section 2.1, the received signal at each antenna is then divided into two parts via a power splitter; one part is for information decoding, and the other is transformed to stored energy. The signal split into the ID receiver at the

where

Define

The SINR of the

where

Let

### 3.2 Problem formulation

To minimize the transmit power under the given QoS constraints, the joint transceiver design and power splitting problem is formulated as

Here, SINR is adopted to measure the QoS of ID. Eqs. (38–41) are nonconvex, and thus, it is very difficult to obtain its optimal solution. Similar to Section 2.1, the AO framework can be adopted to develop an iterative algorithm. In order to achieve this, the concept of interference alignment can be utilized. Therefore, some preliminaries on IA are introduced in the following section.

### 3.3 Interference alignment

IA is a ground-breaking interference management method for IC networks. The idea of IA is to coordinate the transmitters so that the interference received at each receiver can be aligned into a subspace with a small dimension and thus leaves the interference-free subspace for signal [28]. IA has the ability to achieve the maximum degrees of freedom (DoF) of the

As shown in Figure 1, when the EH receivers are removed, the system degenerates into traditional symmetric MIMO IC networks. The DoF of such MIMO IC network is

When the condition (42) is satisfied, the inter-user interference can be completely suppressed. While the condition (43) guarantees that sufficient dimensions are left for signal subspace. For the considered system, in order to achieve IA, the maximum number of streams for each user should no more than

If IA conditions are perfectly satisfied, and the received signal of user

where

Eq. (44) means that the system is equivalent to a traditional point-to-point MIMO system after IA and the ergodic achievable rate of the

In the following sections, the feasibility of the formulated problem (38) will be discussed, and suboptimal schemes solving the problem will be developed.

### 3.4 Feasibility analysis

The feasibility of the formulated problems (38–41) can be given by the following propositions [14].

** Proposition 7**Problem (38) is feasible if and only if the following problem is feasible.

** Proposition 8**Problem (46) is feasible if and only if the following problem is feasible.

where

Proposition 7 and Proposition 8 show that the feasibility of (38–41) is independent of the EH constraints and the PS factors. Proposition 8 is a sufficient and necessary condition for the feasibility of problem (38–41). However, it is not hard to solve (47) [33, 34] directly since SINRs are overconstrained. As an alternative, a sufficient condition for the feasibility of problem (47) is derived based on IA by the following proposition.

Proposition 9 (47) is feasible for any given SINR constraints if the system is interference unlimited, that is, the interference can be completely eliminated by the linear transceivers.

** Proof.**If interference is completely eliminated, given the transceivers

SINR (48) becomes

where

Based on Proposition 9 and the IA feasibility condition (35), problems (38–41) must be feasible if the system is IA feasible. In the following, it is assumed that the considered MIMO IC network is IA feasible.

### 3.5 Alternative optimization solution based on semidefinite programming relaxation

An iterative algorithm for (38–41) can be developed based on AO framework, that is, alternatively optimizing the transmitters

#### 3.5.1 Transmitter and power splitting optimization

When the receivers are fixed, problems (38–41) are reduced to the following joint transmit precoders and PS factors optimization problem

where

According to Lemma 8, Proposition 9, and [35], Proposition 9, problem (52) is feasible if the original problem is feasible. By defining

The convex SDP (53) can be solved efficiently. Moreover, it can be proven that there is

#### 3.5.2 Receiver optimization

When the transmit precoders and PS factors are all fixed, (38–41) become separable with respect to variables

where

#### 3.5.3 Algorithm description

By alternatively optimizing the transmitters together with PS factors and the receivers, an SDP-based joint transceiver design and PS optimization scheme can be obtained, which is summarized in Algorithm 3.

** Algorithm 3**Joint transceiver design and power splitting based on SDP (SDP-JTDPS).

1: Initialize the receivers

2: Solve the convex problem (53) to obtain

3: Recover

4: Update the receiver

5: Repeat 2 to 4 until convergence or the maximum iteration number reached.

The convergence of Algorithm 3 is given by the following proposition [14].

** Proposition 10**If (53) is feasible for the initial receivers

According to Proposition 10, it is important to initialize the receivers

### 3.6 Low-complexity design schemes

Two kinds of low complexity schemes are derived to solve problems (38–41) by separately designing the transceivers and power splitting factors. The transceivers are firstly designed by eigen-decomposing the effective channel matrices generated by interference alignment. Then, the transmit power and receive PS factors are optimized with the precoders and receivers fixed.

As analyzed in the previous section, to ensure that (38–41) are feasible, perfect IA should be realized. To simplify the system design, we assume that the precoders and receive filters are orthogonalized such that

Eq. (56) shows that interference is completely suppressed at the ID receiver by the transceiver design scheme. According to Lemma 7, Lemma 8, and Proposition 9, we know that problems (38–41) are feasible and can be reduced to the following transmit power allocation and power splitting problem

In the following, two different schemes solving (58) is developed by either reformulating it as a convex problem or solved in closed form.

#### 3.6.1 Optimal power allocation and power splitting scheme

After some algebraic manipulations, problem (58) can be reformed as

Problem (59) is convex and thus can be solved optimally. Denote

The proposed IA-based SWIPT scheme with optimal power allocation and power splitting is summarized in Algorithm 4. The computational complexity of Algorithm 4 is mainly from solving (59) in Step 4. When the interior methods are employed, the computational complexity of Algorithm 4 is in the order of

** Algorithm 4**SWIPT design with optimal transmit power allocation and receive power splitting over effective IA channel decomposing (O-PAPS).

1: Obtain IA transceivers

2: Let

3: Let

4: Obtain the optimal transmit power

5: Set

#### 3.6.2 Closed-form power allocation and power splitting scheme

Given the IA solution

According to Proposition 9, (60) is feasible. By further applying Lemma 8, (58) is feasible. Moreover, (60) can be decomposed into

The solution of (61) is then given by

Following Lemma 7, (58) can be optimized by substituting

The closed-form solution of (63) can be derived and given by the following proposition [14].

** Proposition 11**Given the IA transceivers

_{,}and

where

Given

** Algorithm 5**SWIPT design with closed-form transmits power allocation and receive power splitting solutions over the effective IA channel decomposing (CF-PAPS).

1: Obtain IA transceivers

2: Let

3: Let

4: Calculate

5: Obtain

6: Set

The computational complexity of Algorithm 5 is determined by the IA transceiver design process in Step 1. For the famous closed-from IA algorithm, the complexity is

### 3.7 Simulation results and analysis

Simulations are done over the wireless system as described in Section 3.1, by setting the number of users _{,} and

Figure 8 shows the convergence performance of SDP-JTDPS with

The empirical cumulative distribution function (CDF) of the output per-stream SINR for the

Figure 10 shows the total transmit power versus SINR thresholds at different EH thresholds. It is observed that the transmit power will increase along with the increasing of the EH threshold from

Figure 11 shows the relationships between the average transmit power and EH thresholds given different SINR targets. It is seen that the average transmit powers asymptotically tend to be the same as the EH threshold increases for any given SINR value for both O-PAPS and CF-PAPS. For any of the three schemes, higher EH requirement means higher transmit power needed. It is also shown that SDP-JTDPS and O-PAPS achieve the same performance when the SINR threshold is relatively low (e.g.,

Figure 12 compares SDP-JTDPS with DIA proposed [34] in a

The performance of the proposed schemes is further tested in a

Finally, Figure 14 compares the computational complexity of the proposed schemes for different antenna numbers by assuming there are

## 4. Conclusions

The joint transceiver design and power splitting optimization for the simultaneous wireless information and power transfer of the MIMO BC network and IC network are analyzed in this chapter. For the MIMO BC network, a transmit power minimization problem subject to both the EH and MSE constraints is formulated. While for the MIMO IC network, similar transmit power minimization problem is formulated but with the SINR QoS requirements for the ID receivers. Sufficient condition to guarantee the feasibility of nonconvex problems is derived, which reveal that the feasibility of the design problems is not dependent on the PS factors and the EH constraints. Based on the SDP relaxation, alternative solving algorithms are introduced by iteratively optimizing the transmitter together with the PS factors and the receiver. To avoid the high computational complexity of SDP-based schemes, low-complexity algorithms are developed and analyzed. Simulation results have shown the effectiveness of the proposed designs in achieving simultaneous wireless information and power transfer.

## Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (NSFC) under grants 61701269 and 61671278, the National Science Fund of China for Excellent Young Scholars under grant 61622111, the Natural Science Foundation of Shandong Province under grant ZR2017BF012, and the Joint Research Foundation for Young Scholars in the Qilu University of Technology (Shandong Academy of Sciences) under grant 2017BSHZ005.