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 K-user MIMO IC network as shown in Figure 7 is considered. Without loss of generality, a symmetric configuration, that is, each user consists of a pair of

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

Figure 7. MIMO interference channel SWIPT system.

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 by <sup>H</sup>kj <sup>∈</sup>Cnr�nt , ∀k, j∈f g 1, … , K , in which Hkk describes the channel coefficients 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 data vector of the jth transmitter and assumes s <sup>j</sup>s<sup>H</sup> j h i <sup>¼</sup> <sup>I</sup>d. After being precoded by a matrix <sup>V</sup><sup>k</sup> <sup>∈</sup>Cnt�<sup>d</sup> , it will be launched over the wireless channel. The transmit power of the kth transmitter can be calculated as Tr VjV<sup>H</sup> j h i � � <sup>¼</sup> Pj.

The received baseband signal at the kth receiver is written as

$$\mathbf{r}\_k = \mathbf{H}\_{kk}\mathbf{V}\_k\mathbf{s}\_k + \sum\_{j=1,\ j\neq k}^{K} \mathbf{H}\_{kj}\mathbf{V}\_j\mathbf{s}\_j + \mathbf{n}\_k,\tag{33}$$

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

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 user is expressed as

$$\mathbf{r}\_k^{\mathrm{ID}} = \sqrt{\rho\_k} \left( \mathbf{H}\_{kk} \mathbf{V}\_k \mathbf{s}\_k + \sum\_{j=1, j \neq k}^{K} \mathbf{H}\_{kj} \mathbf{V}\_j \mathbf{s}\_j + \mathbf{n}\_k \right) + \mathbf{w}\_k,\tag{34}$$

where <sup>w</sup><sup>k</sup> � CN <sup>0</sup>, <sup>σ</sup><sup>2</sup> w<sup>k</sup> Inr � � is the additive complex Gaussian noise introduced by the power splitter.

Define U<sup>k</sup> to be the receive filter for information decoding at the kth receiver, the detected signal is written as

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

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

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

3. Joint transceiver design and power splitting optimization based on

In this section, we further consider the joint transceiver design and wireless

A K-user MIMO IC network as shown in Figure 7 is considered. Without loss of

generality, a symmetric configuration, that is, each user consists of a pair of

that the low complexity is achieved at the cost of high transit power.

SINR criterion for MIMO IC channel

power transfer for MIMO IC networks.

shown in Figure 5.

Comparison of computational complexity.

Recent Wireless Power Transfer Technologies

Figure 6.

Section 2.3.4.

3.1 System model

134

$$\begin{split} \hat{\mathbf{s}}\_{k} &= \sqrt{\rho\_{k}} \mathbf{U}\_{k}^{H} \mathbf{r}\_{k}^{\mathrm{ID}} \\ &= \underbrace{\sqrt{\rho\_{k}} \mathbf{U}\_{k}^{H} \mathbf{H}\_{kk} \mathbf{V}\_{k} \mathbf{s}\_{k}}\_{\text{Desired Signal}} + \underbrace{\sum\_{j=1, j \neq k}^{K} \sqrt{\rho\_{k}} \mathbf{U}\_{k}^{H} \mathbf{H}\_{kj} \mathbf{V}\_{j} \mathbf{s}\_{j}}\_{\text{Interference}} + \underbrace{\sqrt{\rho\_{k}} \mathbf{U}\_{k}^{H} \mathbf{n}\_{k} + \mathbf{U}\_{k}^{H} \mathbf{w}\_{k}}\_{\text{Noise}}. \end{split} \tag{35}$$

The SINR of the lth data stream of the kth user is defined as

$$\text{SINR}\_{kl} = \frac{\rho\_k \left| \mathbf{u}\_{kl}^H \mathbf{H}\_{kk} \mathbf{v}\_{kl} \right|^2}{\sum\_{(j,m) \neq (k,l)} \rho\_k \left| \mathbf{u}\_{kl}^H \mathbf{H}\_{kj} \mathbf{v}\_{jm} \right|^2 + \left( \rho\_k \sigma\_n^2 + \sigma\_w^2 \right) \mathbf{u}\_{kl}^H \mathbf{u}\_{kl}},\tag{36}$$

where ukl and vkl denote the lth column vector in U<sup>k</sup> and Vk, respectively.

Let ξ<sup>k</sup> ∈ð � 0, 1 be the energy conversion efficiency, the harvested energy at the kth receiver writes

$$P\_k^{\mathrm{EH}} = \xi\_k (\mathbf{1} - \rho\_k) \left[ \sum\_{j=1}^{K} \mathrm{Tr} \left( \mathbf{H}\_{kj} \mathbf{V}\_j \mathbf{V}\_j^H \mathbf{H}\_{kj}^H \right) + n\_r \sigma\_{\mathbf{n}\_k}^2 \right]. \tag{37}$$

#### 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

$$\min\_{\{\mathbf{U}\_k, \mathbf{V}\_k, \rho\_k\}} \quad \sum\_{k=1}^K \sum\_{l=1}^d \|\mathbf{v}\_{kl}\|\_2^2 \tag{38}$$

$$\text{s.t.}\;:\quad\text{SINR}\_{kl}\geq\gamma\_{kl},\tag{39}$$

$$P\_k^{\text{EH}} \ge \varphi\_k,\tag{40}$$

rank U<sup>H</sup>

the maximum number of streams for each user should no more than

^s<sup>k</sup> <sup>¼</sup> <sup>U</sup><sup>H</sup>

MIMO system after IA and the ergodic achievable rate of the kth user is

R<sup>k</sup> ¼ log <sup>2</sup>jI<sup>d</sup> þ

discussed, and suboptimal schemes solving the problem will be developed.

<sup>k</sup> <sup>H</sup>kkV<sup>k</sup> and <sup>n</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup><sup>H</sup>

the effective noise vector at receiver k, respectively.

MMSE algorithm [32].

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

where <sup>H</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup><sup>H</sup>

3.4 Feasibility analysis

propositions [14].

feasible.

feasible.

where

137

SINR<sup>0</sup>

reduces to

<sup>k</sup> HkkV<sup>k</sup>

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

d≤ð Þ nt þ nr =ð Þ K þ 1 [31]. Since the IA condition is over constrained, it is not trivial to develop closed-form solution for it. In literature, a lot of iterative algorithms have been developed, such as the MIL algorithm [30], max-SINR algorithm [30], and

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,

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

Eq. (44) means that the system is equivalent to a traditional point-to-point

<sup>H</sup>kH<sup>H</sup> k σn2 k j

" #

In the following sections, the feasibility of the formulated problem (38) will be

The feasibility of the formulated problems (38–41) can be given by the following

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

Find : f g Uk, Vk, ρ<sup>k</sup> Such that : SINRkl ≥γkl,

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

Find : f g Uk, V<sup>k</sup>

Such that : SINR<sup>0</sup>

kl <sup>¼</sup> <sup>u</sup><sup>H</sup>

ð Þ <sup>j</sup>,<sup>m</sup> 6¼ð Þ <sup>k</sup>,<sup>l</sup> <sup>u</sup><sup>H</sup>

�

dent of the EH constraints and the PS factors. Proposition 8 is a sufficient and

P

0≤ρ<sup>k</sup> ≤1, ∀ð Þ k, l :

klHkkvkl

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

� 2

> <sup>n</sup> þ σ<sup>2</sup> w � �u<sup>H</sup>

� �

�

klHkjvjm

Proposition 7 and Proposition 8 show that the feasibility of (38–41) is indepen-

� �

� � <sup>¼</sup> <sup>d</sup>, (43)

<sup>k</sup> r<sup>k</sup> ¼ Hks<sup>k</sup> þ nk, (44)

<sup>k</sup> n<sup>k</sup> denote the effective channel matrix and

: (45)

kl <sup>≥</sup>γkl, <sup>∀</sup>ð Þ <sup>k</sup>, <sup>l</sup> , (47)

klukl

: (48)

(46)

$$0 \le \rho\_k \le 1, \forall (k, l). \tag{41}$$

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 K-user IC networks.

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 min nt ð Þ , nr K=2 [28, 29]. To achieve IA, the following feasibility condition should be satisfied [30].

$$\mathbf{U}\_k^H \mathbf{H}\_{kj} \mathbf{V}\_j = \mathbf{0} \quad \forall j, k \in \{1, \ldots, K\}, j \neq k \tag{42}$$

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

$$rank\left(\mathbf{U}\_k^H \mathbf{H}\_{kk} \mathbf{V}\_k\right) = d,\tag{43}$$

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 d≤ð Þ nt þ nr =ð Þ K þ 1 [31]. Since the IA condition is over constrained, it is not trivial to develop closed-form solution for it. In literature, a lot of iterative algorithms have been developed, such as the MIL algorithm [30], max-SINR algorithm [30], and MMSE algorithm [32].

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

$$
\hat{\mathbf{s}}\_k = \mathbf{U}\_k^\mathrm{H} \mathbf{r}\_k = \overline{\mathbf{H}}\_k \mathbf{s}\_k + \overline{\mathbf{n}}\_k,\tag{44}
$$

where <sup>H</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup><sup>H</sup> <sup>k</sup> <sup>H</sup>kkV<sup>k</sup> and <sup>n</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup><sup>H</sup> <sup>k</sup> n<sup>k</sup> denote the effective channel matrix and the effective noise vector at receiver k, respectively.

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 kth user is

$$\mathcal{R}\_k = \mathbb{E}\left[\log\_2|\mathbf{I}\_d + \frac{\overline{\mathbf{H}}\_k \overline{\mathbf{H}}\_k^H}{\sigma\_{\mathbf{n}\_k^2}}|\right]. \tag{45}$$

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

^sk¼ ffiffiffiffiffi ρk p U<sup>H</sup> <sup>k</sup> rID k

> ¼ ffiffiffiffiffi ρk p U<sup>H</sup>

kth receiver writes

3.2 Problem formulation

3.3 Interference alignment

condition should be satisfied [30].

136

U<sup>H</sup>

<sup>k</sup> HkkVks<sup>k</sup> |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Desired Signal

Recent Wireless Power Transfer Technologies

PEH

<sup>þ</sup> <sup>X</sup> K

SINRkl <sup>¼</sup> <sup>ρ</sup><sup>k</sup> <sup>u</sup><sup>H</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>X</sup>

transceiver design and power splitting problem is formulated as

inaries on IA are introduced in the following section.

maximum degrees of freedom (DoF) of the K-user IC networks.

min f g Uk, Vk, ρ<sup>k</sup>

P

j¼1, j6¼k

The SINR of the lth data stream of the kth user is defined as

ð Þ <sup>j</sup>,<sup>m</sup> 6¼ð Þ <sup>k</sup>,<sup>l</sup> <sup>ρ</sup><sup>k</sup> <sup>u</sup><sup>H</sup>

�

K

j¼1

ffiffiffiffiffi ρk p U<sup>H</sup>


�

klHkjvjm

� �

where ukl and vkl denote the lth column vector in U<sup>k</sup> and Vk, respectively. Let ξ<sup>k</sup> ∈ð � 0, 1 be the energy conversion efficiency, the harvested energy at the

To minimize the transmit power under the given QoS constraints, the joint

X K

X d

∥vkl∥<sup>2</sup>

s:t: : SINRkl ≥γkl, (39)

0≤ρ<sup>k</sup> ≤1, ∀ð Þ k, l : (41)

<sup>k</sup> ≥ψk, (40)

l¼1

k¼1

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 prelim-

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

As shown in Figure 1, when the EH receivers are removed, the system degener-

<sup>k</sup> HkjV<sup>j</sup> ¼ 0 , ∀j, k∈ f g 1, … ,K , j 6¼ k (42)

ates into traditional symmetric MIMO IC networks. The DoF of such MIMO IC network is min nt ð Þ , nr K=2 [28, 29]. To achieve IA, the following feasibility

PEH

<sup>k</sup> HkjVjs <sup>j</sup>

klHkkvkl

� 2

þ ρkσ<sup>2</sup>

<sup>j</sup> H<sup>H</sup> kj

<sup>n</sup> þ σ<sup>2</sup> w � �u<sup>H</sup>

> <sup>þ</sup> nrσ<sup>2</sup> nk

� �

� 2

Tr HkjVjV<sup>H</sup>

� �

" #

þ ffiffiffiffiffi ρk p U<sup>H</sup>

<sup>k</sup> <sup>n</sup><sup>k</sup> <sup>þ</sup> <sup>U</sup><sup>H</sup>


<sup>k</sup> w<sup>k</sup>

klukl

<sup>2</sup> (38)

: (35)

, (36)

: (37)

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.

$$\begin{aligned} \text{Find } : \quad & \{ \mathbf{U}\_k, \mathbf{V}\_k, \rho\_k \} \\ \text{Such that } : \quad & \text{SINR}\_{kl} \ge \gamma\_{kl}, \\ & \mathbf{0} \le \rho\_k \le \mathbf{1}, \forall (k, l). \end{aligned} \tag{46}$$

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

$$\begin{array}{ll}\text{Find :} & \{\mathbf{U}\_k, \mathbf{V}\_k\} \\\\ \text{Such that :} & \text{SINR}'\_{kl} \ge \gamma\_{kl}, \forall (k,l), \end{array} \tag{47}$$

where

$$\text{SINR}\_{kl}^{\prime} = \frac{\left| \mathbf{u}\_{kl}^{H} \mathbf{H}\_{kk} \mathbf{v}\_{kl} \right|^{2}}{\sum\_{(j,m) \neq (k,l)} \left| \mathbf{u}\_{kl}^{H} \mathbf{H}\_{kj} \mathbf{v}\_{jm} \right|^{2} + \left( \sigma\_{n}^{2} + \sigma\_{w}^{2} \right) \mathbf{u}\_{kl}^{H} \mathbf{u}\_{kl}}. \tag{48}$$

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 Uk, Vk, ∀k, that is,

$$\mathbf{u}\_{kl}^{\rm H} \mathbf{H}\_{kj} \mathbf{v}\_{jm} = \mathbf{0}, \forall (j, m) \neq (k, l), \tag{49}$$

$$\mathbf{u}\_{kl}^{\rm H} \mathbf{H}\_{kk} \mathbf{v}\_{kl} \neq \mathbf{0}, \forall (k, l), \tag{50}$$

According to Lemma 8, Proposition 9, and [35], Proposition 9, problem (52) is

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

kl, Xkl ⪰ 0, problem

(53)

(54)

feasible if the original problem is feasible. By defining <sup>X</sup>kl <sup>¼</sup> <sup>v</sup>klv<sup>H</sup>

Trð Þ Xkl

klHkkXklH<sup>H</sup> kkukl � �

klHkjXjmH<sup>H</sup>

kj h i<sup>≥</sup> <sup>ψ</sup><sup>k</sup>

(52) can be further recovered from the eigen decomposition of Xkl.

max ukl

<sup>m</sup>¼<sup>1</sup>Hkjvjmv<sup>H</sup>

u<sup>H</sup>

jmH<sup>H</sup>

<sup>u</sup>kl <sup>¼</sup> <sup>B</sup>�<sup>1</sup>

∥B�<sup>1</sup>

By alternatively optimizing the transmitters together with PS factors and the receivers, an SDP-based joint transceiver design and PS optimization scheme can be

Algorithm 3 Joint transceiver design and power splitting based on SDP

2: Solve the convex problem (53) to obtain Xkl and power splitting factors

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

3: Recover vkl, ∀ð Þ k, l from Xkl through eigenvalue decomposition.

generalized Rayleigh quotient and its closed-form solution is given by

kjukl � � <sup>≥</sup>γkl <sup>σ</sup><sup>2</sup>

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

The convex SDP (53) can be solved efficiently. Moreover, it can be proven that there is rankð Þ¼ Xkl 1 and the SDP relaxation is tight [34], meaning that the optimal solution of the relaxed problem is also optimal to the original problem. After obtaining the rank-one solution Xkl f g , ∀ð Þ k, l , the optimal solution vkl to problem

When the transmit precoders and PS factors are all fixed, (38–41) become separable with respect to variables ukl f g , ∀ð Þ k, l . Recall that the SINR constraints have been satisfied by the solution of (52), we can further maximize the receive

klHkkvklv<sup>H</sup>

u<sup>H</sup> klBklukl

klH<sup>H</sup> kkukl

klH<sup>H</sup>

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

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

: (55)

� �I<sup>n</sup><sup>r</sup> . Eq. (54) is a

kj � <sup>H</sup>kkvklv<sup>H</sup>

kl Hkkvkl

kl Hkkvkl∥<sup>2</sup>

<sup>n</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w ρk � �∥ukl∥<sup>2</sup>

n,

2,

(52) can be relaxed as the following convex SDP program

X d

l¼1

Tr u~<sup>H</sup>

Tr HkjXjmH<sup>H</sup>

X K

k¼1

<sup>s</sup>:t: : <sup>1</sup> <sup>þ</sup> <sup>γ</sup>kl ð ÞTr <sup>u</sup><sup>H</sup>

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

X d

m¼1

min Xkl, ρ<sup>k</sup> f g , ∀ð Þ k, l

> �γkl<sup>X</sup> K

X K

j¼1

3.5.2 Receiver optimization

per-stream SINR by

where <sup>B</sup>kl <sup>¼</sup> <sup>P</sup><sup>K</sup>

3.5.3 Algorithm description

(SDP-JTDPS).

ρk, ∀ð Þ k, l .

139

j¼1 P<sup>d</sup>

obtained, which is summarized in Algorithm 3.

4: Update the receiver ukl, ∀ð Þ k, l by (55).

1: Initialize the receivers Uk, ∀k.

j¼1

X d

m¼1

0≤ρ<sup>k</sup> ≤1, ∀ð Þ k, l :

SINR (48) becomes

$$\text{SINR}\_{kl}^{\prime} = \frac{\left| \mathbf{u}\_{kl}^{\text{H}} \mathbf{H}\_{kk} \mathbf{v}\_{kl} \right|^2}{\left( \sigma\_n^2 + \sigma\_w^2 \right) \mathbf{u}\_{kl}^{\text{H}} \mathbf{u}\_{kl}} = \frac{p\_{kl} \left| \mathbf{u}\_{kl}^{\text{H}} \mathbf{H}\_{kk} \overline{\mathbf{v}}\_{kl} \right|^2}{\left( \sigma\_n^2 + \sigma\_w^2 \right) \left\| \mathbf{u}\_{kl} \right\|\_2^2},\tag{51}$$

where <sup>v</sup>kl <sup>¼</sup> <sup>v</sup>kl <sup>∥</sup>vkl∥<sup>2</sup> is the normalized precoding vector, pkl is the transmit power along the beamforming direction vkl and vkl ¼ pklvkl. According to (51), the SINR constraints in problem (47) can always be satisfied by increasing the transmit power pkl, if the interference is completely suppressed.

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 Vk, ∀k together with the PS factors ρk, ∀k and the receivers Uk, ∀k.

#### 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

$$\begin{aligned} \min\_{\{\mathbf{v}\_{kl}, \rho\_k, \mathbf{v}(k,l)\}} & \quad \sum\_{k=1}^{K} \sum\_{l=1}^{d} \|\mathbf{v}\_{kl}\|\_{2}^{2} \\ \text{s.t.} & \quad \frac{\left\|\mathbf{u}\_{kl}^{H} \mathbf{H}\_{kk} \mathbf{v}\_{kl}\right\|^{2}}{\mathbf{u}\_{kl} \mathbf{B}\_{kl} \mathbf{u}\_{kl}^{H}} \ge \gamma\_{kl}, \\ & \quad \sum\_{j=1}^{K} \sum\_{m=1}^{d} \|\mathbf{H}\_{kj} \mathbf{v}\_{jm}\|\_{2}^{2} \ge \frac{\Psi\_{k}}{\xi\_{k}(1-\rho\_{k})} - n\_{r} \sigma\_{n}^{2}, \\ & \quad \mathbf{0} \le \rho\_{k} \le \mathbf{1}, \forall (k,l), \end{aligned} \tag{52}$$

where <sup>B</sup>kl <sup>¼</sup> <sup>P</sup><sup>K</sup> j¼1 P<sup>d</sup> <sup>m</sup>¼<sup>1</sup>Hkjvjmv<sup>H</sup> jmH<sup>H</sup> kj � <sup>H</sup>kkvklv<sup>H</sup> klH<sup>H</sup> kk <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>n</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w ρk � �Inr.

138

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

According to Lemma 8, Proposition 9, and [35], Proposition 9, problem (52) is feasible if the original problem is feasible. By defining <sup>X</sup>kl <sup>¼</sup> <sup>v</sup>klv<sup>H</sup> kl, Xkl ⪰ 0, problem (52) can be relaxed as the following convex SDP program

$$\begin{split} & \min\_{\{\mathbf{X}\_{kl},\rho\_{k},\forall(k,l)\}} \quad \sum\_{k=1}^{K} \sum\_{l=1}^{d} \operatorname{Tr}(\mathbf{X}\_{kl}) \\ & \text{s.t.} \quad : \quad (1+\gamma\_{kl}) \operatorname{Tr}\left(\mathbf{u}\_{kl}^{H} \mathbf{H}\_{kk} \mathbf{X}\_{kl} \mathbf{H}\_{kk}^{H} \mathbf{u}\_{kl}\right) \\ & \quad - \gamma\_{kl} \sum\_{j=1}^{K} \sum\_{m=1}^{d} \operatorname{Tr}\left(\hat{\mathbf{u}}\_{kl}^{H} \mathbf{H}\_{kj} \mathbf{X}\_{jm} \mathbf{H}\_{kj}^{H} \mathbf{u}\_{kl}\right) \geq \gamma\_{kl} \left(\sigma\_{n}^{2} + \frac{\sigma\_{w}^{2}}{\rho\_{k}}\right) \|\mathbf{u}\_{kl}\|\_{2}^{2}, \\ & \sum\_{j=1}^{K} \sum\_{m=1}^{d} \operatorname{Tr}\left[\mathbf{H}\_{kj} \mathbf{X}\_{jm} \mathbf{H}\_{kj}^{H}\right] \geq \frac{\Psi\_{k}}{\xi\_{k}(1-\rho\_{k})} - n\_{r}\sigma\_{n}^{2}, \\ & \qquad 0 \leq \rho\_{k} \leq 1, \forall(k,l). \end{split} \tag{53}$$

The convex SDP (53) can be solved efficiently. Moreover, it can be proven that there is rankð Þ¼ Xkl 1 and the SDP relaxation is tight [34], meaning that the optimal solution of the relaxed problem is also optimal to the original problem. After obtaining the rank-one solution Xkl f g , ∀ð Þ k, l , the optimal solution vkl to problem (52) can be further recovered from the eigen decomposition of Xkl.

#### 3.5.2 Receiver optimization

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

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

klHkjvjm ¼ 0, ∀ð Þ j, m 6¼ ð Þ k, l , (49)

klHkkvkl 6¼ 0, ∀ð Þ k, l , (50)

klHkkvkl

� 2

2

, (51)

� �

Proof. If interference is completely eliminated, given the transceivers

klHkkvkl

� 2

along the beamforming direction vkl and vkl ¼ pklvkl. According to (51), the SINR constraints in problem (47) can always be satisfied by increasing the transmit

3.5 Alternative optimization solution based on semidefinite programming

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

An iterative algorithm for (38–41) can be developed based on AO framework, that is, alternatively optimizing the transmitters Vk, ∀k together with the PS factors

When the receivers are fixed, problems (38–41) are reduced to the following

X K

X d

∥vkl∥<sup>2</sup> 2

<sup>2</sup> <sup>≥</sup> <sup>ψ</sup><sup>k</sup>

kj � <sup>H</sup>kkvklv<sup>H</sup>

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

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

klH<sup>H</sup>

n,

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

Inr.

(52)

l¼1

≥ γkl,

k¼1

∥Hkjvjm∥<sup>2</sup>

� 2 klukl

<sup>¼</sup> pkl <sup>u</sup><sup>H</sup>

σ2 <sup>n</sup> þ σ<sup>2</sup> w � �∥ukl∥<sup>2</sup>

<sup>∥</sup>vkl∥<sup>2</sup> is the normalized precoding vector, pkl is the transmit power

�

� �

uH

kl <sup>¼</sup> <sup>u</sup><sup>H</sup>

power pkl, if the interference is completely suppressed.

�

σ2 <sup>n</sup> þ σ<sup>2</sup> w � �u<sup>H</sup>

SINR<sup>0</sup>

considered MIMO IC network is IA feasible.

3.5.1 Transmitter and power splitting optimization

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

�

X K

j¼1

<sup>m</sup>¼<sup>1</sup>Hkjvjmv<sup>H</sup>

joint transmit precoders and PS factors optimization problem

min <sup>v</sup>kl, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup>ð Þ <sup>k</sup>, <sup>l</sup>

klHkkvkl

uklBklu<sup>H</sup> kl

> X d

> m¼1

0≤ρ<sup>k</sup> ≤1, ∀ð Þ k, l ,

jmH<sup>H</sup>

� �

uH

following proposition.

Recent Wireless Power Transfer Technologies

linear transceivers.

Uk, Vk, ∀k, that is,

SINR (48) becomes

where <sup>v</sup>kl <sup>¼</sup> <sup>v</sup>kl

relaxation

ρk, ∀k and the receivers Uk, ∀k.

where <sup>B</sup>kl <sup>¼</sup> <sup>P</sup><sup>K</sup>

138

j¼1 P<sup>d</sup>

When the transmit precoders and PS factors are all fixed, (38–41) become separable with respect to variables ukl f g , ∀ð Þ k, l . Recall that the SINR constraints have been satisfied by the solution of (52), we can further maximize the receive per-stream SINR by

$$\max\_{\mathbf{u}\_{kl}} \frac{\mathbf{u}\_{kl}^H \mathbf{H}\_{kk} \mathbf{v}\_{kl} \mathbf{v}\_{kl}^H \mathbf{H}\_{kk}^H \mathbf{u}\_{kl}}{\mathbf{u}\_{kl}^H \mathbf{B}\_{kl} \mathbf{u}\_{kl}} \tag{54}$$

where <sup>B</sup>kl <sup>¼</sup> <sup>P</sup><sup>K</sup> j¼1 P<sup>d</sup> <sup>m</sup>¼<sup>1</sup>Hkjvjmv<sup>H</sup> jmH<sup>H</sup> kj � <sup>H</sup>kkvklv<sup>H</sup> klH<sup>H</sup> kk <sup>þ</sup> <sup>σ</sup><sup>2</sup> <sup>n</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w ρk � �I<sup>n</sup><sup>r</sup> . Eq. (54) is a generalized Rayleigh quotient and its closed-form solution is given by

$$\mathbf{u}\_{kl} = \frac{\mathbf{B}\_{kl}^{-1} \mathbf{H}\_{kk} \mathbf{v}\_{kl}}{||\mathbf{B}\_{kl}^{-1} \mathbf{H}\_{kk} \mathbf{v}\_{kl}||\_2}. \tag{55}$$

#### 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).


The convergence of Algorithm 3 is given by the following proposition [14]. Proposition 10 If (53) is feasible for the initial receivers Uk, ∀k, the convergence to a locally optimal solution can be guaranteed by Algorithm 3.

3.6.1 Optimal power allocation and power splitting scheme

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

min pkl, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup>k, <sup>l</sup>

<sup>s</sup>:t: : pkl � <sup>σ</sup><sup>2</sup>

X K

X d

m¼1

as its optimal solution, the transmit precoders are <sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffi

0≤ ρ<sup>k</sup> ≤ 1, ∀ð Þ k, l :

j¼1

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

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

X K

X d

l¼1 pkl

<sup>≥</sup> <sup>σ</sup><sup>2</sup>

jm∥<sup>2</sup>

w∥ukl∥<sup>2</sup> <sup>2</sup>γkl

λklρ<sup>k</sup>

<sup>2</sup> <sup>≥</sup> <sup>ψ</sup><sup>k</sup>

<sup>k</sup> , <sup>∀</sup><sup>k</sup> � � that satisfy the IA conditions.

<sup>k</sup> , <sup>∀</sup>k, and decompose <sup>H</sup>kk as <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>kLkV<sup>H</sup>

λ p

,

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

p∗ kl <sup>p</sup> <sup>v</sup>IA kl .

kl and power splitting factors ρ <sup>∗</sup>

<sup>k</sup> , <sup>∀</sup><sup>k</sup> � �, by discarding the EH constraints of (58),

≥γkl, ∀ð Þ k, l :

<sup>k</sup>¼<sup>1</sup><sup>d</sup> parallel subproblems.

<sup>≥</sup>γkl: (61)

2

2

n,

(59)

<sup>k</sup> , ∀k

kl and ρ <sup>∗</sup>

<sup>k</sup> through

<sup>k</sup> , ∀k, l by

(60)

k¼1

nγkl∥ukl∥<sup>2</sup> 2

λkl

pjm∥Hkjv<sup>0</sup>

Problem (59) is convex and thus can be solved optimally. Denote p<sup>∗</sup>

employed, the computational complexity of Algorithm 4 is in the order of <sup>O</sup> ð Þ Kd <sup>3</sup> � � [27], which is significantly lower than that of Algorithm 3.

power splitting over effective IA channel decomposing (O-PAPS).

λk1 <sup>p</sup> , ffiffiffiffiffiffi λk2 <sup>p</sup> , … , ffiffi

<sup>k</sup> <sup>¼</sup> <sup>V</sup>IA

3.6.2 Closed-form power allocation and power splitting scheme

<sup>k</sup> , VIA

we consider the following SINR constrained power optimization problem

X d

l¼1 pkl

According to Proposition 9, (60) is feasible. By further applying Lemma 8, (58)

X K

k¼1

For the l-th data stream of the kth user, the subproblem is expressed as

pkl

<sup>s</sup>:t: : <sup>λ</sup>klpkl σ2 <sup>n</sup> þ σ<sup>2</sup> w � �∥ukl∥<sup>2</sup>

<sup>s</sup>:t: : <sup>λ</sup>klpkl σ2 <sup>n</sup> þ σ<sup>2</sup> w � �∥ukl∥<sup>2</sup>

<sup>k</sup> , VIA

1: Obtain IA transceivers UIA

k � �<sup>H</sup>

SVD, where <sup>Λ</sup><sup>k</sup> <sup>¼</sup> diag ffiffiffiffiffiffi

p∗ kl <sup>p</sup> <sup>v</sup>IA

Given the IA solution UIA

<sup>k</sup> U<sup>k</sup> and V<sup>0</sup>

solving the convex problem (59).

4: Obtain the optimal transmit power p<sup>∗</sup>

kl , ∀k, l.

min pkl f g , <sup>∀</sup>k, <sup>l</sup>

is feasible. Moreover, (60) can be decomposed into P<sup>K</sup>

min pkl

HkkVIA

2: Let <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>IA

3: Let <sup>U</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup>IA

5: Set <sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffi

141

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

Algorithm 4 SWIPT design with optimal transmit power allocation and receive

kd � �.

<sup>k</sup> Vk, ∀k.

According to Proposition 10, it is important to initialize the receivers Uk, ∀k for the success of the algorithm. To guarantee finding a feasible solution to (53), we initialize the receivers by the IA receivers UIA <sup>k</sup> , ∀k in this chapter. Similar to Algorithm 1, the complexity of Algorithm 3 is dominated by the SDP solving process, which is about <sup>O</sup> ð Þ nrKd <sup>4</sup>:<sup>5</sup> log 1ð Þ <sup>=</sup><sup>ϵ</sup> � � for one instance [10]. This complexity becomes prohibitive as the number of antennas or users increases. In the following section, a low-complexity design schemes solving this problem is developed.

#### 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 VIA k � �<sup>H</sup> VIA <sup>k</sup> <sup>¼</sup> <sup>I</sup><sup>d</sup> and <sup>U</sup>IA k � �<sup>H</sup> UIA <sup>k</sup> ¼ Id. Given interference alignment transceivers, the effective channel matrix for user k can be decomposed as <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>IA k � �<sup>H</sup> HkkVIA <sup>k</sup> <sup>¼</sup> <sup>U</sup>kΛkV<sup>H</sup> <sup>k</sup> through singular value decomposition (SVD), where <sup>Λ</sup><sup>k</sup> <sup>¼</sup> diag ffiffiffiffiffiffi λk1 <sup>p</sup> , ffiffiffiffiffiffi λk2 <sup>p</sup> , … , ffiffi λ p kd � � is a diagonal matrix. The transmit precoder matrix can be constructed as V<sup>0</sup> <sup>k</sup> <sup>¼</sup> <sup>V</sup>IA <sup>k</sup> Vk. By further multiplying the power matrix, the transmit precoding vector is then finally constructed as V<sup>k</sup> ¼ V<sup>0</sup> <sup>k</sup>Pk, where P<sup>k</sup> ¼ diag ffiffiffiffiffiffi pk1 p , … , ffiffiffiffiffiffi pkd <sup>p</sup> � � is a diagonal matrix with nonnegative diagonalized elements. At the receiver side, the receive filter is constructed similarly, <sup>U</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup>IA <sup>k</sup> Uk. Then, the following equations are established

$$\mathbf{U}\_k^H \mathbf{H}\_{kj} \mathbf{V}\_j = \mathbf{0} \quad \forall j, k \in \{1, \ldots, K\}, j \neq k,\tag{56}$$

$$\mathbf{U}\_k^H \mathbf{H}\_{kk} \mathbf{V}\_k = \mathbf{A}\_k \mathbf{P}\_k, \forall k. \tag{57}$$

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

$$\begin{aligned} \min\_{\{p\_{kl}, \rho\_{k}, \forall k, l\}} & \quad \sum\_{k=1}^{K} \sum\_{l=1}^{d} p\_{kl} \\ \text{s.t.} & \quad \frac{\rho\_{k} \lambda\_{kl} p\_{kl}}{(\rho\_{k} \sigma\_{n}^{2} + \sigma\_{w}^{2}) \|\mathbf{u}\_{kl}\|\_{2}^{2}} \geq \boldsymbol{\gamma}\_{kl}, \\ & \quad \boldsymbol{\xi}\_{k} (\mathbf{1} - \boldsymbol{\rho}\_{k}) \left( \sum\_{j=1}^{K} \sum\_{m=1}^{d} p\_{jm} \|\mathbf{H}\_{kj} \mathbf{v}\_{jm}^{\prime}\|\_{2}^{2} + n\_{\text{r}} \sigma\_{n}^{2} \right) \geq \boldsymbol{\nu}\_{k}, \\ & \quad \mathbf{0} \leq \boldsymbol{\rho}\_{k} \leq \mathbf{1}, \forall (k, l). \end{aligned} \tag{58}$$

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

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

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

The convergence of Algorithm 3 is given by the following proposition [14]. Proposition 10 If (53) is feasible for the initial receivers Uk, ∀k, the convergence

rithm 1, the complexity of Algorithm 3 is dominated by the SDP solving process,

becomes prohibitive as the number of antennas or users increases. In the following section, a low-complexity design schemes solving this problem is developed.

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

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

> λk1 <sup>p</sup> , ffiffiffiffiffiffi λk2 <sup>p</sup> , … , ffiffi

> > pkd

Given interference alignment transceivers, the effective channel matrix for user k

plying the power matrix, the transmit precoding vector is then finally constructed

p � �

ative diagonalized elements. At the receiver side, the receive filter is constructed

<sup>k</sup> Uk. Then, the following equations are established

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

pk1 p , … , ffiffiffiffiffiffi

HkkVIA

k � �<sup>H</sup>

<sup>k</sup> HkjV<sup>j</sup> ¼ 0 , ∀j, k∈f g 1, … , K , j 6¼ k, (56)

<sup>k</sup> HkkV<sup>k</sup> ¼ ΛkPk, ∀k: (57)

<sup>k</sup> <sup>¼</sup> <sup>U</sup>kΛkV<sup>H</sup>

VIA

λ p kd � � is a diagonal matrix.

<sup>k</sup> <sup>¼</sup> <sup>V</sup>IA

<sup>k</sup> <sup>¼</sup> <sup>I</sup><sup>d</sup> and <sup>U</sup>IA

<sup>k</sup> through singular value

is a diagonal matrix with nonneg-

k � �<sup>H</sup>

<sup>k</sup> Vk. By further multi-

UIA <sup>k</sup> ¼ Id.

According to Proposition 10, it is important to initialize the receivers Uk, ∀k for the success of the algorithm. To guarantee finding a feasible solution to (53), we

<sup>k</sup> , ∀k in this chapter. Similar to Algo-

for one instance [10]. This complexity

to a locally optimal solution can be guaranteed by Algorithm 3.

initialize the receivers by the IA receivers UIA

� �

optimized with the precoders and receivers fixed.

and receive filters are orthogonalized such that VIA

The transmit precoder matrix can be constructed as V<sup>0</sup>

transmit power allocation and power splitting problem

min pkl, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup>k, <sup>l</sup>

<sup>s</sup>:t: : <sup>ρ</sup>kλklpkl ρkσ<sup>2</sup> <sup>n</sup> þ σ<sup>2</sup> w � �∥ukl∥<sup>2</sup>

decomposition (SVD), where <sup>Λ</sup><sup>k</sup> <sup>¼</sup> diag ffiffiffiffiffiffi

<sup>k</sup>Pk, where P<sup>k</sup> ¼ diag ffiffiffiffiffiffi

U<sup>H</sup>

k � �<sup>H</sup>

U<sup>H</sup>

X K

X d

l¼1 pkl

> 2 ≥ γkl,

X d

pjm∥Hkjv<sup>0</sup>

!

jm∥<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>n</sup>rσ<sup>2</sup> n

≥ψk,

(58)

m¼1

In the following, two different schemes solving (58) is developed by either

k¼1

K

j¼1

<sup>ξ</sup><sup>k</sup> <sup>1</sup> � <sup>ρ</sup><sup>k</sup> ð Þ <sup>X</sup>

0≤ ρ<sup>k</sup> ≤ 1, ∀ð Þ k, l :

reformulating it as a convex problem or solved in closed form.

which is about <sup>O</sup> ð Þ nrKd <sup>4</sup>:<sup>5</sup> log 1ð Þ <sup>=</sup><sup>ϵ</sup>

Recent Wireless Power Transfer Technologies

3.6 Low-complexity design schemes

can be decomposed as <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>IA

as V<sup>k</sup> ¼ V<sup>0</sup>

140

similarly, <sup>U</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup>IA

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

$$\begin{aligned} \min\_{\{p\_{kl}, \rho\_k, \psi k, l\}} & \quad \sum\_{k=1}^{K} \sum\_{l=1}^{d} p\_{kl} \\ \text{s.t. } & \quad p\_{kl} - \frac{\sigma\_n^2 \nu\_{kl} ||\mathbf{u}\_{kl}||\_2^2}{\lambda\_{kl}} \ge \frac{\sigma\_w^2 ||\mathbf{u}\_{kl}||\_2^2 \nu\_{kl}}{\lambda\_{kl} \rho\_k}, \\ & \quad \sum\_{j=1}^{K} \sum\_{m=1}^{d} p\_{jm} ||\mathbf{H}\_{kj} \mathbf{v}\_{jm}^{\prime}||\_2^2 \ge \frac{\nu\_k}{\xi\_k (1 - \rho\_k)} - n\_r \sigma\_n^2, \\ & \quad \mathbf{0} \le \rho\_k \le \mathbf{1}, \forall (k, l). \end{aligned} \tag{59}$$

Problem (59) is convex and thus can be solved optimally. Denote p<sup>∗</sup> kl and ρ <sup>∗</sup> <sup>k</sup> , ∀k as its optimal solution, the transmit precoders are <sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffi p∗ kl <sup>p</sup> <sup>v</sup>IA kl .

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 <sup>O</sup> ð Þ Kd <sup>3</sup> � � [27], which is significantly lower than that of Algorithm 3.

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


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

Given the IA solution UIA <sup>k</sup> , VIA <sup>k</sup> , <sup>∀</sup><sup>k</sup> � �, by discarding the EH constraints of (58), we consider the following SINR constrained power optimization problem

$$\begin{array}{ll}\min\_{\{p\_{kl},\Psi k,l\}} & \sum\_{k=1}^{K} \sum\_{l=1}^{d} p\_{kl} \\ \text{s.t.} & \frac{\lambda\_{kl} p\_{kl}}{\left(\sigma\_{n}^{2} + \sigma\_{w}^{2}\right) \|\mathbf{u}\_{kl}\|\_{2}^{2}} \geq \gamma\_{kl}, \forall (k,l). \end{array} \tag{60}$$

According to Proposition 9, (60) is feasible. By further applying Lemma 8, (58) is feasible. Moreover, (60) can be decomposed into P<sup>K</sup> <sup>k</sup>¼<sup>1</sup><sup>d</sup> parallel subproblems. For the l-th data stream of the kth user, the subproblem is expressed as

$$\begin{array}{ll}\min\_{p\_{kl}} & p\_{kl} \\ \text{s.t.} & \frac{\lambda\_{kl} p\_{kl}}{\left(\sigma\_n^2 + \sigma\_w^2\right) \|\mathbf{u}\_{kl}\|\_2^2} \ge \gamma\_{kl} .\end{array} \tag{61}$$

141

The solution of (61) is then given by

$$\hat{p}\_{kl} = \frac{\left(\sigma\_n^2 + \sigma\_w^2\right) \|\mathbf{u}\_{kl}\|\_2^2 \gamma\_{kl}}{\lambda\_{kl}}.\tag{62}$$

complexity is <sup>O</sup> <sup>n</sup><sup>3</sup>

rkl <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>

vergence analysis.

Figure 8.

143

(JTDPS) scheme for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup> network.

t

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

3.7 Simulation results and analysis

. When the max-SINR or MIL algorithm [30] is applied, the

�β

<sup>w</sup> ¼ �50 dBm, ξkl ¼ ξ ¼ 0:8, γkl ¼ γ, and

kj , where rkj is the

complexity is about <sup>O</sup> niter max nt ð Þ , nr <sup>3</sup> . No matter which method is adopted, the complexity of Algorithm 5 is much lower than that of Algorithm 3 and Algorithm 4.

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

Simulations are done over the wireless system as described in Section 3.1, by setting the number of users K ¼ 3 or K ¼ 4. The entries of Hkj are assumed to be i.i.

distance in meters between the jth transmitter and the kth receiver, and β is the path loss factor. The parameters of the symmetric network are set to be nt ¼ nr,

ψ<sup>k</sup> ¼ ψ. For the three-user network, the closed-form MIMO linear IA algorithm given in [28] is adopted to design IA transceivers. For the four-user network, the MIL algorithm [30] is used. The simulation results are obtained by taking the

Figure 8 shows the convergence performance of SDP-JTDPS with γ ¼ f g 0, 20 dB and <sup>ψ</sup> <sup>¼</sup> f g 0, 10 dBm for one channel realization of the 4, 4, 2 ð Þ<sup>3</sup> network. It can be observed that the algorithm converges monotonically, which verifies the con-

The empirical cumulative distribution function (CDF) of the output per-stream SINR for the 4, 4, 2 ð Þ<sup>3</sup> network is shown in Figure 9. The SINR target <sup>γ</sup> is 20 dB, and the EH target is 0 and 30 dBm, respectively. The results show that the achieved SINR values exceed the given 20 dB, meaning that the proposed schemes can satisfy the SINR constraints. The difference is that the achieved SINR value can be greater than the target SINR value for SDP-JTDPS and O-PAPS, while for CF-PAPS, the achieved SINR values are always equal to the SINR target, which implies that the

Convergence property of the semidefinite programming (SDP)-joint transceiver design and power splitting

d. zero mean complex Gaussian random variables with variance r

<sup>n</sup> ¼ �70 dBm, <sup>σ</sup><sup>2</sup>

average of the simulation results of all 100 channel realizations.

EH constraints are satisfied with equality in CF-PAPS.

Following Lemma 7, (58) can be optimized by substituting pkl with αp^kl, where α≥ 1 is a scaling factor to be optimized. Then, (58) is reduced to a problem jointly optimizing α and PS factors ρ<sup>k</sup> under SINR and EH constraints, which is

$$\begin{aligned} \min\_{a,\{\rho\_k,\Psi\_k,l\}} &\quad \sum\_{k=1}^K \sum\_{l=1}^d \alpha \hat{p}\_{kl} \\ \text{s.t.} &\quad \rho\_k \ge \frac{\sigma\_w^2 \|\mathbf{u}\_{kl}\|\_2^2 \nu\_{kl}}{a\lambda\_{kl}\hat{p}\_{kl} - \sigma\_n^2 \nu\_{kl} \|\mathbf{u}\_{kl}\|\_2^2}, \\ &\quad \mathbf{1}-\rho\_k \ge \frac{\Psi\_k}{\xi\_k \left(\sum\_{j=1}^K \sum\_{m=1}^d a\hat{p}\_{jm} \|\mathbf{H}\_{kj}\mathbf{v}\_{jm}^\prime\|\_2^2 + n\_\mathbf{r}\sigma\_n^2\right)}, \\ &\quad \mathbf{0} \le \rho\_k \le \mathbf{1}, \forall (k,l), \\ &\quad a > \mathbf{1}. \end{aligned} \tag{63}$$

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

Proposition 11 Given the IA transceivers UIA <sup>k</sup> , VIA <sup>k</sup> , <sup>∀</sup><sup>k</sup> � �, define akl <sup>¼</sup> <sup>σ</sup><sup>2</sup> w∥ukl∥<sup>2</sup> <sup>2</sup>γkl, bkl <sup>¼</sup> <sup>σ</sup><sup>2</sup> nγkl∥ukl∥<sup>2</sup> 2, ckl ¼ λklp^kl, f <sup>k</sup> ¼ ξ<sup>k</sup> P<sup>K</sup> j¼1 P<sup>d</sup> <sup>m</sup>¼<sup>1</sup>p^jmξk∥Hkjv<sup>0</sup> jm∥<sup>2</sup> 2, and gk <sup>¼</sup> <sup>ξ</sup>knrσ<sup>2</sup> <sup>n</sup>. The optimal solution to (63) is given by

$$a^\* = \max\_{\mathbb{V}(k,l)} a^\*\_{kl},\tag{64}$$

$$
\rho\_k^\* = \max\_{\mathbb{M}} \rho\_{kl}, \tag{65}
$$

where α<sup>∗</sup> kl is the solution to the equation akl <sup>α</sup>ckl�bkl <sup>þ</sup> <sup>ψ</sup><sup>k</sup> α f <sup>k</sup>þgk <sup>¼</sup> <sup>1</sup> ð Þ <sup>α</sup>><sup>1</sup> , <sup>ρ</sup>kl <sup>¼</sup> akl α<sup>∗</sup> ckl�bkl . Given α<sup>∗</sup> and p^kl, the transmit precoders are then determined by <sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup><sup>∗</sup> <sup>p</sup>^kl <sup>p</sup> <sup>v</sup><sup>0</sup> kl. The proposed IA-based SWIPT scheme with the closed-form transmit power allocation and receive power splitting is summarized in Algorithm 5.

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 UIA <sup>k</sup> , VIA <sup>k</sup> , <sup>∀</sup><sup>k</sup> � � that satisfy the IA conditions. 2: Let <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>IA k � �<sup>H</sup> HkkVIA <sup>k</sup> , <sup>∀</sup>k, and decompose <sup>H</sup>kk as <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>kΛkV<sup>H</sup> <sup>k</sup> through SVD, where <sup>Λ</sup><sup>k</sup> <sup>¼</sup> diag ffiffiffiffiffiffi λk1 <sup>p</sup> , ffiffiffiffiffiffi λk2 <sup>p</sup> , … , ffiffi λ p kd � �. 3: Let <sup>U</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup>IA <sup>k</sup> U<sup>k</sup> and V<sup>0</sup> <sup>k</sup> <sup>¼</sup> <sup>V</sup>IA <sup>k</sup> Vk, ∀k. 4: Calculate <sup>p</sup>^kl <sup>¼</sup> <sup>σ</sup><sup>2</sup> <sup>n</sup>þσ<sup>2</sup> ð Þ<sup>w</sup> <sup>∥</sup>ukl∥<sup>2</sup> <sup>2</sup>γkl <sup>λ</sup>kl , ∀ð Þ k, l . 5: Obtain α<sup>∗</sup> and ρ <sup>∗</sup> <sup>k</sup> , ∀k according to (64). 6: Set <sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi <sup>α</sup><sup>∗</sup> <sup>p</sup>^kl <sup>p</sup> <sup>v</sup>IA kl , ∀ð Þ k, l .

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

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

complexity is <sup>O</sup> <sup>n</sup><sup>3</sup> t . When the max-SINR or MIL algorithm [30] is applied, the complexity is about <sup>O</sup> niter max nt ð Þ , nr <sup>3</sup> . No matter which method is adopted, the complexity of Algorithm 5 is much lower than that of Algorithm 3 and Algorithm 4.

### 3.7 Simulation results and analysis

The solution of (61) is then given by

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min <sup>α</sup>, <sup>ρ</sup><sup>k</sup> f g , <sup>∀</sup>k, <sup>l</sup>

α>1:

<sup>2</sup>γkl, bkl <sup>¼</sup> <sup>σ</sup><sup>2</sup>

1: Obtain IA transceivers UIA

k � �<sup>H</sup>

SVD, where <sup>Λ</sup><sup>k</sup> <sup>¼</sup> diag ffiffiffiffiffiffi

α<sup>∗</sup> p^kl p vIA

<sup>k</sup> U<sup>k</sup> and V<sup>0</sup>

HkkVIA

<sup>n</sup>þσ<sup>2</sup> ð Þ<sup>w</sup> <sup>∥</sup>ukl∥<sup>2</sup>

kl , ∀ð Þ k, l .

2: Let <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>IA

3: Let <sup>U</sup><sup>k</sup> <sup>¼</sup> <sup>U</sup>IA

4: Calculate <sup>p</sup>^kl <sup>¼</sup> <sup>σ</sup><sup>2</sup>

5: Obtain α<sup>∗</sup> and ρ <sup>∗</sup>

6: Set <sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi

proposition [14].

and gk <sup>¼</sup> <sup>ξ</sup>knrσ<sup>2</sup>

where α<sup>∗</sup>

<sup>v</sup>kl <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffi α<sup>∗</sup> p^kl p v<sup>0</sup>

Algorithm 5.

(CF-PAPS).

142

w∥ukl∥<sup>2</sup>

akl <sup>¼</sup> <sup>σ</sup><sup>2</sup>

<sup>p</sup>^kl <sup>¼</sup> <sup>σ</sup><sup>2</sup>

optimizing α and PS factors ρ<sup>k</sup> under SINR and EH constraints, which is

αp^kl

w∥ukl∥<sup>2</sup> <sup>2</sup>γkl

X K

X d

l¼1

αλklp^kl � σ<sup>2</sup>

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

k¼1

ξk P<sup>K</sup> j¼1 P<sup>d</sup>

0≤ρ<sup>k</sup> ≤1, ∀ð Þ k, l ,

Proposition 11 Given the IA transceivers UIA

nγkl∥ukl∥<sup>2</sup>

kl is the solution to the equation akl

<sup>n</sup>. The optimal solution to (63) is given by

ρ ∗

Given α<sup>∗</sup> and p^kl, the transmit precoders are then determined by

transmit power allocation and receive power splitting is summarized in

<sup>k</sup> , VIA

λk1 <sup>p</sup> , ffiffiffiffiffiffi λk2 <sup>p</sup> , … , ffiffi

<sup>k</sup> <sup>¼</sup> <sup>V</sup>IA

<sup>2</sup>γkl <sup>λ</sup>kl , ∀ð Þ k, l .

<sup>k</sup> , ∀k according to (64).

Algorithm 5 SWIPT design with closed-form transmits power allocation and receive power splitting solutions over the effective IA channel decomposing

� �.

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

<sup>k</sup> Vk, ∀k.

<sup>α</sup><sup>∗</sup> <sup>¼</sup> max ∀ð Þ k, l α∗

<sup>k</sup> ¼ max

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

<sup>n</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> w � �∥ukl∥<sup>2</sup>

λkl

nγkl∥ukl∥<sup>2</sup> 2 ,

The closed-form solution of (63) can be derived and given by the following

2, ckl ¼ λklp^kl, f <sup>k</sup> ¼ ξ<sup>k</sup>

<sup>m</sup>¼<sup>1</sup>αp^jm∥Hkjv<sup>0</sup>

� � ,

jm∥<sup>2</sup>

<sup>k</sup> , VIA

<sup>α</sup>ckl�bkl <sup>þ</sup> <sup>ψ</sup><sup>k</sup>

<sup>k</sup> , <sup>∀</sup><sup>k</sup> � � that satisfy the IA conditions.

<sup>k</sup> , <sup>∀</sup>k, and decompose <sup>H</sup>kk as <sup>H</sup>kk <sup>¼</sup> <sup>U</sup>kΛkV<sup>H</sup>

λ p kd

kl. The proposed IA-based SWIPT scheme with the closed-form

α f <sup>k</sup>þgk

<sup>k</sup> , <sup>∀</sup><sup>k</sup> � �, define

P<sup>K</sup> j¼1 P<sup>d</sup>

<sup>m</sup>¼<sup>1</sup>p^jmξk∥Hkjv<sup>0</sup>

<sup>¼</sup> <sup>1</sup> ð Þ <sup>α</sup>><sup>1</sup> , <sup>ρ</sup>kl <sup>¼</sup> akl

kl, (64)

<sup>∀</sup><sup>l</sup> <sup>ρ</sup>kl, (65)

<sup>2</sup> þ nrσ<sup>2</sup> n

Following Lemma 7, (58) can be optimized by substituting pkl with αp^kl, where α≥ 1 is a scaling factor to be optimized. Then, (58) is reduced to a problem jointly

<sup>2</sup>γkl

: (62)

(63)

jm∥<sup>2</sup> 2,

α<sup>∗</sup> ckl�bkl .

<sup>k</sup> through

Simulations are done over the wireless system as described in Section 3.1, by setting the number of users K ¼ 3 or K ¼ 4. The entries of Hkj are assumed to be i.i. d. zero mean complex Gaussian random variables with variance r �β kj , where rkj is the distance in meters between the jth transmitter and the kth receiver, and β is the path loss factor. The parameters of the symmetric network are set to be nt ¼ nr, rkl <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> ¼ �70 dBm, <sup>σ</sup><sup>2</sup> <sup>w</sup> ¼ �50 dBm, ξkl ¼ ξ ¼ 0:8, γkl ¼ γ, and ψ<sup>k</sup> ¼ ψ. For the three-user network, the closed-form MIMO linear IA algorithm given in [28] is adopted to design IA transceivers. For the four-user network, the MIL algorithm [30] is used. The simulation results are obtained by taking the average of the simulation results of all 100 channel realizations.

Figure 8 shows the convergence performance of SDP-JTDPS with γ ¼ f g 0, 20 dB and <sup>ψ</sup> <sup>¼</sup> f g 0, 10 dBm for one channel realization of the 4, 4, 2 ð Þ<sup>3</sup> network. It can be observed that the algorithm converges monotonically, which verifies the convergence analysis.

The empirical cumulative distribution function (CDF) of the output per-stream SINR for the 4, 4, 2 ð Þ<sup>3</sup> network is shown in Figure 9. The SINR target <sup>γ</sup> is 20 dB, and the EH target is 0 and 30 dBm, respectively. The results show that the achieved SINR values exceed the given 20 dB, meaning that the proposed schemes can satisfy the SINR constraints. The difference is that the achieved SINR value can be greater than the target SINR value for SDP-JTDPS and O-PAPS, while for CF-PAPS, the achieved SINR values are always equal to the SINR target, which implies that the EH constraints are satisfied with equality in CF-PAPS.

#### Figure 8.

Convergence property of the semidefinite programming (SDP)-joint transceiver design and power splitting (JTDPS) scheme for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup> network.

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 �10 to 30 dBm when the SINR threshold is fixed. This is because more transmit power is needed to support higher EH requirements. When the SINR threshold is low, the SDP-JTDPS performs the best, and the O-PAPS schemes achieve almost the same performance, and both of them outperform the CF-PAPS scheme. However, when the SINR threshold is high, the SDP-JTDPS scheme performs worse than the O-PAPS scheme and even worse than the CF-PAPS scheme. From the derivation of the algorithms, we expect that the SDP-JTDPS achieves the best performance, but it does not at high SINR regime. The reason is that when SINR is high, the convergence becomes slow, but we set the fixed iteration number in our simulations. Moreover, the difference between the CF-PAPS and O-PAPS will tend to zero as the SINR threshold becomes high. This

implies that the performance of the CF-PAPS scheme is asymptotically the same as that of the O-PAPS scheme. The reason can be explained. Specifically, high SINR means high transmit power, and it is well known that the margin reward of the power allocation will tend to zero when the transmit power becomes high.

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

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

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., γ ¼ �20 dB). But when the SINR threshold becomes high, SDP-JTDPS performs inferior to the other two schemes significantly at a low EH threshold. The reason is that the convergence speed of SDP-JTDPS tends to slow at that regime. In addition, the performance curves of SDP-JTDPS and CF-PAPS tend to almost the

Figure 11.

Figure 12.

145

Average total transmit power versus EH thresholds for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup> network.

Total transmit power versus SINR thresholds for the 6ð Þ , <sup>6</sup>, <sup>2</sup> <sup>4</sup> network over one channel realization.

Figure 9.

Empirical distribution of achieved SINR at ID receivers with different SINR and energy harvesting (EH) thresholds for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup> .

Figure 10. Average total transmit power versus SINR thresholds for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup> .

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

implies that the performance of the CF-PAPS scheme is asymptotically the same as that of the O-PAPS scheme. The reason can be explained. Specifically, high SINR means high transmit power, and it is well known that the margin reward of the power allocation will tend to zero when the transmit power becomes high.

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., γ ¼ �20 dB). But when the SINR threshold becomes high, SDP-JTDPS performs inferior to the other two schemes significantly at a low EH threshold. The reason is that the convergence speed of SDP-JTDPS tends to slow at that regime. In addition, the performance curves of SDP-JTDPS and CF-PAPS tend to almost the

Figure 11. Average total transmit power versus EH thresholds for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup> network.

Figure 12. Total transmit power versus SINR thresholds for the 6ð Þ , <sup>6</sup>, <sup>2</sup> <sup>4</sup> network over one channel realization.

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 �10 to 30 dBm when the SINR threshold is fixed. This is because more transmit power is needed to support higher EH requirements. When the SINR threshold is low, the SDP-JTDPS performs the best, and the

Recent Wireless Power Transfer Technologies

O-PAPS schemes achieve almost the same performance, and both of them

Figure 9.

Figure 10.

144

thresholds for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup>

.

Average total transmit power versus SINR thresholds for the 4ð Þ , <sup>4</sup>, <sup>2</sup> <sup>3</sup>

outperform the CF-PAPS scheme. However, when the SINR threshold is high, the SDP-JTDPS scheme performs worse than the O-PAPS scheme and even worse than the CF-PAPS scheme. From the derivation of the algorithms, we expect that the SDP-JTDPS achieves the best performance, but it does not at high SINR regime. The reason is that when SINR is high, the convergence becomes slow, but we set the fixed iteration number in our simulations. Moreover, the difference between the CF-PAPS and O-PAPS will tend to zero as the SINR threshold becomes high. This

Empirical distribution of achieved SINR at ID receivers with different SINR and energy harvesting (EH)

.

same when an extremely high EH threshold and a high SINR threshold (e.g., EH ¼ 30 dBm and γ ¼ 40 dB) are given.

because only one beamforming vector is utilized at each transmitter in the DIA scheme. Multiplexing gain of SDP-JTDPS helps achieve better performance in

Transceiver Design for Wireless Power Transfer for Multiuser MIMO Communication Systems

The performance of the proposed schemes is further tested in a 6, 6, 2 ð Þ<sup>4</sup> network. The average transmit powers versus the EH thresholds over different SINR requirements are shown in Figure 13. Similar to Figure 11, the curves of O-PAPS and CF-PAPS asymptotically tend to be the same for a high EH threshold over all the given SINR thresholds. The curves of DIA also asymptotically tend to be the same over all the given SINR thresholds. The phenomenon reflects that CF-PAPS achieves similar performance with O-PAPS. This is somewhat like the water-filling for the power allocation in traditional MIMO systems where the average power allocation is near optimal at high SNR regimes [36]. Nevertheless, the curves of SDP-JTPDS do not tend to be the same over the observed EH scope. This is because again the convergence speed of SDP-JTPDS will slow down when SINR becomes high. Considering its high computational complexity, the iteration number is set to

Finally, Figure 14 compares the computational complexity of the proposed schemes for different antenna numbers by assuming there are K ¼ 3 users. It can be observed that the computational complexity of SDP-JTDPS increases nonlinearly with M, while those of O-PAPS and CF-PAPS increase linearly. Therefore, CF-PAPS and O-PAPS are of much lower complexity than the SDP-based scheme and thus are

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 simulta-

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

transmit power than DIA.

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

be 10 in all the simulations.

4. Conclusions

Acknowledgements

147

more attractive for practical applications.

neous wireless information and power transfer.

(Shandong Academy of Sciences) under grant 2017BSHZ005.

Figure 12 compares SDP-JTDPS with DIA proposed [34] in a 6, 6, 2 ð Þ<sup>4</sup> network over different EH thresholds at fixed SINR values. Note that restricted by its design mechanism, only one data stream is transmitted in the DIA algorithm in the simulation. It can be seen that SDP-JTDPS performs the best, and O-PAPS performs almost the same for low SINR threshold. But when the SINR threshold becomes high, SDP-JTDPS performs worse. This phenomenon is again because SDP-JTDPS has a slower convergence speed when the SINR is high, while the maximum iteration number is set to be 10 in the simulation for saving calculation time. The DIA scheme consumes more transmit power at any given SINR and EH threshold. This is

Figure 13. Total transmit power versus EH thresholds for the 6ð Þ , <sup>6</sup>, <sup>2</sup> <sup>4</sup> network over one channel realizations.

Figure 14. Average execution time versus M at γ ¼ 10 dB and ψ ¼ 10 dBm.

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

because only one beamforming vector is utilized at each transmitter in the DIA scheme. Multiplexing gain of SDP-JTDPS helps achieve better performance in transmit power than DIA.

The performance of the proposed schemes is further tested in a 6, 6, 2 ð Þ<sup>4</sup> network. The average transmit powers versus the EH thresholds over different SINR requirements are shown in Figure 13. Similar to Figure 11, the curves of O-PAPS and CF-PAPS asymptotically tend to be the same for a high EH threshold over all the given SINR thresholds. The curves of DIA also asymptotically tend to be the same over all the given SINR thresholds. The phenomenon reflects that CF-PAPS achieves similar performance with O-PAPS. This is somewhat like the water-filling for the power allocation in traditional MIMO systems where the average power allocation is near optimal at high SNR regimes [36]. Nevertheless, the curves of SDP-JTPDS do not tend to be the same over the observed EH scope. This is because again the convergence speed of SDP-JTPDS will slow down when SINR becomes high. Considering its high computational complexity, the iteration number is set to be 10 in all the simulations.

Finally, Figure 14 compares the computational complexity of the proposed schemes for different antenna numbers by assuming there are K ¼ 3 users. It can be observed that the computational complexity of SDP-JTDPS increases nonlinearly with M, while those of O-PAPS and CF-PAPS increase linearly. Therefore, CF-PAPS and O-PAPS are of much lower complexity than the SDP-based scheme and thus are more attractive for practical applications.

### 4. Conclusions

same when an extremely high EH threshold and a high SINR threshold (e.g., EH ¼

Total transmit power versus EH thresholds for the 6ð Þ , <sup>6</sup>, <sup>2</sup> <sup>4</sup> network over one channel realizations.

Figure 12 compares SDP-JTDPS with DIA proposed [34] in a 6, 6, 2 ð Þ<sup>4</sup> network over different EH thresholds at fixed SINR values. Note that restricted by its design mechanism, only one data stream is transmitted in the DIA algorithm in the simulation. It can be seen that SDP-JTDPS performs the best, and O-PAPS performs almost the same for low SINR threshold. But when the SINR threshold becomes high, SDP-JTDPS performs worse. This phenomenon is again because SDP-JTDPS has a slower convergence speed when the SINR is high, while the maximum iteration number is set to be 10 in the simulation for saving calculation time. The DIA scheme consumes more transmit power at any given SINR and EH threshold. This is

30 dBm and γ ¼ 40 dB) are given.

Recent Wireless Power Transfer Technologies

Figure 13.

Figure 14.

146

Average execution time versus M at γ ¼ 10 dB and ψ ¼ 10 dBm.

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.

#### Acknowledgements

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.

Recent Wireless Power Transfer Technologies
