Transmit Power Minimization in Multihop AF Relay Systems with Simultaneous Wireless Information and Power Transfer

This paper studies a multihop amplify-and-forward (AF) simultaneous wireless information and power transfer (SWIPT) relaying system. Each relaying node harvests energy using power splitting (PS) scheme from a part of its received signal to amplify and forward the rest the received signal to the next relay. Based on this system model and signal flow, we derived and solved the convex energy minimization problem with optimal PS ratio. The influence of processing cost was then investigated for the AF-SWIPT system with the decode-and-forward SWIPT as benchmark, where AF-SWIPT was found to be superior.

Keywords:

multihop amplify-and-forward (AF) relays

;

power splitting (PS) ratio

;

simultaneous wireless information and power transfer (SWIPT)

Subject:

Engineering - Telecommunications

1. Introduction

The Internet-of-Things (IoT) connects network-enabled devices communicating with each other over the Internet. Hence, the objective of IoT is to integrate the physical world and the virtual world to attain a self-sustaining system [1]. Currently, research into the application of IoT for the creation of smart-cities, smart-homes, smart-energy, intelligent-transportation system, and many more is growing. An IoT network consists of the use of routing schemes, a gateway supporting communication between the different nodes and a central system [1]. These routing schemes facilitate either direct or relaying communication between a gateway and a particular node. The routing/relaying nodes can be opportunistically implemented by non-active user devices [1]. This results in the selected user nodes consuming their power in facilitating communication between the two user pairs [2,3]. This implies that, the selected users sacrifice their resources to facilitate relaying procedures [2,11]. The strain on the relay nodes can be mitigated by employing wireless power transfer (WPT) [1,2,3].

Recently, research into alternative wireless sources (e.g., radio frequency (RF) and light energy harvesting) of energy to power IoT devices is also increasing [1,2,3,4,5,6]. Such alternative wireless power sources are on the market, e.g., wi-charge technology (the wi-charger devices), Pi charger, energous RF chargers and Warp RF wireless charging systems. From literature, WPT using RF can be accomplished by using two different techniques, namely, simultaneous wireless information and power transfer (SWIPT), and wireless powered communication network (WPCN) [1,2,3,5]. SWIPT involves the transmission of wireless information signal and wireless power signal concurrently [1,2,3,5]. Time switching (TS) and power splitting (PS) are the two main techniques in order to implement SWIPT [1,2,3]. The successive transmission of wireless information signal and wireless power signal is used to accomplish WPCN [1,2,3].

Research on RF-based WPT has evolved from point-to-point systems, dual-hop cooperative systems to multi-hop cooperative systems. A few literature on both dual-hop and multi-hop cooperative systems can be found in [1,2,3,4,5,6,7,8,9]. In this paper, we extend the work in [1] which focused on a decode-and-forward (DF)-SWIPT multihop system model to an amplify-and forward (AF)-SWIPT multi-hop system model. Similar to the work in [1], we consider single-antenna nodes, SWIPT PS mode at each relay node, and the optimization problem on energy efficiency. Finally, the closed-form solution are obtained for our optimization problem. In the simulation results, we compare AF-SWIPT multi-hop systems to the DF-SWIPT in [1].

The rest of the paper is organized as follows, Section 2, Section 3, and Section 4 contains the the stepwise process in formulating the AF-SWIPT multi-hop optimization problem, the solutions for our optimization problem, and simulation results and discussion, respectively. Finally, concluding remarks are provided in Section 5.

2. System Model and Problem Formulation

The IoT network consists of the source node, K AF-SWIPT relays, and the destination node, are shown in Figure 1. Each node has a single antenna. The source is a base station (BS) which may consist of the data gateway and the external/central systems of the IoT network [11,12,13,15,16,17]. In order to increase the operation efficiency of the relay node, each relay node operates in SWIPT mode. A RF signal received at each relay is split using the PS ratio where a part of the signal is stored in battery in each relay via energy harvesting (EH). Then, the information processing (i.e., amplification and forwarding) is performed using the remaining RF signals, and then the transmission is performed to the next node with the harvested energy.

The structure of our AF-SWIPT relay is shown in Figure 2. Each relay has a battery, which can store energy and perform retransmission using it. Each AF-SWIPT relay performs EH using a power splitting (PS) scheme and operates in half-duplex mode.

We assume that the source node provides the channel state information (CSI) for all communication nodes. However, each relay and the destination node only knows the CSI for their communication channel. It is assumed that there is no direct link between the current node and the next second or more nodes. For example, there is no direct link between the first node and the third node.

The received RF signal at node

k, \forall k = 1 \dots, K + 1

from the previous node is given as

y_{k} = h_{k} x_{k - 1} + n_{k}

(1)

where

h_{k}

is the channel coefficient between the current node and the previous node, and

n_{k}

∼

C N

(0,

δ_{k}^{2}

) represents the antenna noise at the current node. The channel

h_{k}

is defined as

h_{k} = \sqrt{ζ_{k}} {\tilde{h}}_{k}

, where

ζ_{k} = G_{k} {(d_{k} / d_{0})}^{- α_{k}}

is the large-scale fading coefficient,

G_{k}

is the attenuation constant for a distance

d_{0}

α_{k}

is the pathloss exponent,

d_{k}

is distance between the transmit and receive nodes and

{\tilde{h}}_{k}

∼

C N

(0,1) is the Rayleigh fading component. Next, the signal for EH and harvested energy at the kth relay node are written, respectively, as

y_{k}^{E H} = \sqrt{ρ_{k}} y_{k}

(2)

E_{k} = β_{k} ρ_{k} {|h_{k}|}^{2} E_{k - 1}

(3)

E_{K + 1} = β_{K + 1} ρ_{K + 1} {|h_{K + 1}|}^{2} E_{K} - P_{c}

(4)

where

P_{c}

is the processing power for information decoding at destination node,

E_{0}

is the transmit energy at source node, and signal for information transmission at the kth relay is represented by

y_{k}^{T R} = \sqrt{1 - ρ_{k}} y_{k} + z_{k}

(5)

where

z_{k} \sim C N (0, σ_{k}^{2})

is the additional noise introduced by the information decoding (ID) circuitry. The signal-to-noise ratio (SNR) is expressed as

γ_{k} = \frac{\prod_{j = 1}^{k - 1} β_{j} ρ_{j} \prod_{j = 1}^{k} (1 - ρ_{j}) {|h_{j}|}^{2} E_{0}}{\sum_{i = 1}^{k} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{k - 1} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{k} (1 - ρ_{j}) \prod_{j = i + 1}^{k} {|h_{j}|}^{2}]}

(6)

The signal to be transmitted to the kth relay is represented by

x_{k} = g_{k} y_{k}^{T R}

(7)

where

g_{k} \approx \sqrt{\frac{E_{k}}{(1 - ρ_{k}) E_{k - 1} {|h_{j}|}^{2}}} = \sqrt{\frac{β_{k} ρ_{k}}{1 - ρ_{k}}}

is amplification factor, and

x_{0}

is the information signal at source node.

We now consider the problem of optimizing the PS ratio

{\{ρ_{k}\}}_{k = 1}^{K}

at the relay nodes and energy at the source node

E_{0}

. We aim to minimize the source transmit power, under the SNR of destination node constraint and PS ratio for each of the multihop links as

\begin{matrix} \min_{E_{0}, {\{ρ_{k}\}}_{k = 1}^{K}} E_{0} \\ subject to γ_{K + 1} \geq {\bar{γ}}_{K + 1} \\ 0 \leq ρ_{k} \leq 1, k = 1, \dots, K + 1 \\ E_{0} \geq 0 \end{matrix}

(8)

The solution to the above problem is provided in the following section, and the induction process will be attached to the Appendix A.

3. Problem Solution

In this section, we provide the solution of source transmit power minimization problem. The optimal transmit energy at the source node is expressed to

E_{0} = {\bar{γ}}_{K + 1} \frac{\sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]}{\prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}}

(9)

where

{\bar{γ}}_{K + 1}

is the SNR threshold constraint of the destination node. The optimal PS ratio at the kth node is written as

ρ_{k} = \frac{C_{k} + \sum_{i = k + 1}^{K} D_{i} - \sqrt{F_{k} [\sum_{i = k + 1}^{K + 1} D_{i}]}}{C_{k} + \sum_{i = k + 1}^{K} D_{i} - F_{k}}, k = 1, \dots, K

(10)

where

C_{k}

D_{i}

, and

F_{k}

are defined as

C_{k} = ((1 - ρ_{K + 1}) δ_{K + 1}^{2} + σ_{K + 1}^{2}) \prod_{j = k}^{K} (1 - ρ_{j + 1})

(11)

D_{i} = ((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} \prod_{j = k + 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i}^{K} ρ_{j} {|h_{j + 1}|}^{2}

(12)

F_{k} = \prod_{j = k}^{K} β_{j} \prod_{j = k + 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2} \prod_{j = k + 1}^{K} ρ_{j} σ_{k}^{2}

(13)

4. Simulation Results

This section evaluates the performance of the AF-SWIPT system compared with the DF-SWIPT system as benchmark [1]. For the large-scale fading component, the attenuation constant

G_{0} = - 10

dB, and the pathloss exponent

α = 3

are assumed. The distance between each node is

d = d_{1} = \dots = d_{K} = 2

m, the antenna noise variance

σ^{2} = σ_{1}^{2} = \dots = σ_{K + 1}^{2} = - 80

dBm, and the energy conversion efficiency

β = β_{1} = \dots = β_{K} = 0.7

. We asuume that

P_{c} = 0

at destination node, because the destination node are received the enough power for decoding to the previous node. The PS ratio used for the suboptimal scheme is

ρ_{k} = 0.5

. The simulation results are obtained over

10^{4}

random channel realizations.

Figure 3 shows the effect of each relay circuit power

P_{c}

E_{0}

. When

P_{c}

is low, the effect on

E_{0}

is insignificant, but as

P_{c}

increases,

E_{0}

also increases. Unlike AF-SWIPT, where signals are decoded only at destination nodes, DF-SWIPT has a larger of

E_{0}

value because signals are decoded at all relay nodes before they are re-transmission.

A simulation result based on the influence of increasing number of relays, K, follows in Figure 4. Figure 4 shows that

E_{0}

increases as the number of relays. The influence of

P_{c}

affects all the relays. It is observed that, the AF-SWIPT system has a lower

E_{0}

than the DF-SWIPT system. This is because there is no signal decoding (i.e., processing power) at the relays for AF-SWIPT, unlike in the DF-SWIPT. In the AF-SWIPT signal decoding occurs only at the destination. Therefore, processing power is used only at the destination for the AF-SWIPT, while processing power is used at all nodes in the DF-SWIPT.

Figure 5 shows a plot of the average

E_{0}

versus the SNR threshold constrain

\bar{γ}

(in dB). The optimal PS scheme shows a lower

E_{0}

value than the suboptimal PS scheme. When

\bar{γ}

is low,

E_{0}

is very high, which seems to have a great influence on the

\bar{γ}

with a low

P_{c}

influence.

Figure 6 shows the

E_{0}

against the internode distance,

d_{k}

. As

d_{k}

increases, more transmission power is required.

5. Conclusion

This paper investigated on the AF-SWIPT multi-hop cooperative relaying in IoT networks. Depending on the decoding cost, the AF-SWIPT system is more efficient compared to DF-SWIPT for data relaying. The AF-SWIPT source energy minimization optimization problem is presented in this paper. The idea is to promote energy efficiency through signal transmission energy cost. Simulation results showed the AF-SWIPT has better energy efficiency in terms of decoding cost compared to the DF-SWIPT. Possible extensions of this research are in the area of implementing the time-switching SWIPT protocol, MIMO multi-hop systems, WPCN multi-hop networks and development of routing algorithms for the multi-hop wireless powered networks.

Author Contributions

Conceptualization Asiedu, D.K.P.; Methodology, Original Draft Preparation and Writing Kim, K.J.; Review and editing Prince, A. and Kim, E; Supervision, Lee, K.-J..

Appendix A

The problem 8 is nonconvex with respect to

E_{0}

and

{\{ρ_{k}\}}_{k = 1}^{K}

. By changing the problem, we rewrite problem 8 into and equivalent convex problem.

\begin{matrix} \min_{Q, {\{ρ_{k}\}}_{k = 1}^{K}} 1 / Q \\ subject to γ_{K + 1} \geq {\bar{γ}}_{K + 1} \\ 0 \leq ρ_{k} \leq 1, k = 1, \dots, K + 1 \\ Q \geq 0 \end{matrix}

(A1)

where

Q = (1 / E_{0})

. The Lagrangian of the problem A1 is defined with its KKT conditions given as

L \{Q, λ\} = \frac{1}{Q} + λ ({\bar{γ}}_{K + 1} - \frac{\prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}}{Q \sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]})

(A2)

\begin{matrix} \frac{\partial L}{\partial Q} = - \frac{1}{Q^{2}} \\ + λ ({\bar{γ}}_{K + 1} + \frac{\prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}}{Q^{2} \sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]}) = 0 \end{matrix}

(A3)

λ ({\bar{γ}}_{K + 1} - \frac{\prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}}{Q \sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]}) = 0

(A4)

From A3, we can calculate

λ

λ = \frac{\sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]}{\prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}}

(A5)

and substituting A5 into A3,

\frac{1}{Q} = {\bar{γ}}_{K + 1} \frac{\sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]}{\prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}}

(A6)

Next, we change the problem A1 to find the optimal PS ratio as

\begin{matrix} \min_{x, y, {\{ρ_{k}\}}_{k = 1}^{K}} \frac{x^{2}}{y} \\ subject to {\bar{γ}}_{K + 1} \sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}] \geq x^{2} \\ \prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2} \geq y \end{matrix}

(A7)

and we formulate the Lagrangian of changed problem A7 to

\begin{matrix} L \{x, y, {\{ρ_{k}\}}_{k = 1}^{K}, λ_{1}, λ_{2}\} = \frac{x^{2}}{y} + \\ λ_{1} (x^{2} - {\bar{γ}}_{K + 1} \sum_{i = 1}^{K + 1} [((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} ρ_{j} \prod_{j = 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i + 1}^{K + 1} {|h_{j}|}^{2}]) \\ + λ_{2} (y - \prod_{j = 1}^{K} β_{j} ρ_{j} \prod_{j = 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2}) \end{matrix}

(A8)

The KKT conditions are represented as

\frac{\partial L}{\partial ρ_{k}} = {\bar{γ}}_{K + 1} (- δ_{k}^{2} A_{k} - \sum_{i = k + 1}^{K + 1} B_{i} \frac{1}{ρ_{k}} + B_{k} \frac{1}{1 - ρ_{k}}) = 0, k = 1, . . ., K

(A9)

where

A_{k}

and

B_{k}

are defined as bellows

A_{k} = \frac{1}{\prod_{j = 1}^{k - 1} β_{j} ρ_{j} (1 - ρ_{j}) \prod_{j = 1}^{k} {|h_{j}|}^{2}}

(A10)

B_{k} = ((1 - ρ_{k}) δ_{k}^{2} + σ_{k}^{2}) A_{k}

(A11)

From A5, we can find the each optimal PS ratios to

ρ_{k} = \frac{C_{k} + \sum_{i = k + 1}^{K} D_{i} - \sqrt{F_{k} [\sum_{i = k + 1}^{K + 1} D_{i}]}}{C_{k} + \sum_{i = k + 1}^{K} D_{i} - F_{k}}, k = 1, \dots, K

(A12)

ρ_{K + 1} = 0

(A13)

where the constants

C_{k}

D_{i}

, and

F_{k}

are defined as

C_{k} = ((1 - ρ_{K + 1}) δ_{K + 1}^{2} + σ_{K + 1}^{2}) \prod_{j = k}^{K} (1 - ρ_{j + 1})

(A14)

D_{i} = ((1 - ρ_{i}) δ_{i}^{2} + σ_{i}^{2}) \prod_{j = i}^{K} β_{j} \prod_{j = k + 1, j \neq i}^{K + 1} (1 - ρ_{j}) \prod_{j = i}^{K} ρ_{j} {|h_{j + 1}|}^{2}

(A15)

F_{k} = \prod_{j = k}^{K} β_{j} \prod_{j = k + 1}^{K + 1} (1 - ρ_{j}) {|h_{j}|}^{2} \prod_{j = k + 1}^{K} ρ_{j} σ_{k}^{2}

(A16)

References

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Figure 1. Multihop AF relay systems with SWIPT architecture.

Figure 2. Multihop AF relay node SWIPT architecture.

Figure 3. The influence of increasing each relay

P_{c}

(

K = 3, \bar{γ} = - 5

dB).

Figure 3. The influence of increasing each relay

P_{c}

(

K = 3, \bar{γ} = - 5

dB).

Figure 4. Average

E_{0}

against the number of relay nodes at

\bar{γ} = - 5

dB,

P_{c} = - 50

dBm.

Figure 4. Average

E_{0}

against the number of relay nodes at

\bar{γ} = - 5

dB,

P_{c} = - 50

dBm.

Figure 5. Average

E_{0}

relative to

\bar{γ}

K = 3

P_{c} = - 50

dBm.

Figure 5. Average

E_{0}

relative to

\bar{γ}

K = 3

P_{c} = - 50

dBm.

Figure 6. Average

E_{0}

relative to

\bar{γ}

K = 3

P_{c} = - 50

dBm.

Figure 6. Average

E_{0}

relative to

\bar{γ}

K = 3

P_{c} = - 50

dBm.

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Transmit Power Minimization in Multihop AF Relay Systems with Simultaneous Wireless Information and Power Transfer

Abstract

Keywords:

Subject:

1. Introduction

2. System Model and Problem Formulation

3. Problem Solution

4. Simulation Results

5. Conclusion

Author Contributions

Appendix A

References

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