2.1 System model

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

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 NPhard 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

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

Notations: C represents the complex and positive real field. Bold uppercase and lowercase letters represent matrix and column vectors, respectively. Nonbold italic

represent the Hermitian transpose, transpose, and inverse of A, respectively. Trð Þ A and rankð Þ A are the trace and rank of matrix A, respectively. ∣A∣ denotes the

, and A�<sup>1</sup>

from traditional transceiver design algorithms are developed.

letters represent scalar values. <sup>I</sup><sup>N</sup> is an <sup>N</sup> � <sup>N</sup> identity matrix. <sup>A</sup>H, <sup>A</sup><sup>T</sup>

transmission (WIT) system [5–9].

Recent Wireless Power Transfer Technologies

effectively.

122

A downlink MIMO BC channel network shown in Figure 1, where one base station (BS) serves K mobile stations (MSs) simultaneously through spatial multiplexing, is considered. The number of antennas of the BS and the kth (k∈f g 1, 2, … ,K ) user are denoted as M and Lk, respectively. Nk represents the number of data stream for the kth user, and the total number of data streams served by BS is <sup>d</sup> <sup>¼</sup> <sup>P</sup><sup>K</sup> <sup>k</sup>¼<sup>1</sup>Nk <sup>≤</sup> <sup>M</sup>. Let <sup>s</sup><sup>k</sup> <sup>∈</sup> <sup>C</sup>Nk�<sup>1</sup> denote the data vector transmitted to the user k, the data vector transmitted by BS can be expressed as

<sup>s</sup> <sup>¼</sup> <sup>s</sup><sup>H</sup> <sup>1</sup> , … , s<sup>H</sup> K � �<sup>H</sup> <sup>∈</sup>C<sup>d</sup>�<sup>1</sup> . It is assumed that ss<sup>H</sup> � � <sup>¼</sup> <sup>I</sup>d.

The BS transmits the signal s to users, and the received baseband signal at the kth receiver is

$$\mathbf{r}\_k = \mathbf{H}\_k \mathbf{G} \mathbf{s} + \mathbf{n}\_k,\tag{1}$$

where <sup>G</sup> <sup>∈</sup>C<sup>M</sup>�<sup>d</sup> denote the transmit precoding matrix, <sup>H</sup><sup>k</sup> <sup>∈</sup> <sup>C</sup>Lk�<sup>M</sup> denotes the channel propagation matrix from the BS to the <sup>k</sup>th user and <sup>n</sup><sup>k</sup> <sup>∈</sup> <sup>C</sup>Lk�<sup>1</sup> denotes the noise vector, elements of which are assumed to be independent and identically (i.i.d.) zero mean complex Gaussian random variables with variance σ<sup>2</sup> nk . The total transmit power of the BS can be calculated as <sup>P</sup> <sup>¼</sup> <sup>∥</sup>G∥<sup>2</sup> F.

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 0 ≤ρ<sup>k</sup> ≤1 as the PS factors for the kth user, the signal received at the ID of the kth user is

$$\mathbf{r}\_k^{\mathrm{ID}} = \sqrt{\rho\_k} (\mathbf{H}\_k \mathbf{G} \mathbf{s} + \mathbf{n}\_k) + \mathbf{w}\_k,\tag{2}$$

Figure 1. Downlink MU-MIMO SWIPT system.

where <sup>w</sup><sup>k</sup> <sup>∈</sup>CLk�<sup>1</sup> is the noise caused by power splitter, elements of which are assumed to be i.i.d. zero mean complex Gaussian random variables with variance σ2 w<sup>k</sup> . The signal received by the EH receiver of user k is written as

$$\mathbf{r}\_k^{\rm EH} = \sqrt{\mathbf{1} - \rho\_k} (\mathbf{H}\_k \mathbf{G} \mathbf{s} + \mathbf{n}\_k). \tag{3}$$

2.2.2 Feasibility analysis

feasible:

lem (7) can be given by the following propositions.

min G, Fk, ρ<sup>k</sup> f g , ∀k

DOI: http://dx.doi.org/10.5772/intechopen.89676

min f g G, Fk, ∀k

Some sufficient and necessary conditions for the feasibility of the original prob-

Proposition 1 Problem (7) is feasible if and only if the following problem is 05

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

2 <sup>F</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

2 <sup>F</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

For the sake of brevity, we omit the proof to these propositions. Interested readers are suggested to refer to [12–16]. 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 [17–19]. So, in the following, we assume that problem (9) is feasible under the given

2.3 Alternative optimization solution based on semidefinite programming

By reviewing the MSE expression (5), we know that it is convex with respect to either G or Fk. So, we can develop an iterative algorithm for the original problem based on the optimization (AO) framework, that is, optimizing the transmit

Specifically, when the receiver Fk, ∀k is fixed, the optimization problem is reduced to a joint transmitter design and power splitting (JTDPS) subproblem,

Tr GG<sup>H</sup> � �

0≤ ρ<sup>k</sup> ≤ 1, ∀k ¼ 1, … ,K:

It is noted that problem (10) is not convex in its current form. We will further

s:t: : MSE<sup>k</sup> ≤εk, PEH <sup>k</sup> ≥ψk,

Proposition 2 Problem (8) is feasible if and only if the following problem is feasible:

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> ρk

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � �k k <sup>F</sup><sup>k</sup>

k k F<sup>k</sup> 2 <sup>F</sup> ≤ εk,

> 2 <sup>F</sup> ≤εk,

(8)

(9)

(10)

!

Tr GG<sup>H</sup> � �

Tr GG<sup>H</sup> � �

s:t: : k k FkHkG � Ξ<sup>k</sup>

∀k ¼ 1, … , K:

s:t: : k k FkHkG � Ξ<sup>k</sup>

∀k ¼ 1, … , K:

MSE QoS requirements and focus on how to solve it.

precoder together with PS factors and the receivers iteratively.

min G, ρ<sup>k</sup> f g , ∀k

process it based on SDR techniques in the next subsection.

2.3.1 Alternative optimization framework

relaxation

which is

125

At the ID receiver, a filter <sup>F</sup><sup>k</sup> <sup>∈</sup> <sup>C</sup>Nk�Lk is employed to process the received signal and the detected signal is written as

$$\hat{\mathbf{s}}\_{k} = \frac{1}{\sqrt{\rho\_{k}}} \mathbf{F}\_{k} \mathbf{r}\_{k}^{ID} = \mathbf{F}\_{k} \mathbf{H}\_{k} \mathbf{G} \mathbf{s} + \mathbf{F}\_{k} \mathbf{n}\_{k} + \frac{1}{\sqrt{\rho\_{k}}} \mathbf{F}\_{k} \mathbf{w}\_{k}. \tag{4}$$

It is noted that a scaling factor <sup>1</sup> ffiffiffi ρk <sup>p</sup> is introduced to the received signal, which makes the problem modeling and solving more convenient.

Consequently, the MSE at the kth ID receiver can be expressed as

$$\begin{split} \text{MSE}\_{\mathbf{k}} &= \mathbb{E}\left[ \|\mathbf{\dot{s}}\_{\mathbf{k}} - \mathbf{s}\_{\mathbf{k}}\|\_{2}^{2} \right] \\ &= \text{Tr}(\mathbf{I}\_{\mathbf{k}\_{\mathbf{k}}}) + \text{Tr}\left( \mathbf{H}\_{\mathbf{k}}^{\text{H}} \mathbf{F}\_{\mathbf{k}}^{\text{H}} \mathbf{F}\_{\mathbf{k}} \mathbf{H}\_{\mathbf{k}} \mathbf{G} \mathbf{G}^{\text{H}} \right) - \text{Tr}\left( \mathbf{\Xi}\_{\mathbf{k}} \mathbf{G}^{\text{H}} \mathbf{H}\_{\mathbf{k}}^{\text{H}} \mathbf{F}\_{\mathbf{k}}^{\text{H}} \right) \\ &- \text{Tr}\left( \mathbf{F}\_{\mathbf{k}} \mathbf{H}\_{\mathbf{k}\mathbf{k}} \mathbf{G} \mathbf{E}\_{\mathbf{k}}^{\text{H}} \right) + \left( \sigma\_{\mathbf{n}\_{\mathbf{k}}}^{2} + \frac{\sigma\_{\mathbf{w}\_{\mathbf{k}}}^{2}}{\rho\_{\mathbf{k}}} \right) \text{Tr}\left( \mathbf{F}\_{\mathbf{k}}^{\text{H}} \mathbf{F}\_{\mathbf{k}} \right) . \end{split} \tag{5}$$

where Ξ<sup>k</sup> ¼ 0 Nk� P<sup>k</sup>�<sup>1</sup> <sup>l</sup>¼<sup>1</sup> Nl ,INk , <sup>0</sup>Nk� P<sup>K</sup> l¼kþ1 Nl � �. At the same time, the energy harvested by the kth EH receiver is expressed as

$$P\_k^{\rm EH} = \xi\_k (1 - \rho\_k) \left( \|\mathbf{H}\_k \mathbf{G}\|\_F^2 + L\_k \sigma\_{\mathbf{n}\_k}^2 \right), \tag{6}$$

where 0 ≤ξ<sup>k</sup> ≤1 denotes the energy conversion efficiency.

#### 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

$$\begin{aligned} \min\_{\{\mathbf{G}, \mathbf{F}\_k, \rho\_k, \boldsymbol{\forall} k\}} & \quad \mathrm{Tr}(\mathbf{G} \mathbf{G}^H) \\ \text{s.t.} & \quad \mathrm{MSE}\_k \le \varepsilon\_k, \\ & P\_k^{\mathrm{EH}} \ge \boldsymbol{\forall} \boldsymbol{\epsilon}\_k, \\ & \mathbf{0} < \rho\_k < \mathbf{1}, \forall k = \mathbf{1}, \ldots, K, \end{aligned} \tag{7}$$

where ε<sup>k</sup> >0 and ψ<sup>k</sup> >0 are the ID MSE target and the EH threshold of user k, respectively.

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.

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems DOI: http://dx.doi.org/10.5772/intechopen.89676

#### 2.2.2 Feasibility analysis

where <sup>w</sup><sup>k</sup> <sup>∈</sup>CLk�<sup>1</sup> is the noise caused by power splitter, elements of which are assumed to be i.i.d. zero mean complex Gaussian random variables with variance

At the ID receiver, a filter <sup>F</sup><sup>k</sup> <sup>∈</sup> <sup>C</sup>Nk�Lk is employed to process the received signal

<sup>k</sup> ¼ FkHkGs þ Fkn<sup>k</sup> þ

<sup>p</sup> ð Þ <sup>H</sup>kGs <sup>þ</sup> <sup>n</sup><sup>k</sup> : (3)

1 ffiffiffi <sup>ρ</sup> <sup>p</sup> <sup>k</sup>

<sup>p</sup> is introduced to the received signal, which

Tr F<sup>H</sup> <sup>k</sup> F<sup>k</sup> � �:

. At the same time, the energy

<sup>k</sup> <sup>F</sup>kHkGG<sup>H</sup> � � � Tr <sup>Ξ</sup>kG<sup>H</sup>H<sup>H</sup>

<sup>F</sup> <sup>þ</sup> Lkσ<sup>2</sup> nk

� �

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> ρk

!

Fkwk: (4)

<sup>k</sup> F<sup>H</sup> k

, (6)

(5)

(7)

� �

. The signal received by the EH receiver of user k is written as

<sup>k</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ρ<sup>k</sup>

> ffiffiffi ρk

Consequently, the MSE at the kth ID receiver can be expressed as

<sup>k</sup> F<sup>H</sup>

P<sup>K</sup> l¼kþ1 Nl

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

Tr GG<sup>H</sup> � �

0<ρ<sup>k</sup> <1, ∀k ¼ 1, … , K,

where ε<sup>k</sup> >0 and ψ<sup>k</sup> >0 are the ID MSE target and the EH threshold of user k,

splitters and thus difficult to be solved directly. Before developing an effective

Obviously, (7) is nonconvex with respect to the precoders, receivers, and power

s:t: : MSE<sup>k</sup> ≤εk, PEH <sup>k</sup> ≥ψk,

k � � <sup>þ</sup> <sup>σ</sup><sup>2</sup>

<sup>k</sup> <sup>¼</sup> <sup>ξ</sup><sup>k</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> ð Þ <sup>∥</sup>HkG∥<sup>2</sup>

2

,INk , <sup>0</sup>Nk�

where 0 ≤ξ<sup>k</sup> ≤1 denotes the energy conversion efficiency.

min G, Fk, ρ<sup>k</sup> f g , ∀k

algorithm for it, its feasibility should be analyzed first.

� �

rEH

FkrID

makes the problem modeling and solving more convenient.

� �

<sup>¼</sup> Tr <sup>I</sup>Nk ð Þþ Tr <sup>H</sup><sup>H</sup>

�Tr <sup>F</sup>kHkkGΞ<sup>H</sup>

and the detected signal is written as

Recent Wireless Power Transfer Technologies

^s<sup>k</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffi ρk p

MSEk <sup>¼</sup> <sup>E</sup> <sup>∥</sup>^s<sup>k</sup> � <sup>s</sup>k∥<sup>2</sup>

It is noted that a scaling factor <sup>1</sup>

Nk� P<sup>k</sup>�<sup>1</sup> <sup>l</sup>¼<sup>1</sup> Nl

harvested by the kth EH receiver is expressed as

PEH

2.2 Problem formulation and feasibility analysis

where Ξ<sup>k</sup> ¼ 0

2.2.1 Problem formulation

minimization problem

respectively.

124

σ2 w<sup>k</sup>

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

Proposition 1 Problem (7) is feasible if and only if the following problem is 05 feasible:

$$\begin{aligned} &\min\_{\{\mathbf{G},\mathbf{F}\_k,\rho\_k,\forall k\}} \quad \text{Tr}\left(\mathbf{G}\mathbf{G}^\mathrm{H}\right) \\ &\text{s.t.} \; ||\mathbf{F}\_k\mathbf{H}\_k\mathbf{G} - \Xi\_k||\_F^2 + \left(\sigma\_{\mathbf{n}\_k}^2 + \frac{\sigma\_{\mathbf{w}\_k}^2}{\rho\_k}\right) \|\mathbf{F}\_k\|\_F^2 \le \varepsilon\_k, \\ &\forall k = 1,\ldots,K. \end{aligned} \tag{8}$$

Proposition 2 Problem (8) is feasible if and only if the following problem is feasible:

$$\begin{aligned} &\min\_{\{\mathbf{G}, \mathbf{F}\_k, \forall k\}} \quad \text{Tr}\left(\mathbf{G}\mathbf{G}^\mathrm{H}\right) \\ &\text{ s.t.} : \left\|\mathbf{F}\_k \mathbf{H}\_k \mathbf{G} - \boldsymbol{\Xi}\_k\right\|\_{\mathrm{F}}^2 + \left(\sigma\_{\mathbf{n}\_k}^2 + \sigma\_{\mathbf{w}\_k}^2\right) \left\|\mathbf{F}\_k\right\|\_{\mathrm{F}}^2 \le \varepsilon\_k, \\ &\forall k = \mathbf{1}, \dots, K. \end{aligned} \tag{9}$$

For the sake of brevity, we omit the proof to these propositions. Interested readers are suggested to refer to [12–16]. 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 [17–19]. 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 G or Fk. So, we can develop an iterative algorithm for the original problem based on the optimization (AO) framework, that is, optimizing the transmit precoder together with PS factors and the receivers iteratively.

Specifically, when the receiver Fk, ∀k is fixed, the optimization problem is reduced to a joint transmitter design and power splitting (JTDPS) subproblem, which is

$$\begin{aligned} \min\_{\{\mathbf{G}, \boldsymbol{\rho}\_k, \boldsymbol{\Psi}\}} & \quad \text{Tr}\left(\mathbf{G} \mathbf{G}^H\right) \\ \text{s.t.} & \quad \mathbf{MSE}\_k \le \boldsymbol{\varepsilon}\_k, \\ & P\_k^{\text{EH}} \ge \boldsymbol{\nu}\_k, \\ & \mathbf{0} \le \boldsymbol{\rho}\_k \le \mathbf{1}, \forall k = 1, \ldots, K. \end{aligned} \tag{10}$$

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 k is relevant to Fk, we can optimize the ID receiver by minimizing the MSE. The optimization problem can be formulated as

$$\min\_{\{\mathbf{F}\_k\}} \quad \text{MSE}\_k. \tag{11}$$

MSE<sup>k</sup> <sup>¼</sup> <sup>∥</sup> <sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.89676

min <sup>G</sup>, pk, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup><sup>k</sup>

<sup>s</sup>:t: : <sup>1</sup> pk

� � � �

reformulated as

feasible:

pk

FkHkG � Ξ<sup>k</sup>

<sup>ξ</sup><sup>k</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> ð Þ <sup>∥</sup>HkG∥<sup>2</sup>

problem (15) is given by the following proposition [12].

Tr GG<sup>H</sup> � �

� � � �

2

F <sup>þ</sup> <sup>σ</sup><sup>2</sup>

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

Proposition 5 Fix Fk, ∀k, problem (16) can be reformulated as a convex SDP

ffiffi ε p

k k F<sup>k</sup> <sup>F</sup>, and a<sup>k</sup> ¼ vec FkHkG � pkΞ<sup>k</sup>

� 2 <sup>F</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

� �∥<sup>2</sup>

<sup>k</sup> β<sup>k</sup>

ffiffi ε p I 0

β<sup>k</sup> 0 pk

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

� �

¼ ∥vec FkHkG � pkΞ<sup>k</sup>

2

2 ≤p<sup>2</sup> kε

� � � � �

FkHkG � Ξ<sup>k</sup>

∥G∥<sup>2</sup> F

a<sup>k</sup> pk

<sup>k</sup>MSE<sup>k</sup> ¼ FkHkG � pkΞ<sup>k</sup> �

ak

�" # � � � �

βk

∀k ¼ 1, … ,K:

min <sup>G</sup>, pk f g , <sup>∀</sup><sup>k</sup>

<sup>s</sup>:t: : <sup>1</sup> pk

following proposition is proposed.

s:t: :

q

p2

problem given by

where β<sup>k</sup> ¼

and pk.

127

� � � �

∀k ¼ 1, … , K:

min <sup>G</sup>, pk f g , <sup>∀</sup><sup>k</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2 <sup>n</sup><sup>k</sup> þ σ<sup>2</sup> w<sup>k</sup>

¼

pk ffiffi ε p a<sup>H</sup>

<sup>F</sup>kHk<sup>G</sup> � <sup>Ξ</sup>k∥<sup>2</sup>

Tr GG<sup>H</sup> � �

� � � �

2

F

<sup>F</sup> <sup>þ</sup> Lkσ<sup>2</sup> nk � � <sup>≥</sup>ψk,

<sup>þ</sup> <sup>σ</sup><sup>2</sup>

It can be proved that a sufficient and necessary condition for the feasibility of

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

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

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

<sup>F</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> ρk ! <sup>1</sup>

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � � 1

p2 k k k F<sup>k</sup> 2 <sup>F</sup> ≤ εk,

⪰0 , ∀k ¼ 1, … , K,

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � �k k <sup>F</sup><sup>k</sup>

> <sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � �k k <sup>F</sup><sup>k</sup>

<sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

� � is affine jointly in G

2 F

> 2 F

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> ρk ! k k <sup>F</sup><sup>k</sup>

> p2 k k k F<sup>k</sup> 2 <sup>F</sup> ≤εk,

2 F p2 k

(14)

(15)

(16)

(17)

(18)

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

$$\mathbf{F}\_k = \left(\mathbf{H}\_k \mathbf{G} \mathbf{E}\_k^H\right)^H \left[\mathbf{H}\_k \mathbf{G} \mathbf{G}^H \mathbf{H}\_k^H + \left(\sigma\_{\mathbf{n}\_k}^2 + \frac{\sigma\_{\mathbf{w}\_k}^2}{\rho\_k}\right) \mathbf{I}\right]^{-1}.\tag{12}$$

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.


#### 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 F~k, ∀k, if problem (10) is feasible and its optimal solution can be obtained, then the proposed Algorithm 1 is convergent.

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 Fk, ∀k should be carefully chosen such that proper transmitters and the PS factors can be obtained in the first iteration of the Algorithm 1. This means that it is vital to find the optimal solution for the subproblem (10). Therefore, before proceeding, the feasibility of the subproblem (10) is investigated in the following subsection.

#### 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

$$
\varepsilon\_k - \left(\sigma\_{\mathbf{n}\_k}^2 + \sigma\_{\mathbf{w}\_k}^2\right) \|\mathbf{F}\_k\|\_{\mathcal{F}}^2 \ge \mathbf{0}.\tag{13}
$$

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 pk to the receiver F<sup>k</sup> in the transceiver design model (10), that is, <sup>1</sup> pk Fk. The expression of MSE is then recast as

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems DOI: http://dx.doi.org/10.5772/intechopen.89676

$$\text{MSE}\_{k} = \left\| \frac{\mathbf{1}}{p\_{k}} \mathbf{F}\_{k} \mathbf{H}\_{k} \mathbf{G} - \boldsymbol{\Xi}\_{k} \right\|\_{\text{F}}^{2} + \left( \sigma\_{\mathbf{n}\_{k}}^{2} + \frac{\sigma\_{\mathbf{w}\_{k}}^{2}}{\rho\_{k}} \right) \frac{\left\| \mathbf{F}\_{k} \right\|\_{\text{F}}^{2}}{p\_{k}^{2}} \tag{14}$$

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

$$\begin{split} & \min\_{\left\{ \mathbf{G}, p\_k, \rho\_k, \forall k \right\}} \quad \text{Tr} \left( \mathbf{G} \mathbf{G}^H \right) \\ & \text{s.t.} : \left\| \frac{1}{p\_k} \mathbf{F}\_k \mathbf{H}\_k \mathbf{G} - \boldsymbol{\Xi}\_k \right\|\_{\mathrm{F}}^2 + \left( \sigma\_{\mathbf{n}\_k}^2 + \frac{\sigma\_{\mathbf{w}\_k}^2}{\rho\_k} \right) \frac{1}{p\_k^2} \left\| \mathbf{F}\_k \right\|\_{\mathrm{F}}^2 \le \varepsilon\_k, \\ & \quad \quad \xi\_k (\mathbf{1} - \rho\_k) \left( \| \mathbf{H}\_k \mathbf{G} \|\_{\mathrm{F}}^2 + L\_k \sigma\_{\mathbf{n}\_k}^2 \right) \ge \nu\_k, \\ & \forall k = \mathbf{1}, \ldots, K. \end{split} \tag{15}$$

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:

$$\begin{array}{ll}\min\_{\{\mathbf{G}, p\_k, \mathbb{W}k\}} & \mathrm{Tr}\left(\mathbf{G}\mathbf{G}^{\mathrm{H}}\right) \\\\ \text{s.t.} & \left\|\frac{1}{p\_k}\mathbf{F}\_k\mathbf{H}\_k\mathbf{G}-\boldsymbol{\Xi}\_k\right\|\_{\mathrm{F}}^2 + \left(\sigma\_{\mathbf{n}\_k}^2 + \sigma\_{\mathbf{w}\_k}^2\right)\frac{1}{p\_k^2}||\mathbf{F}\_k||\_{\mathrm{F}}^2 \leq \varepsilon\_k, \\ & \forall k=1,\ldots,K. \end{array} \tag{16}$$

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 Fk, ∀k, problem (16) can be reformulated as a convex SDP problem given by

$$\begin{aligned} \min\_{\{\mathbf{G}, p\_k, \forall k\}} \quad & \|\mathbf{G}\|\_{\rm F}^2\\ \text{s.t.} \quad & \begin{bmatrix} p\_k \sqrt{\varepsilon} & \mathbf{a}\_k^H & \beta\_k\\ \mathbf{a}\_k & p\_k \sqrt{\varepsilon} \mathbf{I} & \mathbf{0}\\ \beta\_k & \mathbf{0} & p\_k \sqrt{\varepsilon} \end{bmatrix} \succeq \mathbf{0} \ , \forall k = 1, \ldots, K, \end{aligned} \tag{17}$$

where β<sup>k</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ2 <sup>n</sup><sup>k</sup> þ σ<sup>2</sup> w<sup>k</sup> q k k F<sup>k</sup> <sup>F</sup>, and a<sup>k</sup> ¼ vec FkHkG � pkΞ<sup>k</sup> � � is affine jointly in G and pk.

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

$$\begin{aligned} \|p\_k^2 \text{MSE}\_k &= \left\| \mathbf{F}\_k \mathbf{H}\_k \mathbf{G} - p\_k \mathbf{E}\_k \right\|\_\text{F}^2 + \left(\sigma\_{\mathbf{n}\_k}^2 + \sigma\_{\mathbf{w}\_k}^2\right) \|\mathbf{F}\_k\|\_\text{F}^2 \\ &= \|\text{vec}\left(\mathbf{F}\_k \mathbf{H}\_k \mathbf{G} - p\_k \mathbf{E}\_k\right)\|\_2^2 + \left(\sigma\_{\mathbf{n}\_k}^2 + \sigma\_{\mathbf{w}\_k}^2\right) \|\mathbf{F}\_k\|\_\text{F}^2 \\ &= \left\| \begin{bmatrix} \mathbf{a}\_k \\ \boldsymbol{\rho}\_k \end{bmatrix} \right\|\_2^2 \le p\_k^2 \varepsilon \end{aligned} \tag{18}$$

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 k is relevant to Fk, we can optimize the ID receiver by minimizing the MSE. The

Problem (11) is the traditional unconstrained MSE minimization problem and its

Therefore, by alternatively optimizing the transmitter together with PS factors according to (10) and the receivers according to (12), an iterative optimization

1: Initialize the receivers <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>F</sup>~k, <sup>∀</sup><sup>k</sup> and the power splitting factors <sup>ρ</sup>k, <sup>∀</sup>k. 2: Optimize the transmitter G and the PS factors ρk, ∀k by solving problem (10).

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

For the AO framework, it is vital to analyze its convergence property. The

Proposition 3 For the initial receivers F~k, ∀k, if problem (10) is feasible and its optimal solution can be obtained, then the proposed Algorithm 1 is convergent. 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 Fk, ∀k should be carefully chosen such that proper transmitters and the PS factors can be obtained in the first iteration of the Algorithm 1. This means that it is vital to find the optimal solution for the subproblem (10). Therefore, before proceeding, the feasibility of the subproblem (10) is inves-

By checking the MSE constraints of (9), a necessary condition for the feasibility

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 pk to the receiver F<sup>k</sup> in the transceiver design

Fk. The expression of MSE is then recast as

k k F<sup>k</sup> 2

<sup>F</sup> ≥0: (13)

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � �

<sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

" #�<sup>1</sup>

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> ρk

!

MSEk: (11)

I

: (12)

min f g F<sup>k</sup>

optimization problem can be formulated as

Recent Wireless Power Transfer Technologies

<sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>H</sup>kGΞ<sup>H</sup>

k � �<sup>H</sup> <sup>H</sup>kGG<sup>H</sup>H<sup>H</sup>

framework is established and is summarized in Algorithm 1.

3: Optimize the receive filters Fk, ∀k according to (12).

Algorithm 1 Alternating optimization framework for JTDPS.

closed-form solution can be given by

reached.

2.3.2 Convergence analysis

following proposition reveals this property.

tigated in the following subsection.

of problem (9) is established as

pk

model (10), that is, <sup>1</sup>

126

2.3.3 Feasibility of the transmitter design subproblem

<sup>ε</sup><sup>k</sup> � <sup>σ</sup><sup>2</sup>

1: Given the MSE requirements εk, ∀k. 2: Generate random matrix F~<sup>k</sup> ∈ C<sup>M</sup>�<sup>N</sup>, ∀k. 3: By fixing <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>F</sup>~k, solve (17) to obtain the optimized receivers <sup>G</sup> and PS factors pk, <sup>∀</sup>k. 4: Return <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup> pk F~k, ∀k.

#### Table 1.

The initializing procedure for algorithm 1.

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

$$
\begin{bmatrix}
p\_k \sqrt{\varepsilon} & \mathbf{a}\_k^H & \beta\_k \\
\mathbf{a}\_k & p\_k \sqrt{\varepsilon} \mathbf{I} & \mathbf{O} \\
\beta\_k & \mathbf{O} & p\_k \sqrt{\varepsilon}
\end{bmatrix} \succeq \mathbf{0}
\quad . \tag{19}
$$

min G, ρ<sup>k</sup> f g ,ck, dk, ∀k

s:t: : Tr G<sup>H</sup>H<sup>H</sup>

Tr G<sup>H</sup>H<sup>H</sup>

ck σ<sup>w</sup> σ<sup>w</sup> ρ<sup>k</sup>

" #

dk

relaxed as

s:t: : Tr H<sup>H</sup>

Tr H<sup>H</sup>

ck σ<sup>w</sup> σ<sup>w</sup> ρ<sup>k</sup> " #

dk

X G G<sup>H</sup> I<sup>d</sup> " #

[23, 26].

129

ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> <sup>p</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> " #

min <sup>X</sup> <sup>⪰</sup> <sup>0</sup>, <sup>G</sup>, <sup>ρ</sup><sup>k</sup> f g ,ck, dk, <sup>∀</sup><sup>k</sup>

<sup>k</sup> F<sup>H</sup>

<sup>k</sup> <sup>H</sup>k<sup>X</sup> � � <sup>þ</sup> Lkσ<sup>2</sup>

⪰0,

0≤ ρ<sup>k</sup> ≤1, ∀k ¼ 1, … ,K,

⪰0:

problem (22), if <sup>X</sup>SDR <sup>¼</sup> <sup>G</sup>SDRG<sup>H</sup>

solving problem (10) is finally obtained.

ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> p

ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> <sup>p</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup>

0≤ρ<sup>k</sup> ≤1, ∀k ¼ 1, … ,K:

relaxed as X ⪰ GGH, which is equivalent to

Trð Þ X

<sup>n</sup><sup>k</sup> ≥dk,

⪰0,

<sup>k</sup> F<sup>H</sup> <sup>k</sup> Ξ<sup>k</sup> � � � Tr <sup>Ξ</sup><sup>H</sup>

Problem (22) is a convex SDP with respect to <sup>G</sup> <sup>∈</sup>C<sup>M</sup>�<sup>d</sup>

Hermitian symmetric variable <sup>X</sup> <sup>∈</sup>C<sup>M</sup>�<sup>M</sup> and nonnegative variables <sup>ρ</sup>k,ck, dk, <sup>∀</sup><sup>k</sup> and thus can be solved efficiently by using traditional convex optimization techniques

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 XSDR and GSDR denote the optimal solution of the SDR

not yet proven, simulation results show that the relaxation is always tight, that is, the equality in the relaxation is always satisfied. Replacing G in step 2 of Algorithm 1 with GSDR results in an SDP-based JTDPS (SDP-JTDPS) algorithm, a practical algorithm

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

solution accuracy ϵ> 0, the computational complexity solving SDP is about

<sup>k</sup> <sup>F</sup>kHk<sup>X</sup> � � � Tr <sup>G</sup>HH<sup>H</sup>

<sup>n</sup><sup>k</sup> þ ck � �Tr <sup>F</sup><sup>H</sup>

<sup>þ</sup> <sup>σ</sup><sup>2</sup>

Tr GG<sup>H</sup> � �

<sup>k</sup> F<sup>k</sup>

<sup>k</sup> <sup>F</sup>kHk<sup>G</sup> � � � Tr <sup>Ξ</sup>kG<sup>H</sup>H<sup>H</sup>

� �<sup>≤</sup> <sup>ε</sup><sup>k</sup> � Nk,

<sup>n</sup><sup>k</sup> ≥dk,

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

Problem (21) is a nonconvex inhomogeneous quadratically constrained quadratic program (QCQP) [20, 21], which are NP-hard [10, 22–25]. In order to solve it effectively, SDR is utilized. Specifically, a new variable <sup>X</sup> <sup>¼</sup> GG<sup>H</sup> is defined and

X G G<sup>H</sup> I<sup>d</sup>

<sup>k</sup> <sup>F</sup>kHkk<sup>G</sup> � � <sup>þ</sup> <sup>σ</sup><sup>2</sup>

� �⪰0, problem (21) can be

<sup>n</sup><sup>k</sup> þ ck � �Tr <sup>F</sup><sup>H</sup>

SDR, then GSDR must be optimal for (21). Although

<sup>k</sup> F<sup>k</sup> � �<sup>≤</sup> <sup>ε</sup><sup>k</sup> � Nk,

, positive semidefinite

(22)

<sup>k</sup> F<sup>H</sup> k � � � Tr <sup>F</sup>kHkkGΞ<sup>H</sup>

k � �

(21)

<sup>k</sup> F<sup>H</sup>

DOI: http://dx.doi.org/10.5772/intechopen.89676

<sup>k</sup> <sup>H</sup>k<sup>G</sup> � � <sup>þ</sup> Lkσ<sup>2</sup>

⪰0,

ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> p

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 F<sup>k</sup> can be initialized by any randomly generated matrix, that is, <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>F</sup>~<sup>k</sup> <sup>∈</sup>CLk�<sup>M</sup>, <sup>∀</sup>k. Then, by solving (17), pk is obtained. Finally, the receiver F<sup>k</sup> is constructed to be <sup>1</sup> pk F~k, such that the problem (16) is feasible. The initialization process is summarized in Table 1.

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 ck and dk with <sup>σ</sup><sup>2</sup> wk ρk ≤ck and ψk <sup>ξ</sup><sup>k</sup> <sup>1</sup>�ρ<sup>k</sup> ð Þ <sup>≤</sup> dk, <sup>∀</sup><sup>k</sup> <sup>¼</sup> 1, … , <sup>K</sup>, (10) can be rewritten as

$$\begin{aligned} &\min\_{\{\mathbf{G}, \rho\_k, c\_k, d\_k, \psi\_k\}} \quad \text{Tr}\left(\mathbf{G} \mathbf{G}^H\right) \\ &\text{ s.t. } : \text{Tr}\left(\mathbf{G}^H \mathbf{H}\_k^H \mathbf{F}\_k^H \mathbf{F}\_k \mathbf{H}\_k \mathbf{G}\right) - \text{Tr}\left(\mathbf{E}\_k \mathbf{G}^H \mathbf{H}\_k^H \mathbf{F}\_k^H\right) - \text{Tr}\left(\mathbf{F}\_k \mathbf{H}\_{kk} \mathbf{G} X\_k^H\right) \\ &\quad + \left(\sigma\_{\mathbf{n}\_k}^2 + c\_k\right) \text{Tr}\left(\mathbf{F}\_k^H \mathbf{F}\_k\right) \le \varepsilon\_k - N\_k, \\ &\quad \text{Tr}\left(\mathbf{G}^H \mathbf{H}\_k^H \mathbf{H}\_k \mathbf{G}\right) + L\_k \sigma\_{\mathbf{n}\_k}^2 \ge d\_k, \\ &\quad \frac{\sigma\_{\mathbf{n}\_k}^2}{\rho\_k} \le c\_k, \\ &\quad \frac{\Psi\_k}{\xi\_k \left(1 - \rho\_k\right)} \le d\_k, \\ &\quad 0 \le \rho\_k \le 1, \forall k = 1, \dots, K. \end{aligned} \tag{20}$$

Adopting the Schur complement lemma, the constraints <sup>σ</sup><sup>2</sup> wk ρk ≤ ck and <sup>ψ</sup><sup>k</sup> <sup>ξ</sup><sup>k</sup> <sup>1</sup>�ρ<sup>k</sup> ð Þ <sup>≤</sup>dk can be reformulated as ck <sup>σ</sup><sup>w</sup> σ<sup>w</sup> ρ<sup>k</sup> � �⪰<sup>0</sup> and dk ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> p ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> <sup>p</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> " #⪰0, respectively. Then, problem (21) can be further rewritten as

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems DOI: http://dx.doi.org/10.5772/intechopen.89676

min G, ρ<sup>k</sup> f g ,ck, dk, ∀k Tr GG<sup>H</sup> � � s:t: : Tr G<sup>H</sup>H<sup>H</sup> <sup>k</sup> F<sup>H</sup> <sup>k</sup> <sup>F</sup>kHk<sup>G</sup> � � � Tr <sup>Ξ</sup>kG<sup>H</sup>H<sup>H</sup> <sup>k</sup> F<sup>H</sup> k � � � Tr <sup>F</sup>kHkkGΞ<sup>H</sup> k � � <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>n</sup><sup>k</sup> þ ck � �Tr <sup>F</sup><sup>H</sup> <sup>k</sup> F<sup>k</sup> � �<sup>≤</sup> <sup>ε</sup><sup>k</sup> � Nk, Tr G<sup>H</sup>H<sup>H</sup> <sup>k</sup> <sup>H</sup>k<sup>G</sup> � � <sup>þ</sup> Lkσ<sup>2</sup> <sup>n</sup><sup>k</sup> ≥dk, ck σ<sup>w</sup> σ<sup>w</sup> ρ<sup>k</sup> " #⪰0, dk ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> p ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> <sup>p</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> 2 6 6 4 3 7 7 5 ⪰0, 0≤ρ<sup>k</sup> ≤1, ∀k ¼ 1, … ,K: (21)

Problem (21) is a nonconvex inhomogeneous quadratically constrained quadratic program (QCQP) [20, 21], which are NP-hard [10, 22–25]. In order to solve it effectively, SDR is utilized. Specifically, a new variable <sup>X</sup> <sup>¼</sup> GG<sup>H</sup> is defined and relaxed as X ⪰ GGH, which is equivalent to X G G<sup>H</sup> I<sup>d</sup> � �⪰0, problem (21) can be

relaxed as

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

<sup>k</sup> β<sup>k</sup>

ffiffi ε p

> wk ρk

<sup>k</sup> F<sup>H</sup> k � � � Tr <sup>F</sup>kHkkGX<sup>H</sup>

dk

ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> <sup>p</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup>

" #

≤ck and

k

(20)

<sup>ξ</sup><sup>k</sup> <sup>1</sup>�ρ<sup>k</sup> ð Þ <sup>≤</sup>dk

� �

wk ρk

ffiffiffiffiffiffiffiffiffiffiffiffi ψk=ξ<sup>k</sup> p

≤ ck and <sup>ψ</sup><sup>k</sup>

⪰0, respectively.

3 7

<sup>5</sup><sup>⪰</sup> <sup>0</sup> : (19)

pk F~k,

ffiffi ε p I 0

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 F<sup>k</sup> can be initialized by any randomly generated matrix, that is, <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>F</sup>~<sup>k</sup> <sup>∈</sup>CLk�<sup>M</sup>, <sup>∀</sup>k. Then,

β<sup>k</sup> 0 pk

3: By fixing <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>F</sup>~k, solve (17) to obtain the optimized receivers <sup>G</sup> and PS factors pk, <sup>∀</sup>k.

by solving (17), pk is obtained. Finally, the receiver F<sup>k</sup> is constructed to be <sup>1</sup>

ing, the SDP relaxation is adopted to reform it in to convex form.

By introducing two variables ck and dk with <sup>σ</sup><sup>2</sup>

<sup>ξ</sup><sup>k</sup> <sup>1</sup>�ρ<sup>k</sup> ð Þ <sup>≤</sup> dk, <sup>∀</sup><sup>k</sup> <sup>¼</sup> 1, … , <sup>K</sup>, (10) can be rewritten as

Tr F<sup>H</sup> <sup>k</sup> F<sup>k</sup>

<sup>k</sup> F<sup>H</sup>

<sup>k</sup> <sup>H</sup>k<sup>G</sup> � � <sup>þ</sup> Lkσ<sup>2</sup>

0≤ρ<sup>k</sup> ≤1, ∀k ¼ 1, … , K:

Then, problem (21) can be further rewritten as

<sup>n</sup><sup>k</sup> þ ck � � Tr GG<sup>H</sup> � �

<sup>k</sup> <sup>F</sup>kHk<sup>G</sup> � � � Tr <sup>Ξ</sup>kG<sup>H</sup>H<sup>H</sup>

Adopting the Schur complement lemma, the constraints <sup>σ</sup><sup>2</sup>

⪰0 and

σ<sup>w</sup> ρ<sup>k</sup> � �

� �≤ε<sup>k</sup> � Nk,

<sup>n</sup><sup>k</sup> ≥dk,

such that the problem (16) is feasible. The initialization process is summarized in

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 follow-

pk ffiffi ε p a<sup>H</sup>

2 6 4

The proposition is obtained.

1: Given the MSE requirements εk, ∀k. 2: Generate random matrix F~<sup>k</sup> ∈ C<sup>M</sup>�<sup>N</sup>, ∀k.

Recent Wireless Power Transfer Technologies

pk F~k, ∀k.

The initializing procedure for algorithm 1.

4: Return <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>

Table 1.

2.3.4 Algorithm description

min G, ρ<sup>k</sup> f g ,ck, dk, ∀k

s:t: : Tr G<sup>H</sup>H<sup>H</sup>

<sup>þ</sup> <sup>σ</sup><sup>2</sup>

σ2 w<sup>k</sup> ρk

Tr G<sup>H</sup>H<sup>H</sup>

ψk <sup>ξ</sup><sup>k</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> ð Þ <sup>≤</sup> dk,

can be reformulated as ck <sup>σ</sup><sup>w</sup>

≤ck,

Table 1.

ψk

128

a<sup>k</sup> pk

$$\begin{aligned} &\min\_{\begin{subarray}{c}\mathbf{X}\in\mathcal{Q},\boldsymbol{\sigma}\_{k},\boldsymbol{c}\_{k},d\_{k},\forall k\end{subarray}}\text{Tr}(\mathbf{X})\\ &\text{ s.t.:}\,\mathbf{Tr}\left(\mathbf{H}\_{k}^{\mathrm{H}}\mathbf{H}\_{k}^{\mathrm{H}}\mathbf{H}\_{k}\mathbf{X}\_{k}\right)-\mathrm{Tr}\left(\mathbf{G}^{\mathrm{H}}\mathbf{H}\_{k}^{\mathrm{H}}\mathbf{F}\_{k}^{\mathrm{H}}\mathbf{E}\_{k}\right)-\mathrm{Tr}\left(\boldsymbol{\Xi}\_{k}^{\mathrm{H}}\mathbf{F}\_{k}\mathbf{H}\_{k}\mathbf{G}\right)+\left(\sigma\_{\mathbf{n}\_{k}}^{2}+c\_{k}\right)\mathrm{Tr}\left(\mathbf{F}\_{k}^{\mathrm{H}}\mathbf{F}\_{k}\right)\leq e\_{k}-N\_{k},\\ &\text{Tr}\left(\mathbf{H}\_{k}^{\mathrm{H}}\mathbf{H}\_{k}\mathbf{X}\right)+L\_{k}\sigma\_{\mathbf{n}\_{k}}^{2}\geq d\_{k},\\ &\left[\begin{matrix}c\_{k}&\sigma\_{\sigma}\\ \sigma\_{w}&\rho\_{k}\end{matrix}\right]\geq\mathbf{0},\\ &\left[\begin{matrix}d\_{k}&\sqrt{\boldsymbol{\nu}\_{k}/\boldsymbol{\xi}\_{k}}\\ \sqrt{\boldsymbol{\nu}\_{k}/\boldsymbol{\xi}\_{k}}&1-\rho\_{k}\end{matrix}\right]\geq\mathbf{0},\\ &0\leq\rho\_{k}\leq 1,\forall k=1,\ldots,K,\\ &\left[\begin{matrix}\mathbf{X}\quad\mathbf{G}\quad\mathbf{}\\ \mathbf{G}^{\mathrm{H}}\quad\mathbf{I}\_{d}\end{matrix}\right]\geq\mathbf{0}.\tag{22} \end{aligned} \tag{23}$$

Problem (22) is a convex SDP with respect to <sup>G</sup> <sup>∈</sup>C<sup>M</sup>�<sup>d</sup> , positive semidefinite Hermitian symmetric variable <sup>X</sup> <sup>∈</sup>C<sup>M</sup>�<sup>M</sup> and nonnegative variables <sup>ρ</sup>k,ck, dk, <sup>∀</sup><sup>k</sup> and thus can be solved efficiently by using traditional convex optimization techniques [23, 26].

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 XSDR and GSDR denote the optimal solution of the SDR problem (22), if <sup>X</sup>SDR <sup>¼</sup> <sup>G</sup>SDRG<sup>H</sup> SDR, then GSDR must be optimal for (21). Although not yet proven, simulation results show that the relaxation is always tight, that is, the equality in the relaxation is always satisfied. Replacing G in step 2 of Algorithm 1 with GSDR results in an SDP-based JTDPS (SDP-JTDPS) algorithm, a practical algorithm solving problem (10) is finally obtained.

The complexity of Algorithm 1 is mainly introduced by the SDP (22). Given a solution accuracy ϵ> 0, the computational complexity solving SDP is about

<sup>O</sup> <sup>N</sup>IterM4:<sup>5</sup> log 1ð Þ <sup>=</sup><sup>ϵ</sup> � � [10], where <sup>N</sup>Iter is the iteration number. Therefore, the computational complexity of the algorithm is prohibitively high when the system becomes large in number of antennas and users. So, it is necessary to develop low-complexity algorithms.

0<ρ<sup>k</sup> <1, ∀k ¼ 1, … , K: (30)

� �. Since the MSE constraints are satisfied with

<sup>n</sup><sup>k</sup> ¼ ck, and

<sup>k</sup> <sup>¼</sup> Bk<sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 <sup>k</sup>�4AkCk

k .

<sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup> wk α<sup>∗</sup> ck�σ<sup>2</sup> nk , we

<sup>k</sup> according to (24)

<sup>2</sup>Ak .

(32)

p

<sup>w</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

(31)

By summing (27) and (28) for user k, problem (26–30) is equivalent to

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

nk

wk ck�σ<sup>2</sup> nk

When <sup>α</sup> <sup>&</sup>gt;1, the function Fkð Þ¼ <sup>α</sup> 1 has a unique solution <sup>α</sup><sup>∗</sup>

min f g<sup>α</sup> <sup>α</sup> s:t: : α≥ α<sup>∗</sup>

The optimal solution of problem (32) is given by <sup>α</sup><sup>∗</sup> <sup>¼</sup> max <sup>∀</sup><sup>k</sup>α<sup>∗</sup>

<sup>k</sup> . Thus, α<sup>∗</sup> , ρ <sup>∗</sup>

<sup>α</sup><sup>∗</sup> <sup>p</sup> <sup>G</sup>^ , <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>ffiffiffiffi

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

constraint power minimization algorithm proposed in [17]. 2: Optimize the optimal scaling factor α<sup>∗</sup> and the PS factors ρ <sup>∗</sup>

wk α<sup>∗</sup> ck�σ<sup>2</sup> nk

s:t: : Fkð Þ α ≤1, ∀k ¼ 1, … ,K,

<sup>¼</sup> 1 or <sup>σ</sup><sup>2</sup>

� � <sup>&</sup>gt; 1. It is known that Fkð Þ <sup>α</sup> will monotonically decrease

. Thus, Fkð Þ α decreases monotonically when α >1.

<sup>k</sup> , ∀k:

<sup>þ</sup> <sup>ψ</sup><sup>k</sup> ξ<sup>k</sup> α<sup>∗</sup> dkþLkσ<sup>2</sup>

<sup>α</sup><sup>∗</sup> <sup>p</sup> <sup>F</sup>^k, <sup>∀</sup><sup>k</sup>

nk � � <sup>≤</sup>1. Let <sup>ρ</sup> <sup>∗</sup>

n o, the transceiver which is feasible to problem (7) can

<sup>α</sup><sup>∗</sup> <sup>p</sup> <sup>G</sup>^ , <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>ffiffiffiffi

The main computational complexity of Algorithm 2 comes from solving problem (9), which is of <sup>O</sup> <sup>N</sup>Iterd<sup>3</sup> � � [17]. Thus, the complexity of Algorithm 2 is quite lower

The performance of the described algorithms is validated through simulations. The user number is set to be K ¼ 3. The channel matrices Hk, ∀k are set as i.i.d. zero

n o. The design process is summarized

<sup>k</sup> , <sup>∀</sup><sup>k</sup> � � is an optimal solution for (26–30).

n o based on the traditional MSE QoS

<sup>α</sup><sup>∗</sup> <sup>p</sup> <sup>F</sup>^k, <sup>ρ</sup> <sup>∗</sup>

n o to problem (7).

<sup>k</sup> , ∀k

min f g<sup>α</sup> <sup>α</sup>

<sup>þ</sup> <sup>ψ</sup><sup>k</sup> ξ<sup>k</sup> αdkþLkσ<sup>2</sup>

where Fkð Þ¼ <sup>α</sup> <sup>σ</sup><sup>2</sup>

Fkð Þ¼ <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>ψ</sup><sup>k</sup>

nk ck <sup>¼</sup> <sup>σ</sup><sup>2</sup> nk σ2 nk þσ<sup>2</sup> wk

when <sup>α</sup><sup>&</sup>gt; <sup>σ</sup><sup>2</sup>

have <sup>ψ</sup><sup>k</sup>

in Algorithm 2.

and (25).

than that of Algorithm 1.

131

2.5 Simulation results and analysis

ξ<sup>k</sup> α<sup>∗</sup> dkþLkσ<sup>2</sup>

nk � � <sup>≤</sup> <sup>1</sup> � <sup>ρ</sup> <sup>∗</sup>

Given α<sup>∗</sup> and G^ , F^k, ∀k

be determined by <sup>G</sup> <sup>¼</sup> ffiffiffiffiffiffi

wk αck�σ<sup>2</sup> nk

DOI: http://dx.doi.org/10.5772/intechopen.89676

equality when <sup>α</sup> <sup>¼</sup> 1, there exists <sup>σ</sup><sup>2</sup>

ξ<sup>k</sup> dkþLkσ<sup>2</sup> nk

Problem (31) is then equivalent to

For the optimal <sup>α</sup><sup>∗</sup> , Fk <sup>α</sup><sup>∗</sup> ð Þ¼ <sup>σ</sup><sup>2</sup>

1: Solve problem (9) to obtain G^ , F^k, ∀k

3: Return the feasible solution <sup>G</sup> <sup>¼</sup> ffiffiffiffiffiffi

α> 1,

#### 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 G^ , F^k, ∀k n o denote its solution, amplify the precoder G^ by a positive scaling factor ffiffiffi <sup>α</sup> <sup>p</sup> <sup>&</sup>gt;1, and decrease the receiver F^<sup>k</sup> by the factor 1= ffiffiffi α p , the problem (9) can be rewritten as

$$\begin{aligned} &\min\_{\{\boldsymbol{\alpha},\boldsymbol{\rho}\_{k},\boldsymbol{\Psi}\}} \quad \boldsymbol{\alpha} \text{Tr} \left( \hat{\mathbf{G}} \hat{\mathbf{G}}^{\text{H}} \right) \\ &\text{ s.t. } : \left\| \hat{\mathbf{F}}\_{k} \mathbf{H}\_{k} \hat{\mathbf{G}} - \boldsymbol{\Xi}\_{k} \right\|\_{\text{F}}^{2} + \left( \sigma\_{\mathbf{n}\_{k}}^{2} + \frac{\sigma\_{\mathbf{w}\_{k}}^{2}}{\rho\_{k}} \right) \frac{1}{a} \left\| \hat{\mathbf{F}}\_{k} \right\|\_{\text{F}}^{2} \leq \epsilon\_{k}, \\ &\quad \xi\_{k} (1 - \rho\_{k}) \left( a \| \hat{\mathbf{H}}\_{k} \hat{\mathbf{G}} \|\_{F}^{2} + L\_{k} \sigma\_{\mathbf{n}\_{k}}^{2} \right) \geq \nu\_{k}, \\ &\quad a > 1, \\ &\quad 0 \leq \rho\_{k} \leq 1, \\ &\forall k = 1, \ldots, K, \end{aligned} \tag{23}$$

where the scaling factor α and the PS factors in (7) are jointly optimized to satisfy both the MSE and EH constraints. Problem (23) can be solved in closed form, which is shown by the following proposition.

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

$$a^\* = \max\_{\mathbb{M}} a^\*\_{k},\tag{24}$$

$$
\rho\_k^\* = \frac{\sigma\_{\mathbf{w}\_k}^2}{a^\* c\_k - \sigma\_{\mathbf{n}\_k}^2},
\tag{25}
$$

$$\begin{split} \text{where } a\_{k}^{\*} &= \frac{B\_{k} + \sqrt{\boldsymbol{\mu}\_{k}^{2} + 4\boldsymbol{A}\_{k}\boldsymbol{\Sigma}\_{k}}}{2A\_{k}} \text{with } c\_{k} = \frac{\boldsymbol{\nu}\_{k} - \left\|\boldsymbol{\hat{\boldsymbol{\nu}}}\_{k}\right\|\_{\mathrm{F}}^{2}}{\left\|\boldsymbol{\hat{\boldsymbol{\nu}}}\_{k}\right\|\_{\mathrm{F}}^{2}}, d\_{k} = \left\|\boldsymbol{\hat{\boldsymbol{\mu}}}\_{k}\boldsymbol{\hat{\boldsymbol{\sigma}}}\right\|\_{\mathrm{F}}^{2}, A\_{k} = c\_{k}\boldsymbol{\xi}\_{k}d\_{k}, \\ B\_{k} = \boldsymbol{\xi}\_{k}d\_{k} \left(\sigma\_{\mathbf{n}\_{k}}^{2} + \sigma\_{\mathbf{w}\_{k}}^{2}\right) + c\_{k}\boldsymbol{\mu}\_{k} - c\_{k}\boldsymbol{\xi}\_{k}L\_{k}\sigma\_{\mathbf{n}\_{k}}^{2}, \text{ and } C\_{k} = \boldsymbol{\nu}\_{k}\sigma\_{\mathbf{n}\_{k}}^{2} - \boldsymbol{\xi}\_{k}L\_{k}\sigma\_{\mathbf{n}\_{k}}^{2} \left(\sigma\_{\mathbf{n}\_{k}}^{2} + \sigma\_{\mathbf{w}\_{k}}^{2}\right). \\ \textbf{Proof. Problem(23) can be transformed to} \end{split}$$

$$\min\_{\{a,\rho\_k,\forall k\}}\quad a\tag{26}$$

$$\text{s.t.} : \rho\_k \ge \frac{\sigma\_{\mathbf{w}\_k}^2}{ac\_k - \sigma\_{\mathbf{n}\_k}^2},\tag{27}$$

$$\mathbf{1} - \rho\_k \ge \frac{\boldsymbol{\Psi}\_k}{\xi\_k \left(\boldsymbol{ad}\_k + L\_k \sigma\_{\mathbf{n}\_k}^2\right)},\tag{28}$$

$$a > 1,\tag{29}$$

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems DOI: http://dx.doi.org/10.5772/intechopen.89676

$$0 < \rho\_k < 1, \forall k = 1, \ldots, K. \tag{30}$$

By summing (27) and (28) for user k, problem (26–30) is equivalent to

$$\begin{aligned} \min\_{\{a\}} &\quad a\\ \text{s.t.} &: F\_k(a) \le 1, \forall k = 1, \dots, K, \\ &\quad a > 1, \end{aligned} \tag{31}$$

where Fkð Þ¼ <sup>α</sup> <sup>σ</sup><sup>2</sup> wk αck�σ<sup>2</sup> nk <sup>þ</sup> <sup>ψ</sup><sup>k</sup> ξ<sup>k</sup> αdkþLkσ<sup>2</sup> nk � �. Since the MSE constraints are satisfied with

equality when <sup>α</sup> <sup>¼</sup> 1, there exists <sup>σ</sup><sup>2</sup> wk ck�σ<sup>2</sup> nk <sup>¼</sup> 1 or <sup>σ</sup><sup>2</sup> <sup>w</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>n</sup><sup>k</sup> ¼ ck, and Fkð Þ¼ <sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>ψ</sup><sup>k</sup> ξ<sup>k</sup> dkþLkσ<sup>2</sup> nk � � <sup>&</sup>gt; 1. It is known that Fkð Þ <sup>α</sup> will monotonically decrease when <sup>α</sup><sup>&</sup>gt; <sup>σ</sup><sup>2</sup> nk ck <sup>¼</sup> <sup>σ</sup><sup>2</sup> nk σ2 nk þσ<sup>2</sup> wk . Thus, Fkð Þ α decreases monotonically when α >1.

When <sup>α</sup> <sup>&</sup>gt;1, the function Fkð Þ¼ <sup>α</sup> 1 has a unique solution <sup>α</sup><sup>∗</sup> <sup>k</sup> <sup>¼</sup> Bk<sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 <sup>k</sup>�4AkCk p <sup>2</sup>Ak . Problem (31) is then equivalent to

$$\begin{array}{ll}\min\limits\_{\{a\}} & a\\ \text{s.t.} : a \ge a\_k^\*, \forall k. \end{array} \tag{32}$$

The optimal solution of problem (32) is given by <sup>α</sup><sup>∗</sup> <sup>¼</sup> max <sup>∀</sup><sup>k</sup>α<sup>∗</sup> k .

For the optimal <sup>α</sup><sup>∗</sup> , Fk <sup>α</sup><sup>∗</sup> ð Þ¼ <sup>σ</sup><sup>2</sup> wk α<sup>∗</sup> ck�σ<sup>2</sup> nk <sup>þ</sup> <sup>ψ</sup><sup>k</sup> ξ<sup>k</sup> α<sup>∗</sup> dkþLkσ<sup>2</sup> nk � � <sup>≤</sup>1. Let <sup>ρ</sup> <sup>∗</sup> <sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup> wk α<sup>∗</sup> ck�σ<sup>2</sup> nk , we

have <sup>ψ</sup><sup>k</sup> ξ<sup>k</sup> α<sup>∗</sup> dkþLkσ<sup>2</sup> nk � � <sup>≤</sup> <sup>1</sup> � <sup>ρ</sup> <sup>∗</sup> <sup>k</sup> . Thus, α<sup>∗</sup> , ρ <sup>∗</sup> <sup>k</sup> , <sup>∀</sup><sup>k</sup> � � is an optimal solution for (26–30).

Given α<sup>∗</sup> and G^ , F^k, ∀k n o, the transceiver which is feasible to problem (7) can be determined by <sup>G</sup> <sup>¼</sup> ffiffiffiffiffiffi <sup>α</sup><sup>∗</sup> <sup>p</sup> <sup>G</sup>^ , <sup>F</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>ffiffiffiffi <sup>α</sup><sup>∗</sup> <sup>p</sup> <sup>F</sup>^k, <sup>∀</sup><sup>k</sup> n o. The design process is summarized in Algorithm 2.

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


The main computational complexity of Algorithm 2 comes from solving problem (9), which is of <sup>O</sup> <sup>N</sup>Iterd<sup>3</sup> � � [17]. Thus, the complexity of Algorithm 2 is quite lower than that of Algorithm 1.

#### 2.5 Simulation results and analysis

The performance of the described algorithms is validated through simulations. The user number is set to be K ¼ 3. The channel matrices Hk, ∀k are set as i.i.d. zero

<sup>O</sup> <sup>N</sup>IterM4:<sup>5</sup> log 1ð Þ <sup>=</sup><sup>ϵ</sup> � � [10], where <sup>N</sup>Iter is the iteration number. Therefore, the computational complexity of the algorithm is prohibitively high when the system becomes large in number of antennas and users. So, it is necessary to develop

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

It is noted that the MSE-constrained transceiver design problem (9) can be

α p , the problem (9) can be rewritten as

1 <sup>α</sup> <sup>F</sup>^<sup>k</sup> � � � � 2 <sup>F</sup> ≤εk,

<sup>k</sup> , (24)

, dk <sup>¼</sup> <sup>∥</sup>H^ <sup>k</sup>G^ <sup>∥</sup><sup>2</sup>

<sup>α</sup>, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup><sup>k</sup> <sup>α</sup> (26)

� � , (28)

α >1, (29)

<sup>n</sup><sup>k</sup> � <sup>ξ</sup>kLkσ<sup>2</sup>

, (25)

<sup>n</sup><sup>k</sup> σ<sup>2</sup>

, (27)

<sup>F</sup>, Ak ¼ ckξkdk,

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � �

.

≥ψk,

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> ρk

<sup>F</sup> <sup>þ</sup> Lkσ<sup>2</sup> nk

!

n o

<sup>α</sup> <sup>p</sup> <sup>&</sup>gt;1, and decrease

denote

(23)

solved efficiently by existing methods proposed in [17]. So, let G^ , F^k, ∀k

� 2 <sup>F</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup>

� �

where the scaling factor α and the PS factors in (7) are jointly optimized to satisfy both the MSE and EH constraints. Problem (23) can be solved in closed form,

<sup>α</sup><sup>∗</sup> <sup>¼</sup> max

<sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

min

<sup>s</sup>:t: : <sup>ρ</sup><sup>k</sup> <sup>≥</sup> <sup>σ</sup><sup>2</sup>

<sup>1</sup> � <sup>ρ</sup><sup>k</sup> <sup>≥</sup> <sup>ψ</sup><sup>k</sup>

ρ ∗

<sup>þ</sup> ckψ<sup>k</sup> � ckξkLkσ<sup>2</sup>

<sup>2</sup>Ak with ck <sup>¼</sup> <sup>ε</sup>k� <sup>F</sup>^kH<sup>k</sup> k k <sup>G</sup>^�Ξ<sup>k</sup>

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

<sup>∀</sup><sup>k</sup> <sup>α</sup><sup>∗</sup>

w<sup>k</sup> α<sup>∗</sup> ck � σ<sup>2</sup>

> <sup>F</sup>^ k k<sup>k</sup> 2 F

<sup>n</sup><sup>k</sup> , and Ck <sup>¼</sup> <sup>ψ</sup>kσ<sup>2</sup>

w<sup>k</sup> αck � σ<sup>2</sup> nk

nk

ξ<sup>k</sup> αdk þ Lkσ<sup>2</sup>

nk

2 F

its solution, amplify the precoder G^ by a positive scaling factor ffiffiffi

low-complexity algorithms.

computational complexity.

the receiver F^<sup>k</sup> by the factor 1= ffiffiffi

min

α >1, 0≤ρ<sup>k</sup> ≤1, ∀k ¼ 1, … , K,

which is shown by the following proposition.

<sup>k</sup> <sup>¼</sup> Bk<sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2 <sup>k</sup>�4AkCk

Proof. Problem(23) can be transformed to

<sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w<sup>k</sup> � �

p

where α<sup>∗</sup>

Bk <sup>¼</sup> <sup>ξ</sup>kdk <sup>σ</sup><sup>2</sup>

130

<sup>α</sup>, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup><sup>k</sup> <sup>α</sup>Tr <sup>G</sup>^G^ <sup>H</sup> � �

<sup>s</sup>:t: : <sup>F</sup>^kHkG^ � <sup>Ξ</sup><sup>k</sup> �

� �

<sup>ξ</sup><sup>k</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> ð Þ <sup>α</sup>∥H^ <sup>k</sup>G^ <sup>∥</sup><sup>2</sup>

2.4 Low-complexity design scheme

Recent Wireless Power Transfer Technologies

mean complex Gaussian random variables with variance r �β <sup>k</sup> , where rk is the distance in meter between the jth transmitter and the kth receiver, and β is the path loss factor. The following parameters are set unless otherwise noted, M ¼ 8, Lk <sup>¼</sup> <sup>L</sup> <sup>¼</sup> 4, Nk <sup>¼</sup> <sup>N</sup> <sup>¼</sup> <sup>L</sup>=2, rk <sup>¼</sup> <sup>r</sup> <sup>¼</sup> 5, <sup>β</sup> <sup>¼</sup> <sup>2</sup>:7, <sup>σ</sup><sup>2</sup> <sup>n</sup><sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup> <sup>n</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup> , and σ2 <sup>w</sup><sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup> <sup>w</sup> <sup>¼</sup> <sup>10</sup>�<sup>2</sup> . The MSE and EH thresholds are set nonuniformly as ε<sup>1</sup> ¼ 0:01, ε<sup>2</sup> ¼ 0:02, ε<sup>3</sup> ¼ 0:2, ψ<sup>1</sup> ¼ 20, ψ<sup>2</sup> ¼ 25, and ψ<sup>3</sup> ¼ 30dBm. CVX toolbox [26] is adopted to solve SDP problems. The traditional MSE QoS constraint (MSE-QoS-TRAD) power minimization algorithm [17] is adopted as a performance benchmark.

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 explana-

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

Figure 3 shows the per-user MSE and harvested energy of the proposed scheme

that the SDP solution of the relaxed problem (22) is also optimal for the joint

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

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

Comparison of per-user MSE and harvested energy achieved by the proposed algorithms. The titles show the QoS

SDR is always satisfied during the simulations. This verifies

tion that <sup>X</sup>SDR <sup>¼</sup> <sup>G</sup>SDRG<sup>H</sup>

Figure 4.

Figure 5.

133

Transmit power versus number of trials.

targets.

transmitter and PS subproblem (10).

DOI: http://dx.doi.org/10.5772/intechopen.89676

requirements can also be satisfied.

Figure 2. Transmit power and PS factors versus iterations with the SDP-JTDPS algorithm.

Figure 3.

Achieved per-user MSE and harvested energy versus iterations by the proposed SDP-TDPS scheme. The titles show the QoS targets.

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems DOI: http://dx.doi.org/10.5772/intechopen.89676

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 <sup>X</sup>SDR <sup>¼</sup> <sup>G</sup>SDRG<sup>H</sup> SDR is always satisfied during the simulations. This verifies that the SDP solution of the relaxed problem (22) is also optimal for the joint transmitter and PS subproblem (10).

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

Figure 4.

mean complex Gaussian random variables with variance r

Lk <sup>¼</sup> <sup>L</sup> <sup>¼</sup> 4, Nk <sup>¼</sup> <sup>N</sup> <sup>¼</sup> <sup>L</sup>=2, rk <sup>¼</sup> <sup>r</sup> <sup>¼</sup> 5, <sup>β</sup> <sup>¼</sup> <sup>2</sup>:7, <sup>σ</sup><sup>2</sup>

Recent Wireless Power Transfer Technologies

σ2 <sup>w</sup><sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

benchmark.

Figure 2.

Figure 3.

132

show the QoS targets.

<sup>w</sup> <sup>¼</sup> <sup>10</sup>�<sup>2</sup>

tance in meter between the jth transmitter and the kth receiver, and β is the path loss factor. The following parameters are set unless otherwise noted, M ¼ 8,

ε<sup>2</sup> ¼ 0:02, ε<sup>3</sup> ¼ 0:2, ψ<sup>1</sup> ¼ 20, ψ<sup>2</sup> ¼ 25, and ψ<sup>3</sup> ¼ 30dBm. CVX toolbox [26] is adopted to solve SDP problems. The traditional MSE QoS constraint (MSE-QoS-

TRAD) power minimization algorithm [17] is adopted as a performance

Transmit power and PS factors versus iterations with the SDP-JTDPS algorithm.

Achieved per-user MSE and harvested energy versus iterations by the proposed SDP-TDPS scheme. The titles

�β

<sup>n</sup> <sup>¼</sup> <sup>10</sup>�<sup>3</sup>

<sup>n</sup><sup>k</sup> <sup>¼</sup> <sup>σ</sup><sup>2</sup>

. The MSE and EH thresholds are set nonuniformly as ε<sup>1</sup> ¼ 0:01,

<sup>k</sup> , where rk is the dis-

, and

Comparison of per-user MSE and harvested energy achieved by the proposed algorithms. The titles show the QoS targets.

Figure 5. Transmit power versus number of trials.

transmitters with nt transmit antennas and nr receive antennas and each user transmits d data streams from its transmitter to its receiver, is assumed. For simplicity, the MIMO IC network is henceforth denoted by ð Þ nt, nr, <sup>d</sup> <sup>K</sup>. It is assumed that the transceiver pairs share the same frequency band and operate in SWIPT mode. The channel matrix from transmitter j to receiver k is denoted

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

of the desired direct link of the kth user pair, and Hkj, ∀j 6¼ k constitutes channel coefficients of all interference links. It is assumed that <sup>s</sup> <sup>j</sup> <sup>∈</sup>C<sup>d</sup>�<sup>1</sup> denotes the

K

j¼1, j6¼k

where <sup>n</sup><sup>k</sup> <sup>∈</sup> <sup>C</sup>nr�<sup>1</sup> is the noise vector at the <sup>k</sup>th receiver, whose elements are

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 kth

K

j¼1, j6¼k

Define U<sup>k</sup> to be the receive filter for information decoding at the kth receiver,

HkjVjs <sup>j</sup> þ n<sup>k</sup>

� � is the additive complex Gaussian noise introduced

1

assumed to be i.i.d. complex Gaussian random variables with variance σ<sup>2</sup>

channel. The transmit power of the kth transmitter can be calculated as

The received baseband signal at the kth receiver is written as

<sup>r</sup><sup>k</sup> <sup>¼</sup> <sup>H</sup>kkVks<sup>k</sup> <sup>þ</sup> <sup>X</sup>

<sup>p</sup> <sup>H</sup>kkVks<sup>k</sup> <sup>þ</sup> <sup>X</sup>

0 @

w<sup>k</sup> Inr

data vector of the jth transmitter and assumes s <sup>j</sup>s<sup>H</sup>

precoded by a matrix <sup>V</sup><sup>k</sup> <sup>∈</sup>Cnt�<sup>d</sup>

MIMO interference channel SWIPT system.

DOI: http://dx.doi.org/10.5772/intechopen.89676

, ∀k, j∈f g 1, … , K , in which Hkk describes the channel coefficients

j

, it will be launched over the wireless

h i <sup>¼</sup> <sup>I</sup>d. After being

HkjVjs <sup>j</sup> þ nk, (33)

nk .

A þ wk, (34)

by <sup>H</sup>kj <sup>∈</sup>Cnr�nt

Figure 7.

Tr VjV<sup>H</sup>

j h i � � <sup>¼</sup> Pj.

user is expressed as

rID <sup>k</sup> ¼ ffiffiffiffiffi ρk

where <sup>w</sup><sup>k</sup> � CN <sup>0</sup>, <sup>σ</sup><sup>2</sup>

the detected signal is written as

by the power splitter.

135

Figure 6. Comparison of computational complexity.

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 <sup>M</sup> with <sup>a</sup> <sup>¼</sup> <sup>2</sup>:<sup>662</sup> � <sup>10</sup>�7, b ¼ 4:608, and c ¼ 2:408. This is exactly coherent with the complexity analysis in Section 2.3.4.
