Energy harvesting assisted cognitive radio: random location-based transceivers scheme and performance analysis

Abstract

We consider spectrum-sharing scenario where coexist two communication networks including primary network and secondary network using the same spectrum. While the primary network includes directional multi-transceivers, the secondary network consists of relaying-based transceiver forwarding signals by energy harvesting assisted relay node. In cognitive radio, signals transmitted from secondary network are sufficiently small so that all of primary network receivers have signal to noise ratio (SNR) greater than a given threshold. In contrast, the transmitted signals from primary network cause increasing noise which is difficult to demodulate at secondary network nodes and hence it leads to the peak power constraint. In this paper, we focus on the influence of random location of transceivers at primary network using decode-and-forward protocol. Specifically, we derive closed-form outage probability expression of the secondary network under random location of transceivers and peak power constraint of primary network. This investigation shows the relationship between the fraction of energy harvesting time $\alpha$ of time switching-based relaying protocol on outage probability of secondary network and throughput. In addition, we analyse the influence of the number of primary network transceivers as well as primary network’s SNR threshold on secondary network. Furthermore, the trade-off between increasing energy harvesting and rate was investigated under the effect of energy conversion efficiency. The accuracy of the expressions is validated via Monte-Carlo simulations. Numerical results highlight the trade-offs associated with the various energy harvesting time allocations as a function of outage performance.

1 Introduction

Recently, energy harvesting (EH) attracted the attention of researchers, starting from the simple physical model, in which one source directly transmits to one destination without noise, until source transmits via an relay node using amplify-and-forward (AF) as well as decode-and-forward (DF) protocols [1,2,3,4,5]. In particular, the authors in [2] analyzed a EH assisted cognitive radio network. In the literature, some models were developed into complicated systems, where a receiver in system has novel design of energy scavenger from other transmitters using time-switching scheme (TS) and power-switching scheme (PS) [4, 5]. As extended work, the hybrid protocol for energy harvesting considering both time allocation and power splitting is proposed in [6]. The authors in [7, 8] also built a AF two-way relaying network (TWRN) and extended to other complex cooperative protocols as in [8]. In [9], the authors investigated the power constraints in cognitive network under EH. More specifically, energy harvesting is performed at secondary receivers. However, most existing energy harvesting solutions rely on natural energy sources, such as wind and solar power and simultaneous wireless information and power transfer (SWIPT) technique can distribute energy harvesting in popular indoor environments in practice.

Considering cognitive radio (CR) networks, the previous works assumed that the system with one primary receiver, one cognitive transmitter-receiver pair, and one energy harvesting relay as in [10,11,12,13]. In order to cope with the stringent demand and scarcity of the radio spectrum in the next generation of wireless communication, it is believed that CR is promising to do so. The authors derived analytical expressions for the outage probability, as well as their high signal-to-noise ratio (SNR) approximations in closed form as [10, 12]. Furthermore, the authors in [11] proposed a dynamic power splitting (DPS) scheme in which the received signal is divided into two streams with adjustable power levels for energy harvesting and information processing separately with the consideration of the instantaneous channel condition. If the interference temperature is under a controllable level [14], or the use of the spectrum is unavailable, access to the spectrum licensed to primary users (PUs) from a secondary user (SU) is permitted. A new CR paradigm has just been suggested from the concept of cooperative communication for cognitive radio networks (CRN), where parts of spectral sources are leased to SUs by the primary network so as to cooperate [15,16,17,18].

According to the aforementioned studies, harvesting technology’s development is in the beginning. For that reason, numerous deployments and system models are found in order to introduce applications in various circumstances. It is believed that wireless communication also enjoys one of a state-of-the-art method, including the mixture of relay transmission and cognitive radio, because cognitive radio enhances spectrum use by sharing between primary users (PUs) and secondary users (SUs) while relay transmission helps improve throughput and extends cover age over conventional point-to-point transmissions. Additionally, the difficulty of throughput optimization in terms of energy harvesting in relay systems is only affected by only one time slot, including relay-destination link and the data transmission time of the source-relay link. In addition, there are some other investigations into the area of cognitive networks in [19,20,21]. Scavenging energy from ambient sources or cognitive radio transfer systems is available, the derived throughput of the secondary transmitter was studied in [19]. Meanwhile, [20] examined random access for a cognitive radio, where energy harvesting secondary terminal can be cooperated in signal transmission. Furthermore, so as to derive the upper bound of the total throughput, an ideal spectrum access policy energy harvesting CR was outlined in [21]. The optimization of the spatial SU throughput subject to SU transmit power and SU density in a given geography utilizing stochastic geometry is noticed by the authors in [22]. It is assumed that if a SUs is close enough to a PU node, the former can scavenge power from the latter. Any SU in harvesting zone is capable of scavenging the PU transmitted energy because each PU transmitter contains a harvesting zone with a small radius satisfying the threshold energy harvesting circuit’s requirements.

It is noting that transmitters and receivers are conventional located at anywhere, received signal in arbitrary node are suffered simultaneously by the influence of unexpected transmitters which becomes the main focus of this paper. In particular, this paper employs RF signal to transmit information and harvest energy from radio wave in cognitive network with assume that all the channels are subject to slowly varying Rayleigh fading. In principle, there are two independent communication networks operating on the same spectrum: primary and secondary. Primary network (called $P\_Net$) has many independent information links. Secondary network (called $S\_Net$) has only one information source, which transmits through a relay node to the destination receiver. All of transmitters are supplied by independent energy source (i.e. grid power) except relay node. Radio wave supplies energy to the relay node in $S\_Net$. Continuously, relay node still uses this energy to transmit information to the destination. It is noting that the more signals to relay node, the more energy to support transmission. It means that communication distance is prolonged. However, when the power transmitted from source or relay node becomes too strong, it causes noise at the destinations of $P\_Net$. Consequently, the performance of wireless communication on $P\_Net$ becomes worse. Therefore, both the transmitted power of relay node and source node in $S\_Net$ are also limited by a given threshold that $P\_Net$ can transmit information successfully. Consequently, received signals have exponential distribution form with different scales.

The main contributions of this paper can be summarized as in the follow:

• This paper derives the secondary network’s closed-form outage probability expression under random location of transceivers and the peak power constraint of primary network. In general, the location of $P\_Net$’s transceivers is arbitrary. Therefore, signal at a receiver is not simply a sum of many identically received signals despite the fact that all of primary network transmitters are assumed the same power.

• By simulation, it can be found in TSR protocol the off-line optimal fraction of block time on secondary network to achieve better outage performance.

• This paper investigates the effect of random location of transceivers on $S\_Net$’s outage probability. Obviously, a re-arrangement policy leads to guarantee reliable communication.

• We analyse the influence of primary network’s threshold and the number of transceivers on secondary network outage probability. As a result, increasing the number of transceivers leads to slightly less outage performance.

• We consider the trade-off between rate and energy harvesting. It can be seen that more harvested energy not only increase rate in the first phase due to satisfying transmitting power but also decrease rate in second phase, especially in fraction of block time is larger than optimal value.

Fig. 1

$S\_Net$ and $P\_Net$ under spectrum-sharing scenario

Fig. 2

TSR protocol structure

Notations: $E\left\{ \right\}$ denotes expectation operation, $\mathrm{Pr}\left\{ {} \right\}$ is outage probability, and ${f_X}\left\{ {} \right\}$ is probability density functions (PDF), ${F_X}\left\{ {} \right\}$ is cumulative distribution functions (CDF).

The remainder of the paper is organized as follows. Section 2 describes the considered system model and Sect. 3 presents some fundamental preliminaries regarding the secondary network performance analysis. Specifically, Sect. 3 examine the trade-off in the outage probability and throughput performance in scenario of energy harvesting. Numerical results and useful insights are provided in Sect. 4, followed by our conclusions in Sect. 5.

2 System model and power constraint condition

2.1 System model

As shown in Fig. 1, two networks coexist simultaneously in a space and operate on the same spectrum. The first network is denoted as $P\_Net$ consists of L transmitter nodes and L receiver nodes, respectively. The other known as $S\_Net$ has one transmitter node (ST) which transfers information to destination node (SD) through a relay node (RL).

In spectrum-sharing scenario, a network needs to control its transmitting power that not to completely disrupt communication from another one. In particular, RF signals from ST and RL are sufficiently small so that receiver nodes PRe[i] have SNR[i] ($i=1,2,\ldots ,L$) that are better than given threshold $\gamma _{P}$. Otherwise, signals from $P\_Net$ cause noise to demodulate at RL and SD. Thus, SNR decreases substantially at these nodes. However, harvested energy significantly increases at RL node that support higher rate on second hop.

2.2 Power constraint condition

We consider the effect of transmitting power from ST on a receiver nodes PRe[i] with assuming that all of PRe[i] have the same SNR threshold $\gamma _{P}$. This power must be small enough that all of outage probability on $i-th$ link is smaller primary outage threshold $\varPsi$ in the $P\_Net$.

$$\begin{aligned} {P_{\mathrm{out - PRe}}} = \Pr \left( {\mathop {Max}\limits _{i = 1:L} \;{P_{out,\mathrm{PRe}[i]}}} \right) \le \varPsi \end{aligned}$$

(1)

To ensure that completely successful communication in $P\_Net$, outage probability on node satisfies as

$$\begin{aligned} {P_{out,\mathrm{{PRe}}[i]}} = \Pr \left( {Bw lo{g_2}\left( {1 + {\gamma _{\mathrm{PRe}[i]}}} \right) \le {R_P}} \right) \le \varPsi \end{aligned}$$

(2)

where $\gamma _{PRe[i]}$ is the SNR on PRe[i], ${R_{P}}$ is primary rate (assumed equal value of each primary link), Bw is bandwidth of a channel. It can be seen that (2) shows the relationship between $\gamma _{PRe[i]}$ and ${R_{P}}$ in accordance with known probability $\varPsi$. It means that all of $\gamma _{PRe[i]}$ must be greater than ${\gamma _p} = {2^{{R_\mathrm{P}}/Bw}} - 1$. Conventionally, PRe[i] suffers from noise from nearby transmitters and ST, RL. In this paper, $S\_Net$ employs TSR protocol as Fig. 2, which consists of three phases: (a) energy harvesting, in which ST transmits, (b) information transmission from ST to RL, (c) information transmission from RL to SD. In another words, ST and RL must not transmit concurrently. Thus, arbitrary PRe[i] is not affected from both ST and RL simultaneously.

It is assumed that $i-th$ link in $P\_Net$ is only influenced from ST, SNR of PRe[i] is given by

$$\begin{aligned} {\gamma _{\mathrm{PRe}[i]}} = \frac{{{{\mathop {\hbox {P}}\nolimits } _{PT[i]}}{{\left| {{h_{pp[i]}}} \right| }^2}}}{{\sum \nolimits _{j = 1,j \ne i}^L {{P_{PT[j]}}{{\left| {{h_{pp[j]}}} \right| }^2}} + {P_{ST}}{{\left| {{h_{sp[i]}}} \right| }^2} + {\sigma ^2}_{\mathrm{Pri}}}}\nonumber \\ \end{aligned}$$

(3)

where $h_{pp[i]}$, $h_{sp[i]}$ is channel gains of $i-th$ directional link and interference link from ST, respectively. $P_{ST}$ is power from ST. $P_{PT[i]}$, which is transmitting power from PTr[i], is assumed equal value of each transceivers, called $P_{T}$. Obviously, the contribution of noise is much less than signal power. Thus, denominator in (3) can re-write with noise term eliminated. Because of concentrating on $S\_Net$, we assume that the influence of signals inside $P\_Net$ is ignored. Thus, (3) is reduced as follow

$$\begin{aligned} {\gamma _{\mathrm{PRe[i]}}} = \frac{{{{\mathop {\hbox {P}}\nolimits } _T}{{\left| {{h_{pp[i]}}} \right| }^2}}}{{{P_{ST}}{{\left| {{h_{sp[i]}}} \right| }^2}}} \end{aligned}$$

(4)

Without loss of generality, we only consider one receiver node known as PRe, and $\lambda _{pp}$ stands for the channel gain $h_{pp[i]}$. In conventional cognitive radio, the power constraint at ST is shown as

$$\begin{aligned}&\Pr \left( {{\gamma _{\mathrm{PRe}}} \le {\gamma _p}} \right) = 1 - {\left( {\frac{{{P_T}{\lambda _{pp}}}}{{{P_{ST}}{\lambda _{sp}}{\gamma _p} + {P_T}{\lambda _{pp}}}}} \right) ^L} \le \varPsi \end{aligned}$$

(5)

$$\begin{aligned}&{P_{ST}} \le \frac{{{P_T}{\lambda _{pp}}\left( {1 - \root L \of {{1 - \varPsi }}} \right) }}{{{\lambda _{sp}}{\gamma _p}\root L \of {{1 - \varPsi }}}} \end{aligned}$$

(6)

Similarly, the power constraint at RL is given by

$$\begin{aligned} {P_{R}} \le \frac{{{P_T}{\lambda _{pp}}\left( {1 - \root L \of {{1 - \varPsi }}} \right) }}{{{\lambda _{sp}}{\gamma _p}\root L \of {{1 - \varPsi }}}} \end{aligned}$$

(7)

3 Performance analysis of secondary network

3.1 Outage probability performance

We first consider the outage probability at relay node. Based on system model shown in the previous section, SNR at RL is given by

$$\begin{aligned} {\gamma _{RL}} = \frac{{{{\mathop {\hbox {P}}\nolimits } _{S\min }}{{\left| {{h_{sr}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {{P_{PT[i]}}{{\left| {{h_{pr[i]}}} \right| }^2}} + \sigma _{RL}^2}} \end{aligned}$$

(8)

where ${P_{S\min }} = Min\left\langle {{P_{ST}},{P_T}} \right\rangle$ is real transmitting power from ST, $h_{pr[i]}$ is the channel gain of $i-{th}$ interference link from $i-{th}$ transmitter to RL, $h_{sr}$ is the channel gain between ST and RL, $\sigma _{RL}^2$ is typically internal noise characterized as Gaussian distribution with zero-mean and variance $\sigma _{RL}^2 = {\sigma ^2}$. For simplicity in computations, we assume that ${P_{PT[i]}} = {P_T}$.

$$\begin{aligned} {\gamma _{RL}} = \frac{{\frac{{{{\mathop {\hbox {P}}\nolimits } _{S\min }}}}{{{\sigma ^2}}}{{\left| {{h_{sr}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_T}}}{{{\sigma ^2}}}{{\left| {{h_{pr[i]}}} \right| }^2}} + 1}} \end{aligned}$$

(9)

Thus, the outage probability can be expressed as

$$\begin{aligned} {F_{RL}}\left( {{\gamma _s}} \right) = Pr\left( {\frac{{\frac{{{{\mathop {\hbox {P}}\nolimits } _{S\min }}}}{{{\sigma ^2}}}{{\left| {{h_{sr}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_T}}}{{{\sigma ^2}}}{{\left| {{h_{pr[i]}}} \right| }^2}} + 1}} \le {\gamma _s}} \right) \end{aligned}$$

(10)

Lemma 1

Let ${\left( {{X_i}} \right) _{i = 1 \div L,{} \;L \ge 2}}$ be independent exponential distributed random variables (ERVs) that have

$$\begin{aligned} pdf\left( {{X_i}} \right) = \left\{ {\begin{array}{ll} \frac{1}{{{\lambda _i}}}{e^{ - \frac{x}{{{\lambda _i}}}}}&{}\quad x \ge 0\\ 0 &{}\quad x < 0\quad \end{array}} \right. \end{aligned}$$

(11)

where ${\lambda _i}$ is the scale of ${X_i}$. Then probability density function (PDF) of their sum is

$$\begin{aligned} {f_{\sum \limits _{i = 1}^L {{X_i}} }}(x) = \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _i}}}} } \right] \times \sum \limits _{j = 1}^L {\frac{{{e^{ - \frac{x}{{{\lambda _j}}}}}}}{{\prod \limits _{k = 1,k \ne j}^L {\left( {\frac{1}{{{\lambda _k}}} - \frac{1}{{{\lambda _j}}}} \right) } }}} \end{aligned}$$

(12)

And cumulative distribution function (CDF) of their sum is

$$\begin{aligned} \Pr \left( {\sum \limits _{i = 1}^L {{X_i}} \le x} \right)= & {} \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _i}}}} } \right] \sum \limits _{j = 1}^L {\frac{{\Pr \left( {{X_j} < x} \right) }}{{\frac{1}{{{\lambda _j}}}\prod \nolimits _{k = 1,k \ne j}^L {\left( {\frac{1}{{{\lambda _k}}} - \frac{1}{{{\lambda _j}}}} \right) } }}}\nonumber \\ \end{aligned}$$

(13)

Proof

The PDF was found in [23]. The CDF is given by:

$$\begin{aligned} \Pr \left( {\sum \limits _{i = 1}^L {{X_i}} \le x} \right)= & {} \int _0^x {{f_{\sum \limits _{i = 1}^L {{X_i}} }}(u)du} = \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _i}}}} } \right] \nonumber \\&\times \sum \limits _{j = 1}^L {\frac{{\int \nolimits _0^x {\frac{1}{{{\lambda _j}}}{e^{ - \frac{u}{{{\lambda _j}}}}}du} }}{{\frac{1}{{{\lambda _j}}}\prod \nolimits _{k = 1,k \ne j}^L {\left( {\frac{1}{{{\lambda _k}}} - \frac{1}{{{\lambda _j}}}} \right) } }}} \end{aligned}$$

(14)

$\square$

Proposition 1

The cumulative distribution function ${F_{ RL}}\left( {{\gamma _s}} \right)$ is computed as

$$\begin{aligned} {F_{RL}}\left( {{\gamma _s}} \right)= & {} 1 - \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \nonumber \\&\times \sum \limits _{j = 1}^L {\frac{{{e^{ - \;\frac{{{\gamma _s}}}{{{\lambda _{sr}}}}}}\left( {\frac{{{\lambda _{sr}}}}{{{\lambda _{sr}} + {\gamma _s}.{\lambda _{pr[j]}}}}} \right) \times {\lambda _{pr[j]}}}}{{\prod \nolimits _{k = 1,k \ne j}^L {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr[j]}}}}} \right) } }}} \end{aligned}$$

(15)

where ${\lambda _{sr}} = \frac{{{P_{S\min }}}}{{{\sigma ^2}}}$, ${\lambda _{pr[i]}} = \frac{{{P_T}}}{{{\sigma ^2}}}$, ${\gamma _s} = {2^{\frac{{2{R_S}}}{{Bw.\left( {1 - \alpha } \right) .L}}}} - 1$ .

Proof

The proof is proven in “Appendix A”.

Similarly, the expression of outage probability at destination node is derived. In decode-and-forward mode, signal from source to relay node is received and decoded to extract information. Then the information is re-encoded and transmitted to the final destination and hence SNR at destination node is given by

$$\begin{aligned} {\gamma _{SD}} = \frac{{{{\mathop {\hbox {P}}\nolimits } _{R\min }}.{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {{P_{PT[i]}}{{\left| {{h_{pd[i]}}} \right| }^2}} + {\sigma ^2}_{SD}}} \end{aligned}$$

(16)

where ${P_{R\min }} = Min\left\langle {{P_{REH}},{P_R}} \right\rangle$ is the real power that RL transmits to SD, $\sigma _{SD}^2$ denotes as the noise at SD. Since RL has no external power supply, its output power totally depends on the received energy ${E_{REH}}$, which contributes to produce ${P_{REH}}$. However, when ${P_{REH}}$ is large enough, it can cause transmitting signal to influence to $P\_Net$ as mentioned above. Thus, real output power at RL is constrained by minimum ${P_{REH}}$ and ${P_{R}}$, which was referred to (7). For simplicity, we assume that noise at RL and SD have Gaussian distribution with zero mean and variance ${\sigma ^2}$.

In principle, harvested energy is described as

$$\begin{aligned} {E_{REH}} = \delta \alpha T\left( {{{\mathop {\hbox {P}}\nolimits } _{S\min }}{{\left| {{h_{sr}}} \right| }^2} + \sum \limits _{i = 1}^L {{P_{PT[i]}}{{\left| {{h_{pr[i]}}} \right| }^2} + \,} {\sigma ^2}} \right) \nonumber \\ \end{aligned}$$

(17)

where $\delta$ is energy conversion efficiency depending on energy harvesting circuitry.

Therefore, the harvested power at RL is given by

$$\begin{aligned} {P_{REH}} = \frac{{2\alpha \delta }}{{1 - \alpha }}\left( {{{\mathop {\hbox {P}}\nolimits } _{S\min }}{{\left| {{h_{sr}}} \right| }^2} + \sum \limits _{i = 1}^L {{P_{PT[i]}}{{\left| {{h_{pr[i]}}} \right| }^2} + \,} {\sigma ^2}} \right) \nonumber \\ \end{aligned}$$

(18)

It is noting that real output power at RL is known as

$$\begin{aligned} {P_{R\min }} = Min\left\langle {{P_{REH}},{P_R}} \right\rangle \end{aligned}$$

(19)

And then expression of outage probability at SD is derived by

$$\begin{aligned} {F_{SD}}({\gamma _{s}})= & {} Pr\left( {{\gamma _{SD}} \le {\gamma _s}} \right) \nonumber \\= & {} \Pr \left( {\frac{{Min\left\langle {{P_{REH}},{P_R}} \right\rangle {{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {{P_{PT[i]}}{{\left| {{h_{pd[i]}}} \right| }^2}} + {\sigma ^2}}} \le {\gamma _s}} \right) \end{aligned}$$

(20)

$\square$

Proposition 2

The cumulative distribution function ${F_{SD}}({\gamma _s})$ is given by

$$\begin{aligned} {F_{SD}}({\gamma _{s}}) = \left( {1 - {\mathfrak {K}}} \right) \times {\mathfrak {M}} + {\mathfrak {K}}\times {\mathfrak {N}} \end{aligned}$$

(21)

where

$$\begin{aligned} {\mathfrak {K}}= & {} \left[ {\prod \limits _{i = 1}^{L + 1} {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \sum \limits _{j = 1}^{L + 1} {\frac{{1 - {e^{ - \;\frac{{\hat{w}}}{{{\lambda _{pr[j]}}}}}}}}{{\frac{1}{{{\lambda _{pr[j]}}}}\prod \nolimits _{k = 1,k \ne j}^{L + 1} {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr[j]}}}}} \right) } }}}\nonumber \\ \end{aligned}$$

(22)

$$\begin{aligned} {\mathfrak {M}}= & {} 1 - \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _{pd[i]}}}}} } \right] \sum \limits _{j = 1}^L {\frac{{{e^{ - \frac{{{\gamma _s}}}{{{\lambda _{nomi}}}}}}\left[ {\frac{{{\lambda _{nomi}}}}{{{\lambda _{pd[j]}}.{\gamma _s} + {\lambda _{nomi}}}}} \right] }}{{\frac{1}{{{\lambda _{pd[j]}}}}\prod \nolimits _{k = 1,k \ne j}^L {\left( {\frac{1}{{{\lambda _{pd[k]}}}} - \frac{1}{{{\lambda _{pd[j]}}}}} \right) } }}} \nonumber \\\end{aligned}$$

(23)

$$\begin{aligned} {\mathfrak {N}}= & {} \left[ {\prod \limits _{i = 1}^{L + 1} {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _{pd[i]}}}}} } \right] \frac{1}{{{\lambda _{rdn}}}}\nonumber \\&\times \sum \limits _{m = 1}^{L + 1} {\frac{{{e^{ + \;\frac{1}{{{\lambda _{pr[m]}}}}}}}}{{\prod \nolimits _{k = 1,k \ne m}^{L + 1} {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr}}_{[m]}}}} \right) } }}} \nonumber \\&\times \sum \limits _{n = 1}^L {\frac{{I\left( {m,n} \right) }}{{\prod \nolimits _{k = 1,k \ne n}^L {\left( {\frac{1}{{{\lambda _{pd[k]}}}} - \frac{1}{{{\lambda _{pd[n]}}}}} \right) } }}} \end{aligned}$$

(24)

with $\hat{w} = \frac{{{P_R}\left( {1 - \alpha } \right) }}{{2\alpha \delta {\sigma ^2}}} - 1$, ${\lambda _{rd}} = \frac{{{P_R}}}{{{\sigma ^2}}}$, ${\lambda _{pd[i]}} = \frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}$, ${\lambda _{^{nomi}}} = \frac{{{P_R}}}{{{\sigma ^2}}} P{L_{rd}}$, in which $P{L_{rd}}$ is path-loss factor from RL to SD, and $I\left( {m,n} \right) = \Gamma \left( 2 \right) {e^{ + \;\frac{1}{{{\lambda _{pd[n]}}}}}}{\lambda _{rdn}}{\lambda _{pr[m]}}{\lambda _{pd[n]}}{e^{\frac{{{\gamma _s}{\lambda _{pd[n]}}}}{{2{\lambda _{rdn}}{\lambda _{pr[m]}}}}}}{W_{ - 1,\; - \frac{1}{2}}}\left( {\frac{{{\gamma _s}{\lambda _{pd[n]}}}}{{{\lambda _{rdn}}{\lambda _{pr[m]}}}}} \right)$ where $\Gamma \left( . \right)$ is the Gamma function, $W_{.,\,.} \left( . \right)$ is the Whittaker function [24].

Proof

The proof is given in “Appendix B”.

Next, outage probability of overall system can be calculated based on the previous results. To transmit from source to destination, information must go through two hops. Then

$$\begin{aligned} {P_{out}}\left( {{\gamma _s}} \right) = \Pr \left( {Min\left\langle {{\gamma _{RL}},{\gamma _{SD}}} \right\rangle } \right) \le {\gamma _s} \end{aligned}$$

(25)

where ${\gamma _s} = {2^{\frac{{2.{R_S}}}{{\left( {1 - \alpha } \right) .Bw.L}}}} - 1$ under TSR protocol. In fact, probability to transmit successfully information from ST to SD depends on the ability of successful transmission on each hop: ST $\rightarrow$ RL, RL $\rightarrow$ SD.

Applying complementary probability theory, probability to transmit successfully information from source to relay is $1 - {F_{RL}}\left( {{\gamma _s}} \right)$. Similar argument is applied for second hop. The successful probability is $\left[ {1 - {F_{RL}}\left( {{\gamma _s}} \right) } \right] \times \left[ {1 - {F_{SD}}\left( {{\gamma _s}} \right) } \right]$ . Otherwise, outage probability is shown in following expression. The outage probability of the considered model is found as

$$\begin{aligned} {P_{out}} = {F_{RL}}\left( {{\gamma _s}} \right) + {F_{SD}}\left( {{\gamma _s}} \right) - {F_{RL}}\left( {{\gamma _s}} \right) \times {F_{SD}}\left( {{\gamma _s}} \right) \end{aligned}$$

(26)

where ${F_{RL}}$, ${F_{SD}}$ calculated as in (15) and (21), respectively. $\square$

3.2 Throughput performance in delay-limited transmission mode

As assumed above, all of channels in model are slowly varying fading channels. This means that parameters of a channel are virtually changeless during a block time. Outage capacity is defined as the maximum rate of transferred information, which is possible to maintain when being transmitted through channels under slow fading and given outage probability threshold. The outage capacity unit were standardized according to bandwidth is bit/s/Hz. It is affected by the harvested energy process at RL and followed TSR protocol in [4]. The throughput in delay-limited transmission mode can be written as

$$\begin{aligned} {T_S} = \frac{{\left( {1 - \alpha } \right) }}{2}{R_S}\left[ {1 - {P_{out}}\left( {{\gamma _s}} \right) } \right] \end{aligned}$$

(27)

In this context, throughput in (27) shows the relationship between throughput $S\_Net$ and $P_{out}$.

3.3 Rate and energy trade-off

In this section, we investigate the relationship between energy harvesting at RL and ergodic capacity of $S\_Net$. Based on [13], the average harvested energy at RL is shown as

$$\begin{aligned} {E_{RL}} = E\left\{ {\alpha \delta {E_{REH}}} \right\} \end{aligned}$$

(28)

The ergodic capacity is given by

$$\begin{aligned} {R} = E\left\{ {\frac{1}{2}\left( {1 - \alpha } \right) {{\log }_2}\left( {1 + {\gamma _{SD}}} \right) + {\frac{1}{2}{\log }_2}\left( {1 + {\gamma _{RL}}} \right) } \right\} \nonumber \\ \end{aligned}$$

(29)

The characteristic of rate energy (R-E) trade-off shows the relationship between ergodic capacity and the average harvested energy, is presented as below

$$\begin{aligned} {C_{R - E}}= & {} \left\{ \left( {R,{E_{RL}}} \right) :{E_{REH}} \geqslant {E_{Opt}},R \right. \nonumber \\\leqslant & {} E\left\{ \frac{1}{2}\left( {1 - \alpha } \right) {{\log }_2}\left( {1 + {\gamma _{SD}}} \right) \left. +{\frac{1}{2}{\log }_2}\left( {1 + {\gamma _{RL}}} \right) \right\} \right\} \nonumber \\ \end{aligned}$$

(30)

4 Numerical results and discussion

For numerical results, the allocation of transceivers in $P\_Net$ is provided as below: the first transmitter is placed at (1, 1), the next ones are located at points with progressive difference from the previous one 0.25, according to both vertical and horizontal coordinates. Similarity, first receiver is located at (1, 2) and progressive difference is used as above. ST transmitter in $S\_Net$ at the origin (0, 0), relay at (0, 0.5) and destination receiver at (0, 1). This arrangement ensures that all of transmitters in $P\_Net$ have completely differential communication distance when they influence to receivers in $S\_Net$. In addition, ${R_P}=1.5$ bits/s/Hz, transmitting power ${P_T}=20$ dB, bandwidth $Bw = 1$ Hz for both networks, energy conversion factor $\delta = 0.5$, ${R_S}=0.2$ bits/s/Hz, employed path loss model as $P{L_{AB}} = d_{AB}^{ - \rho }$ in condition of suburban area with $\rho = 4$, $d_{AB}$ is the distance from A to B.

Fig. 3

Effect of $\alpha$ and L on outage probability

4.1 Impact of energy harvesting on outage and throughput performance

Figure 3 shows a plot of the outage probability versus time switching fraction in EH protocol. It can be observed that as considering fraction of block time $\alpha$ in the range of zero to 1, $P_{out}$ is decreased rapidly when $\alpha$ begins increasing from zero value. In addition, Fig. 3 also shows that in case of the number of transceivers L = 3, outage probability is lower than the case that L = 7. Clearly, more transceivers would reduce SNR in $S\_Net$ and contribute to raise outage probability. However, this difference is very small.

Fig. 4

Effect of threshold $\varPsi$ on outage probability

Fig. 5

Effect of $\alpha$ and $\varPsi$ on throughput

Figures 4 and 5 evaluate the performance of outage and throughput at secondary network. Regarding outage performance in Fig. 4, due to $\alpha$ is too small, the energy obtained from the radio waves is insignificant and insufficient to supply energy to transmit on second hop within the reach of low outage probability. It is noting that $\alpha$ is increased means that more energy is supplied to RL. Energy harvesting occurs speedily because there are so many signals coming to RL that leads to the outage probability on second hop decreased very rapidly. Furthermore, because energy harvesting happens quickly and hence there is less time to reach the limited-energy, which produce power constrained in (7). As observation, harvested energy accumulates to the limited-energy, which is equivalents with the fraction of block time has optimal value, called $\alpha _{opt}$. After that, although $\alpha$ continues increasing, $P_{out}$ decreases. the reason is that larger number of $\alpha$ results in shorten time for communication in TSR protocol. Moreover, RL is being subjected to power constraint that could not support to increase rate. It can be seen that in case of $\alpha =1$, there is no longer time to transfer information and the result is $P_{out}$ almost equal 1, meaning that it’s impossible to communicate in $S\_Net$. Whereas, Fig. 5 shows that throughput is also approximately zero at the beginning of considered range of $\alpha$. However, harvested energy increases rapidly after changing $\alpha$. When $\alpha$ is larger $\alpha _{opt}$, throughput starts decreasing due to reducing communication time. Especially, $\alpha$ approximately equal 1, $T_S$ approaches to zero.

Fig. 6

Rate energy trade-off, $P_{T} = 20$ dB and $\delta =0.5$ and 1

Figure 6 illustrates rate and energy trade-off. Evidently, there exist trade-offs in assigning the energy harvesting time ratio $\alpha$ to balance between maximizing the average harvested energy for power transfer and maximizing the ergodic capacity for information transfer. As a result, the Rate-Energy (R-E) region is used to characterize such trade-off, which includes all the achievable ergodic capacity and average harvested energy pairs. Moreover, the impact of the energy conversion efficiency $\delta$ on the R-E trade-off is also illustrated in this figure. The figure reveals that the harvested energy is more sensitive to $\delta$ than ergodic capacity. This is due to the fact that harvested energy is linear with $\delta$.

Fig. 7

The outage performance of $S\_Net$ versus $d_1$

Fig. 8

Effect of relocation in outage performance of $S\_Net$

4.2 Effect of transceiver allocation

According to mathematical results as above, transceivers are not constrained by place inside both networks. We investigate the effect of transceiver arrangement on both.

Figure 7 shows the relationship between d1 distance and outage probability in $S\_Net$, in which d1 is space between ST and RL. In the range of coverage distance from 0.1 to 0.9, the optimal outage capacity is reached at $d1 = 0.55$. In addition, this figure also shows that the higher $\varPsi$, the smaller outage probability in $S\_Net$.

We observe trend of the outage probability in terms of relocation primary transceivers in Fig. 8. To make clear study, we compare two typologies, in which secondary network is unchanged: (a) line arrangement as mentioned above with L = 4, (b) rhumboid arrangement: first transceiver at (0, 1.5) and (1.5, 1.5), second at (0.25, 1.5) and (1.75, 1.5), third at (0.25, 1.75) and (1.75, 1.75), forth at (0.5, 1.75) and (2, 1.75). Figure 8 show the outage probability change in terms of relocation primary transceivers. Result also indicates that if channel was approximated by gamma distribution, we would not have achieved success.

5 Conclusion

In this paper, we investigate energy harvesting in cognitive network, which consists of two networks sharing frequency spectrum together: $P\_Net$ includes multiple independent communication links, which all of transmitters arranged with any location, while $S\_Net$ consists of one transmitter node, one receiver node and one relay node, which employs energy harvesting and forwards information concurrently. Our research derives closed-form of secondary network outage probability expression. By using TSR protocol, the paper shows the optimal fraction of block time to achieve better outage performance. In addition, we investigate the effect of random location primary transceivers and relay node on $S\_Net$’s outage probability as well as analyse the influence of the threshold on $P\_Net$ and the number of transceivers on secondary network outage probability. We conclude that outage performance is slightly influenced by the number of transceivers in $P\_Net$. It is also confirmed that harvested energy and rate must not go along together at all.

Appendices

Appendix A: Proof of Proposition 1

Firstly, we can rewrite (9) as below expression

$$\begin{aligned} {\gamma _{SR}} = \frac{X}{{Y + 1}} \end{aligned}$$

(31)

where $X = \frac{{{P_{S\min }}}}{{{\sigma ^2}}}{\left| {{h_{sr}}} \right| ^2}$, $Y = \sum _{i = 1}^L {\frac{{{P_T}}}{{{\sigma ^2}}}} {\left| {{h_{pr[i]}}} \right| ^2}$. X is the exponential distributed random variable (ERV) with mean ${\lambda _{sr}} = \frac{{{P_{S\min }}}}{{{\sigma ^2}}}P{L_{sr}}$, Y is the sum of ERVs, in which each part has mean ${\lambda _{pr[i]}} = \frac{{{P_T}}}{{{\sigma ^2}}}P{L_{pr[i]}}$,respectively. By applying Lemma 1, the pdf(Y) and cdf(Y) are proven. Probability of (31) in (10) is solved by separating two cases:

$$\begin{aligned} \Pr \left( {\frac{X}{{Y + 1}} \le \gamma _s } \right)= & {} \left\{ {\begin{array}{*{20}l} {\Pr \left( {Y \ge \frac{{X - \gamma _s }}{{\gamma _s }}} \right) , \qquad X \ge \gamma _s } \\ {\Pr \left( {Y \ge \frac{{X - \gamma _s }}{{\gamma _s }}} \right) = 1 ,X< \gamma _s } \\ \end{array}} \right. \quad \nonumber \\= & {} 1 - \int \limits _{\gamma _s }^\infty {f_X \left( x \right) } \Pr \left( {Y < \frac{{X - \gamma _s }}{{\gamma _s }}} \right) dx\nonumber \\ \end{aligned}$$

(32)

and vice versa.

After reducing (32), we obtain the express as in (15).

Appendix B: Proof of Propostion 2

From (20) we can write:

$$\begin{aligned}&{F_{SD}}({\gamma _{SD}}) \nonumber \\&\quad = \underbrace{\Pr \left( {{P_{REH}} > {P_R}} \right) }_{1 - {\mathfrak {K}}}\underbrace{\Pr \left( {\frac{{\frac{{{P_R}}}{{{\sigma ^2}}}{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} + 1}} \leqslant {\gamma _s}} \right) }_{\mathfrak {M}}\nonumber \\&\qquad + \underbrace{\Pr \left( {{P_{REH}} \leqslant {P_R}} \right) }_{\mathfrak {K}}\underbrace{\Pr \left( {\frac{{\frac{{{P_{REH}}}}{{{\sigma ^2}}}{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} + 1}} \leqslant {\gamma _s}} \right) }_{\mathfrak {N}}\nonumber \\ \end{aligned}$$

(33)

We need to solve three subcases: (22), (23) and (24). Firstly, substituting the value of ${P_{REH}}$ in (18) to $\mathfrak {K}$ form in (33), we receive

$$\begin{aligned} {\mathfrak {K}}&= \Pr \left( {\frac{{2\alpha \delta {\sigma ^2}}}{{1 - \alpha }}\left( {{X_1} + {X_2} +\cdots + {X_L} + {X_{L + 1}} + 1} \right) \leqslant {P_R}} \right) \end{aligned}$$

(34)

$$\begin{aligned}&= \Pr \left( {{X_1} + {X_2} + \cdots + {X_L} + {X_{L + 1}} \leqslant \hat{w}} \right) \end{aligned}$$

(35)

where ${X_i}$: $i = 1 \div L$: $f\left( {{X_i}} \right) = \frac{1}{{{\lambda _{pr[i]}}}}{e^{ - \;\frac{x}{{{\lambda _{pr[i]}}}}}}$, ${X_{L + 1}}$: $f\left( {{X_{L + 1}}} \right) = \frac{1}{{{\lambda _{sr}}}}{e^{ - \;\frac{x}{{{\lambda _{sr}}}}}}$, $\hat{w} = \frac{{{P_R}\left( {1 - \alpha } \right) }}{{2\alpha \delta {\sigma ^2}}} - 1$. Applying Lemma 1, we obtain (22).

Secondly, $\mathfrak {M}$ form in (33) is given by

$$\begin{aligned} \mathfrak {M} = \Pr \left( {\frac{U}{{V + 1}} \leqslant {\gamma _s}} \right) \end{aligned}$$

(36)

where

$$\begin{aligned} V = \sum \limits _{i = 1}^L {{V_i}} = \sum \limits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} \end{aligned}$$

(37)

is the sum of ERVs, $U = \frac{{{P_R}}}{{{\sigma ^2}}}{\left| {{h_{rd}}} \right| ^2}$ is ERV. (36) and (31) have similar in form. Thus, we use the same way to obtain (23).

Finally, $\mathfrak {N}$ form in (33) is expressed as

$$\begin{aligned} \mathfrak {N} = \Pr \left( {\frac{{\frac{{{P_{REH}}}}{{{\sigma ^2}}}{{\left| {{h_{rd}}} \right| }^2}}}{{\sum \nolimits _{i = 1}^L {\frac{{{P_T}}}{{{\sigma ^2}}}{{\left| {{h_{pd[i]}}} \right| }^2}} + 1}} \leqslant {\gamma _s}} \right) \end{aligned}$$

(38)

Let us denote $T = \frac{{2\alpha \delta }}{{\left( {1 - \alpha } \right) }}{\left| {{h_{rd}}} \right| ^2}$, $Z = \left( \frac{{{{\text {P}} _{S\min }}}}{{{\sigma ^2}}}{{\left| {{h_{sr}}} \right| }^2}\right. \left. + \sum \nolimits _{i = 1}^L {\frac{{{P_{PT[i]}}}}{{{\sigma ^2}}}{{\left| {{h_{pr[i]}}} \right| }^2} + \,} 1 \right)$,${P_{REH}}$ in (18), V in (37) can be substituted, we obtain

$$\begin{aligned} \mathfrak {N} = \Pr \left( {\frac{{TZ}}{{V + 1}} \leqslant {\gamma _s}} \right) \end{aligned}$$

(39)

Applying Rohatgi’s well-known results in [25] and formula Eq. 3.471.9 in [24], we have PDF of ($U=TZ$)

$$\begin{aligned} {f_U}(u)= & {} \left[ {\prod \limits _{i = 1}^{L + 1} {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \frac{1}{{{\lambda _{rdn}}}}\sum \limits _{j = 1}^{L + 1} {\frac{{2{e^{ + \;\frac{1}{{{\lambda _{pr[j]}}}}}}.{K_0}\left( {2\sqrt{\frac{u}{{{\lambda _{rdn}}.{\lambda _{pr[j]}}}}} } \right) }}{{\prod \nolimits _{k = 1,k \ne j}^{L + 1} {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr}}_{[j]}}}} \right) } }}}\nonumber \\ \end{aligned}$$

(40)

where ${\lambda _{^{rdn}}} = P{L_{rd}} \times \frac{{2\alpha \delta }}{{\left( {1 - \alpha } \right) }}$ and pdf(V) is similarly proved in Lemma 1. We can write $\mathfrak {N}$ as:

$$\begin{aligned} \mathfrak {N}= & {} \left[ {\prod \limits _{i = 1}^{L + 1} {\frac{1}{{{\lambda _{pr[i]}}}}} } \right] \left[ {\prod \limits _{i = 1}^L {\frac{1}{{{\lambda _{pd[i]}}}}} } \right] \frac{1}{{{\lambda _{rdn}}}}\nonumber \\&\times \sum \limits _{m = 1}^{L + 1} {\frac{{{e^{ + \;\frac{1}{{{\lambda _{pr[m]}}}}}}}}{{\prod \nolimits _{k = 1,k \ne m}^{L + 1} {\left( {\frac{1}{{{\lambda _{pr[k]}}}} - \frac{1}{{{\lambda _{pr}}_{[m]}}}} \right) } }}} \nonumber \\&\times \sum \limits _{n = 1}^L {\frac{{\overbrace{\int _0^\infty {\overbrace{\int _0^{{\gamma _s}v} {2{K_0}\left( {2\sqrt{\frac{u}{{{\lambda _{rdn}}{\lambda _{pr[m]}}}}} } \right) } }^{{I_1}}} \,{e^{ - \;\frac{{v - 1}}{{{\lambda _{pd[n]}}}}}}}^{{I_2}}.dudv}}{{\prod \nolimits _{k = 1,k \ne n}^L {\left( {\frac{1}{{{\lambda _{pd[k]}}}} - \frac{1}{{{\lambda _{pd[n]}}}}} \right) } }}}\nonumber \\ \end{aligned}$$

(41)

in which ${I_1}$ is calculated when using Eq. 5.522 in [24] where $p = -1$ and ${K_{ - v}} = {K_v}$.

Next, ${I_2}$ is solved when conducted from Eq. 6.643.6 in [24], where $m = 1$ and $\alpha = \frac{{{\lambda _{rdn}}{\lambda _{pr[m]}}}}{{{\gamma _s}{\lambda _{pd[n]}}}}$. After solving (22), (23), (24), Eq. (21) is clearly derived.