# Performance Analysis of RF-Powered Cognitive Radio Networks with Integrated Ambient Backscatter Communications

Authors: Longteng Xu; Kun Zhu; Ran Wang; Shimin Gong
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

Integrating ambient backscatter communications into RF-powered cognitive radio networks has been shown to be a promising method for achieving energy and spectrum efficient communications, which is very attractive for low-power or no-power communications. In such scenarios, a secondary user (SU) can operate in either transmission mode or backscatter mode. Specifically, an SU can directly transmit data if sufficient energy has been harvested (i.e., transmission mode). Or an SU can backscatter ambient signals to transmit data (i.e., backscatter mode). In this paper, we investigate the performance of such systems. Specifically, channel inversion power control and an energy store-and-reuse mechanism for secondary users are adopted for efficient use of harvested energy. We apply stochastic geometry to analyze coverage probability and achievable rates for both primary and secondary users considering both communication modes. Analytical tractable expressions are obtained. Extensive simulations are performed and the numerical results show the validity of our analysis. Furthermore, the results indicate that the performance of secondary systems can be improved with the integration of both communication modes with only limited impact on the performance of primary systems.


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## Body

## 1. Introduction

In recent years, the demand for smart systems (e.g., on-body sensing for e-Healthy) is growing fast. For such systems, the deployed sensors usually need to work continuously and transmit collected data for upper layer applications. Since most sensors have limited battery and limited spectrum resources, energy-efficient and spectrum efficient wireless communications are required.Several techniques have been developed for achieving low-power or even no-power communications in a spectrum efficient manner, among which ambient backscatter communications and radio-frequency (RF) powered cognitive communications are two remarkable ones. In [1], authors investigate practical backscatter for on-body sensors by using the signals from Wi-Fi or Bluetooth. Such a backscatter system based on Wi-Fi is referred to as Wi-Fi backscatter [2]. Be different from RF identification (RFID) which needs a dedicated signal emitter (RFID reader) [3], Wi-Fi backscatter does not need dedicated reader. However, it does not perform well in outdoor environment. While ambient backscatter communication [4] is also a type of passive communication which utilizes ambient RF signals (e.g., TV signals) to transmit data, no dedicated signal source is required, which makes no-power wireless communications possible. However, the communications are vulnerable since backscattered signals are usually weak and volatile.Another technique, harvesting energy from ambient RF signals, has been proposed to support energy-efficient communications [5–7]. Besides, RF-powered cognitive radio network (CRN) offers a method to utilize primary transmitter (PTs) signals as the energy source for secondary transmitters (STs). A main problem is that the transmission opportunities of RF-powered STs are limited by the harvested energy and channel availability.In this work, we consider the integration of ambient backscatter communications into RF-powered cognitive radio networks in a similar way to [8]. In this case, these two techniques could complement each other to jointly achieve the advantages while overcoming the individual shortcomings. Specifically, in such scenarios, the secondary transmitters can operate in two modes, that is, transmission mode and backscatter mode. In transmission mode, an ST can directly transmit data to its receiver if sufficient energy has been harvested and the channel is available. When the channel is busy, an ST can switch to backscatter mode which backscatters the ambient signals to transmit its own data.In other words, in our cognitive radio network, the transmission of secondary transmitters falls into interweave paradigm to utilize white spaces of specific channels [9], and the needed energy is harvested from existing signals. As for backscatter communication, in some extent it falls into overlay paradigm since the backscattered signals from secondary transmitters to primary receivers can be ignored which will be stated in more detail in the following [9–11]. As for the primary user activity model, which has a vital influence on cognitive users’ performance, that is, directly determining spectrum access time of cognitive users, in this paper follows a simplified ON/OFF model [12] wherein total active (ON) duration and inactive (OFF) duration of a primary user are fixed. Besides, the details of spectrum sensing are omitted in this paper. Other complex and widely used primary user activity models and spectrum analysis can be found in [12–15].Note that several existing works have been done for the integration of backscatter communications with cognitive radio networks or cellular networks. In [16], authors give an overview of backscatter assisted wireless powered communications and introduce a multiple access scheme in cognitive radio networks. The tradeoff analysis in RF-powered backscatter CRNs is provided in [17]. However, only one single cell is considered and no performance analysis is provided. In [18], the integration of backscatter communications with heterogeneous cellular networks is proposed and analyzed. In [19], a backscatter network is analyzed by using stochastic geometry, but dedicated power beacons are deployed to support the communication, while, in [20, 21], a single hybrid transmitter harvests energy from multiple ambient transmitters, transmits its own signal, or backscatters existing signals to a hybrid receiver, and its performance is analyzed.In this paper, we investigate the performance of ambient backscatter communications in RF-powered cognitive radio network. Specifically, we propose an analytical framework based on stochastic geometry [22–24], with which the tractable expressions for coverage probability and achievable rates for both primary and secondary users considering both communication modes are obtained. We perform extensive simulations and the numerical results demonstrate the validity of the theoretical analysis. Also, the results indicate that secondary systems can achieve improved performance while having only limited impact on the primary systems, which show the effectiveness of integration.The rest of the paper is organized as follows: Section2 presents the comprehensive system model. Section 3 presents channel inversion power control and energy storage and reusing. Analytical expressions are given in Section 4. In Section 5, numerical results from analysis and simulations are described. Finally, Section 6 draws the conclusions.


## 2. System Model

### 2.1. Network Model

We consider a cognitive cellular network in which macro base stations (MBSs, i.e., PTs,Y) serve primary cellular users (PRs, U) in the downlink while overlaid by cognitive secondary users. Each PR will connect to the nearest MBS. Besides, there are secondary transmitters (STs, X) equipped with energy storage and secondary receivers (SRs, Z). An ST can communicate with an SR by either backscattering signals or emitting its own signals. Since backscattered signal is weak, the distance from backscattering node (i.e., ST) to receiving node (i.e., SR) is limited. It is shown in [4] that, for achieving 1 kbps information rate in outdoor environment, 2.5 feet is the maximum distance. Therefore, in this paper, for ease of analysis, we assume that an SR is at a constant small distance d to its associated ST in an isotropic direction [19].MBSs and PRs are modeled by homogeneous Poisson point processes (HPPP)Φ and Φ~U with intensities λ and λ~U, respectively. STs are uniformly distributed in annular regions with radii Rm and RM centered at each MBS. Each ST can be loosely seen as a result of random and independent displacement of the MBS. In each annular region, the number of STs is NST~π(Λ), where π(Λ) is the Poisson distribution with parameter Λ. The distribution model of STs then is similar to a Matern cluster process where Φ is the parent process [25]. An inner radius Rm>0 of the annular region is considered for avoiding singularity of integral in the derivation process and the outer radius RM is related tocircuit power constraint described in the following part.We assume each PR is associated with its nearest MBS. The probability density function (pdf) of distancer from a PR to its nearest MBS is [23](1)fr=2πλre-πλr2.If a point is uniformly distributed within a circle with radius R and r is the distance to the center, the pdf of r is fo(r)=2r/R2 [26]. By using conditional probability, we can get the pdf of distance r from an ST to its MBS as(2)FR∣r≥Rm=PRm≤r≤RPr≥Rm=R2-Rm2RM2-Rm2,f≥Rmr≜fr∣r≥Rm=dFR∣r≥Rm2dr=2rRM2-Rm2.Without loss of generality, according to Slivnyak’s theorem [27], we analyze a typical PR (U0) and a typical SR (Z0) located at the origin. The typical SR’s corresponding ST and MBS are also similarly typical ST (X0) and typical MBS (Y0), respectively. Figure 1 illustrates randomly generated positions of MBSs, PRs, and STs. Figure 2 shows the system model.Figure 1
Network composition illustration. Centers of dual-circles (annular regions), red dots, and blue triangles represent MBSs, primary receivers, and secondary transmitters, respectively. To ensure visibility, only one ST is showed in an annular region and all SRs are omitted. (Radii of annular regions in the illustration are enlarged.)Figure 2
System model of our network.h, g, q refer to three types of fading described in Section 2.2.


### 2.2. Channel Model

All MBSs share the same available channel setC=c1,c2,…,cC, where C=Nch is the number of channels. We assume λ~U≫λ and there is one and only one active PR in each channel of a cell, so active PRs in a generic channel form an HPPP ΦU with intensity λU=λ by independent thinning.We also assume that STs in each cell have equal probabilities to access a channel when it is idle, and no two STs share the same channel. However, STs in different cells may not have equal probabilities since numbers of STs in different cells may differ from each other. This means the thinned processes are not HPPPs. But for ease of analysis, in this paper, similar to the assumptions in [6, 28–30], we assume STs in different cells in a generic channel constitute an HPPP by random displacement and thinning.Besides, if there areNST STs in one annular region, the probability that a channel in C is used by an ST is NST/Nch, while if NST>Nch, the probability is 1. Since NST~π(Λ), the probability that a channel is used averaged over NST is(3)pch=∑k=0NchPNST=kkNch+∑k=Nch+1∞PNST=k·1=1+Λ-NchΓNch,Λ-e-ΛΛNchΓ1+Nch,where Γ(z)=∫0∞tz-1e-tdt is the Gamma function and Γ(z,a)=∫a∞tz-1e-tdt is the upper incomplete Gamma function. So STs in a generic channel form an HPPP ΦST with intensity λST=pchλ by independent thinning and random displacement. And pch is termedchannel in use probability.An extreme case is thatΛ is high enough and pch=1 holds. This equals the setting that only one channel is considered in the network and channels of all cells are used. In this case, besides pch, other details of the network remain unchanged, so does the analysis.We assume that each channel experiences a constant noise powerW and exponential path-loss l(r)=r-α (or l(X-Y)=X-Y-α) with a uniform exponent α>2, where r is a distance, X,Y∈R are two points, and · is the Euclidean norm. Independent Rayleigh fading is considered which remains constant within one time slot. The fading from a PR/ST to its corresponding MBS is h~exp⁡(μh), fading from an ST to its SR is q~exp⁡(μq), and interference fading from an MBS/PR/ST to a PR/SR is g~exp⁡(μg). Moreover, for convenience we set a time slot duration to be a unit time.


### 2.3. Communication Model

When an MBS serves a PR, it may turn into busy or idle mode during each time slot. We further divide each time slot intoM minislots [19] and all minislots are synchronized among MBSs. In addition, we assume that each MBS randomly and independently turns into idle mode in one of the minislots. With such assumptions, when the MBS is transmitting, we consider that an ST performs energy harvesting (EH) and ambient backscatter communication (BC) in DE and DB minislots, respectively, and we have DE+DB+1=M. Minislots for EH and BC of each ST are also randomly and independently selected with equal probability. Besides, the ST performs traditional information transmission (IT) using the harvested energy when the MBS is idle. In the following, these two communication modes are termed BC mode and IT mode, respectively. An example of minislots assignment and selection when M=7 is given in Figure 3.Figure 3
An example of minislots assignment and selection for different mode when number of minislots is 7.In a generic minislot, STs performing energy harvesting form an HPPPΦeh with intensity λeh=(DE/M)λST, which can be seen as the thinning of ΦST since each ST may work in EH mode within a minislot with probability DE/M. Similarly, STs in BC mode and IT mode form HPPPs Φbc and Φ~it with intensities λbc=(DB/M)λST and λ~it=λST/M, respectively. Moreover, MBSs in busy mode also form an HPPP Φbs with intensity λbs=(1-1/M)λ.We further assume that the SR knows about its ST’s work mode so as to perform corresponding decoding. Saturation condition is also assumed where a data packet is always ready for transmission.Besides, we assume that when an ST performs BC, a portionβ of the received power is used for BC, and the backscatter efficiency is ηb. So the backscatter power of a generic ST is(4)Pbc=βηbX-Y-αh,where X-Y is the distance from the MBS to the ST, while the remaining portion 1-β is stored as energy [19] along with received power in EH mode. And the conversion efficiency is η. In this case, the harvested energy within a time slot is(5)EH=ηDEM+1-βDBMr-αh=DebhMrα,where Deb=η(DE+(1-β)DB).In order to perform IT, an ST must satisfy the circuit power constraint thatEH is more than that consumed EC by the circuit, while the remaining power EH-EC is used for data transmission. Since a time slot lasts a unit time, we will also describe EH and EC as the power. The average harvested energy can be obtained as(6)EhEH=∫0∞μhe-μhhDebhMrαdh=DebμhMrα,and the limitation of distance r from an ST to an MBS, that is, RM, can be derived as(7)EhEH-EC>0⟺r<RM≜DebμhMEC1/α,where RM is based on the setting that STs distributed in the annular region should satisfy the circuit power constraint. When fading is considered, the probability that harvested energy EH is above demand is(8)Gx≜PEH>x=PDebhMrα>x=∫RmRM∫Mrαx/Deb∞f≥Rmr·fhdhdr=2Hx,Rm-Hx,RMαRM2-Rm2,where x is a parameter representing energy demand, r is the distance from an ST to its MBS, f≥Rm(r) is the pdf given in (2), h is the fading, and(9)Hx,y=y2Eα-2α,Jxyα,Jx=xμhMDeb,En,z=∫1∞e-ztt-ndtexponentialintegral,where α, μh, M, and Deb have been presented. Therefore, the constraint satisfaction probability psat of an ST is(10)psat≜PEH-EC>0=GEC.In the following, psat will also be termedpower satisfaction probability. Taking this probability into account, STs in IT mode that have enough energy form an HPPP Φit with intensity λit=psatλ~it=psatλST/M. In the following, these STs are calledworking STs.When performing IT, the available energy isEH-EC, but for ease of analysis, in this paper we adopt the setting that a portion ξ of the harvested energy EH can be used for active information transmission [31] and energy not used is ignored in different time slots. So the transmit power of a generic ST is(11)Pit=ξEH1/M=ξDebX-Y-αh,where X-Y is the distance from the MBS to the ST.


### 2.4. Interference Model

In [32], authors mentionedinterference regeneration that a backscatter node reflects all incident signals which leads to a square number of interference components for each SR. One effective solution is to adopt spread spectrum techniques. In this paper, we simplify this problem and assume that an ST backscatters only the signal from its corresponding MBS.


#### 2.4.1. Backscattered Signal to an SR in IT Mode

Backscattered signal from an ST to an SR in IT mode is considered as an extra path. We assume an SR in IT mode has the ability to eliminate multipath effect like PRs [4], so such interference is ignored.


#### 2.4.2. Transmitted Signal to an SR in BC Mode

Since an SR in BC mode senses and decodes backscattered signal by sensing changes in the signal caused by backscattering [4], we also ignore such interference.


#### 2.4.3. Backscattered Interference to an SR in BC Mode

Such interference comes from other STs (Xi) in BC mode whose corresponding MBSs (Yi) are in busy mode. Interfering STs come from Φbc∖{X0}. Such interference is firstly transmitted from an MBS to its ST, encountering path-loss l(Xi-Yi) and Rayleigh fading h~exp⁡(μh), then it is backscattered from an ST to the typical SR, encountering path-loss l(Xi-Z0) and Rayleigh fading g~exp⁡(μg). The interference is denoted as(12)Ibc=∑Xi∈Φbc∖X0PbciXi-Z0-αgi,where Pbci is the backscatter power of an ST in BC mode given in (4).


#### 2.4.4. Transmitted Interference to an SR in IT Mode

Such interference comes from other MBSs in busy mode and other STs in IT mode whose corresponding MBSs are in idle mode. Interfering MBSs come fromΦbs∖{Y0}. Such interference is transmitted from an MBS to the typical SR, encountering path-loss l(Yi-Z0) and Rayleigh fading g~exp⁡(μg). Interfering STs come from Φit\{X0}. Such interference is firstly transmitted from an MBS to its ST, encountering path-loss l(Xi-Yi) and Rayleigh fading h~exp⁡(μh), then it is backscattered from the ST to the SR, encountering path-loss l(Xi-Z0) and Rayleigh fading g~exp⁡(μg). The interference is denoted as(13)Isum=Iit+Ibs=∑Xi∈Φit∖X0PitiXi-Z0-αgi+∑Yi∈Φbs∖Y0Yi-Z0-αgi,where Iit, Ibs represent two interference components, respectively, and Piti is the transmit power of an ST in IT mode given in (11).


#### 2.4.5. Transmitted Interference to a PR

All interference to a PR is identical to the interference to an SR in IT mode. We will analyze performance of a PR located at the origin encountering no interference from STs to make a comparison to reveal the effect on PRs.


### 2.5. Rate Model for Backscatter Link

In [4], the bit rate (alternating sequence of ones and zeros) of the ambient backscatter prototype is related to the setting of circuit elements. Similar settings are also used in [8, 17, 18] and will be used in this paper, too. Besides, we assume if the signal-to-interference-plus-noise-ratio (SINR) of backscatter communication is above a threshold, the predesigned rate can be achieved [20].Notations used in this paper are listed in the Notations.


## 2.1. Network Model

We consider a cognitive cellular network in which macro base stations (MBSs, i.e., PTs,Y) serve primary cellular users (PRs, U) in the downlink while overlaid by cognitive secondary users. Each PR will connect to the nearest MBS. Besides, there are secondary transmitters (STs, X) equipped with energy storage and secondary receivers (SRs, Z). An ST can communicate with an SR by either backscattering signals or emitting its own signals. Since backscattered signal is weak, the distance from backscattering node (i.e., ST) to receiving node (i.e., SR) is limited. It is shown in [4] that, for achieving 1 kbps information rate in outdoor environment, 2.5 feet is the maximum distance. Therefore, in this paper, for ease of analysis, we assume that an SR is at a constant small distance d to its associated ST in an isotropic direction [19].MBSs and PRs are modeled by homogeneous Poisson point processes (HPPP)Φ and Φ~U with intensities λ and λ~U, respectively. STs are uniformly distributed in annular regions with radii Rm and RM centered at each MBS. Each ST can be loosely seen as a result of random and independent displacement of the MBS. In each annular region, the number of STs is NST~π(Λ), where π(Λ) is the Poisson distribution with parameter Λ. The distribution model of STs then is similar to a Matern cluster process where Φ is the parent process [25]. An inner radius Rm>0 of the annular region is considered for avoiding singularity of integral in the derivation process and the outer radius RM is related tocircuit power constraint described in the following part.We assume each PR is associated with its nearest MBS. The probability density function (pdf) of distancer from a PR to its nearest MBS is [23](1)fr=2πλre-πλr2.If a point is uniformly distributed within a circle with radius R and r is the distance to the center, the pdf of r is fo(r)=2r/R2 [26]. By using conditional probability, we can get the pdf of distance r from an ST to its MBS as(2)FR∣r≥Rm=PRm≤r≤RPr≥Rm=R2-Rm2RM2-Rm2,f≥Rmr≜fr∣r≥Rm=dFR∣r≥Rm2dr=2rRM2-Rm2.Without loss of generality, according to Slivnyak’s theorem [27], we analyze a typical PR (U0) and a typical SR (Z0) located at the origin. The typical SR’s corresponding ST and MBS are also similarly typical ST (X0) and typical MBS (Y0), respectively. Figure 1 illustrates randomly generated positions of MBSs, PRs, and STs. Figure 2 shows the system model.Figure 1
Network composition illustration. Centers of dual-circles (annular regions), red dots, and blue triangles represent MBSs, primary receivers, and secondary transmitters, respectively. To ensure visibility, only one ST is showed in an annular region and all SRs are omitted. (Radii of annular regions in the illustration are enlarged.)Figure 2
System model of our network.h, g, q refer to three types of fading described in Section 2.2.


## 2.2. Channel Model

All MBSs share the same available channel setC=c1,c2,…,cC, where C=Nch is the number of channels. We assume λ~U≫λ and there is one and only one active PR in each channel of a cell, so active PRs in a generic channel form an HPPP ΦU with intensity λU=λ by independent thinning.We also assume that STs in each cell have equal probabilities to access a channel when it is idle, and no two STs share the same channel. However, STs in different cells may not have equal probabilities since numbers of STs in different cells may differ from each other. This means the thinned processes are not HPPPs. But for ease of analysis, in this paper, similar to the assumptions in [6, 28–30], we assume STs in different cells in a generic channel constitute an HPPP by random displacement and thinning.Besides, if there areNST STs in one annular region, the probability that a channel in C is used by an ST is NST/Nch, while if NST>Nch, the probability is 1. Since NST~π(Λ), the probability that a channel is used averaged over NST is(3)pch=∑k=0NchPNST=kkNch+∑k=Nch+1∞PNST=k·1=1+Λ-NchΓNch,Λ-e-ΛΛNchΓ1+Nch,where Γ(z)=∫0∞tz-1e-tdt is the Gamma function and Γ(z,a)=∫a∞tz-1e-tdt is the upper incomplete Gamma function. So STs in a generic channel form an HPPP ΦST with intensity λST=pchλ by independent thinning and random displacement. And pch is termedchannel in use probability.An extreme case is thatΛ is high enough and pch=1 holds. This equals the setting that only one channel is considered in the network and channels of all cells are used. In this case, besides pch, other details of the network remain unchanged, so does the analysis.We assume that each channel experiences a constant noise powerW and exponential path-loss l(r)=r-α (or l(X-Y)=X-Y-α) with a uniform exponent α>2, where r is a distance, X,Y∈R are two points, and · is the Euclidean norm. Independent Rayleigh fading is considered which remains constant within one time slot. The fading from a PR/ST to its corresponding MBS is h~exp⁡(μh), fading from an ST to its SR is q~exp⁡(μq), and interference fading from an MBS/PR/ST to a PR/SR is g~exp⁡(μg). Moreover, for convenience we set a time slot duration to be a unit time.


## 2.3. Communication Model

When an MBS serves a PR, it may turn into busy or idle mode during each time slot. We further divide each time slot intoM minislots [19] and all minislots are synchronized among MBSs. In addition, we assume that each MBS randomly and independently turns into idle mode in one of the minislots. With such assumptions, when the MBS is transmitting, we consider that an ST performs energy harvesting (EH) and ambient backscatter communication (BC) in DE and DB minislots, respectively, and we have DE+DB+1=M. Minislots for EH and BC of each ST are also randomly and independently selected with equal probability. Besides, the ST performs traditional information transmission (IT) using the harvested energy when the MBS is idle. In the following, these two communication modes are termed BC mode and IT mode, respectively. An example of minislots assignment and selection when M=7 is given in Figure 3.Figure 3
An example of minislots assignment and selection for different mode when number of minislots is 7.In a generic minislot, STs performing energy harvesting form an HPPPΦeh with intensity λeh=(DE/M)λST, which can be seen as the thinning of ΦST since each ST may work in EH mode within a minislot with probability DE/M. Similarly, STs in BC mode and IT mode form HPPPs Φbc and Φ~it with intensities λbc=(DB/M)λST and λ~it=λST/M, respectively. Moreover, MBSs in busy mode also form an HPPP Φbs with intensity λbs=(1-1/M)λ.We further assume that the SR knows about its ST’s work mode so as to perform corresponding decoding. Saturation condition is also assumed where a data packet is always ready for transmission.Besides, we assume that when an ST performs BC, a portionβ of the received power is used for BC, and the backscatter efficiency is ηb. So the backscatter power of a generic ST is(4)Pbc=βηbX-Y-αh,where X-Y is the distance from the MBS to the ST, while the remaining portion 1-β is stored as energy [19] along with received power in EH mode. And the conversion efficiency is η. In this case, the harvested energy within a time slot is(5)EH=ηDEM+1-βDBMr-αh=DebhMrα,where Deb=η(DE+(1-β)DB).In order to perform IT, an ST must satisfy the circuit power constraint thatEH is more than that consumed EC by the circuit, while the remaining power EH-EC is used for data transmission. Since a time slot lasts a unit time, we will also describe EH and EC as the power. The average harvested energy can be obtained as(6)EhEH=∫0∞μhe-μhhDebhMrαdh=DebμhMrα,and the limitation of distance r from an ST to an MBS, that is, RM, can be derived as(7)EhEH-EC>0⟺r<RM≜DebμhMEC1/α,where RM is based on the setting that STs distributed in the annular region should satisfy the circuit power constraint. When fading is considered, the probability that harvested energy EH is above demand is(8)Gx≜PEH>x=PDebhMrα>x=∫RmRM∫Mrαx/Deb∞f≥Rmr·fhdhdr=2Hx,Rm-Hx,RMαRM2-Rm2,where x is a parameter representing energy demand, r is the distance from an ST to its MBS, f≥Rm(r) is the pdf given in (2), h is the fading, and(9)Hx,y=y2Eα-2α,Jxyα,Jx=xμhMDeb,En,z=∫1∞e-ztt-ndtexponentialintegral,where α, μh, M, and Deb have been presented. Therefore, the constraint satisfaction probability psat of an ST is(10)psat≜PEH-EC>0=GEC.In the following, psat will also be termedpower satisfaction probability. Taking this probability into account, STs in IT mode that have enough energy form an HPPP Φit with intensity λit=psatλ~it=psatλST/M. In the following, these STs are calledworking STs.When performing IT, the available energy isEH-EC, but for ease of analysis, in this paper we adopt the setting that a portion ξ of the harvested energy EH can be used for active information transmission [31] and energy not used is ignored in different time slots. So the transmit power of a generic ST is(11)Pit=ξEH1/M=ξDebX-Y-αh,where X-Y is the distance from the MBS to the ST.


## 2.4. Interference Model

In [32], authors mentionedinterference regeneration that a backscatter node reflects all incident signals which leads to a square number of interference components for each SR. One effective solution is to adopt spread spectrum techniques. In this paper, we simplify this problem and assume that an ST backscatters only the signal from its corresponding MBS.


### 2.4.1. Backscattered Signal to an SR in IT Mode

Backscattered signal from an ST to an SR in IT mode is considered as an extra path. We assume an SR in IT mode has the ability to eliminate multipath effect like PRs [4], so such interference is ignored.


### 2.4.2. Transmitted Signal to an SR in BC Mode

Since an SR in BC mode senses and decodes backscattered signal by sensing changes in the signal caused by backscattering [4], we also ignore such interference.


### 2.4.3. Backscattered Interference to an SR in BC Mode

Such interference comes from other STs (Xi) in BC mode whose corresponding MBSs (Yi) are in busy mode. Interfering STs come from Φbc∖{X0}. Such interference is firstly transmitted from an MBS to its ST, encountering path-loss l(Xi-Yi) and Rayleigh fading h~exp⁡(μh), then it is backscattered from an ST to the typical SR, encountering path-loss l(Xi-Z0) and Rayleigh fading g~exp⁡(μg). The interference is denoted as(12)Ibc=∑Xi∈Φbc∖X0PbciXi-Z0-αgi,where Pbci is the backscatter power of an ST in BC mode given in (4).


### 2.4.4. Transmitted Interference to an SR in IT Mode

Such interference comes from other MBSs in busy mode and other STs in IT mode whose corresponding MBSs are in idle mode. Interfering MBSs come fromΦbs∖{Y0}. Such interference is transmitted from an MBS to the typical SR, encountering path-loss l(Yi-Z0) and Rayleigh fading g~exp⁡(μg). Interfering STs come from Φit\{X0}. Such interference is firstly transmitted from an MBS to its ST, encountering path-loss l(Xi-Yi) and Rayleigh fading h~exp⁡(μh), then it is backscattered from the ST to the SR, encountering path-loss l(Xi-Z0) and Rayleigh fading g~exp⁡(μg). The interference is denoted as(13)Isum=Iit+Ibs=∑Xi∈Φit∖X0PitiXi-Z0-αgi+∑Yi∈Φbs∖Y0Yi-Z0-αgi,where Iit, Ibs represent two interference components, respectively, and Piti is the transmit power of an ST in IT mode given in (11).


### 2.4.5. Transmitted Interference to a PR

All interference to a PR is identical to the interference to an SR in IT mode. We will analyze performance of a PR located at the origin encountering no interference from STs to make a comparison to reveal the effect on PRs.


## 2.4.1. Backscattered Signal to an SR in IT Mode

Backscattered signal from an ST to an SR in IT mode is considered as an extra path. We assume an SR in IT mode has the ability to eliminate multipath effect like PRs [4], so such interference is ignored.


## 2.4.2. Transmitted Signal to an SR in BC Mode

Since an SR in BC mode senses and decodes backscattered signal by sensing changes in the signal caused by backscattering [4], we also ignore such interference.


## 2.4.3. Backscattered Interference to an SR in BC Mode

Such interference comes from other STs (Xi) in BC mode whose corresponding MBSs (Yi) are in busy mode. Interfering STs come from Φbc∖{X0}. Such interference is firstly transmitted from an MBS to its ST, encountering path-loss l(Xi-Yi) and Rayleigh fading h~exp⁡(μh), then it is backscattered from an ST to the typical SR, encountering path-loss l(Xi-Z0) and Rayleigh fading g~exp⁡(μg). The interference is denoted as(12)Ibc=∑Xi∈Φbc∖X0PbciXi-Z0-αgi,where Pbci is the backscatter power of an ST in BC mode given in (4).


## 2.4.4. Transmitted Interference to an SR in IT Mode

Such interference comes from other MBSs in busy mode and other STs in IT mode whose corresponding MBSs are in idle mode. Interfering MBSs come fromΦbs∖{Y0}. Such interference is transmitted from an MBS to the typical SR, encountering path-loss l(Yi-Z0) and Rayleigh fading g~exp⁡(μg). Interfering STs come from Φit\{X0}. Such interference is firstly transmitted from an MBS to its ST, encountering path-loss l(Xi-Yi) and Rayleigh fading h~exp⁡(μh), then it is backscattered from the ST to the SR, encountering path-loss l(Xi-Z0) and Rayleigh fading g~exp⁡(μg). The interference is denoted as(13)Isum=Iit+Ibs=∑Xi∈Φit∖X0PitiXi-Z0-αgi+∑Yi∈Φbs∖Y0Yi-Z0-αgi,where Iit, Ibs represent two interference components, respectively, and Piti is the transmit power of an ST in IT mode given in (11).


## 2.4.5. Transmitted Interference to a PR

All interference to a PR is identical to the interference to an SR in IT mode. We will analyze performance of a PR located at the origin encountering no interference from STs to make a comparison to reveal the effect on PRs.


## 2.5. Rate Model for Backscatter Link

In [4], the bit rate (alternating sequence of ones and zeros) of the ambient backscatter prototype is related to the setting of circuit elements. Similar settings are also used in [8, 17, 18] and will be used in this paper, too. Besides, we assume if the signal-to-interference-plus-noise-ratio (SINR) of backscatter communication is above a threshold, the predesigned rate can be achieved [20].Notations used in this paper are listed in the Notations.


## 3. Channel Inversion Power Control and Energy Storage and Reusing

Since energy is precious for secondary users, the harvested energy should be used more efficiently. So in this section we apply channel inversion power control to ST’s active information transmission to avoid poor signal transmitted from STs. Moreover, since power control is applied, there is a higher probability that secondary users do not use up its energy. Therefore, we also propose a simple energy storage and reusing mechanism, to improve the utilization of harvested energy.


### 3.1. Channel Inversion Power Control

In this part, we use channel inversion power control to let STs make less interference to primary users and conserve energy, while keeping their active transmission reliable. To be specific, an ST in IT mode will not use up its energy but transmit at a power to invert the path-loss to make sure that the average received power at its SR is equal to an SR’s sensitivityρ in IT mode [33]. Specifically, if the available energy is enough to support channel inversion power control, the ST will transmit at power ρdα, where d is the distance between a secondary pair. So ρ will highly impact the performance of STs in IT mode. Besides, if the available energy cannot support the power control to achieve the SR’s sensitivity, the ST will not transmit. So an ST’s transmit power using channel inversion power control is(14)Pρ=ρdα,EH-EC1/M>ρdα,0,EH-EC1/M≤ρdα,which can be rewritten as(15)Pρ=ρdα,EH>ρdαM+EC,0,EH≤ρdαM+EC.The probability that an ST transmits at power Pρ, which takes circuit power constraint into account, is(16)pρ=PEH≥ρdαM+EC=GρdαM+EC,where G(·) is given in (8). For convenience, EH>ρdα/M+EC is termedpower constraint with sensitivity and pρ is termedsensitivity satisfaction probability. Since STs are independent from each other, working STs in IT mode adopting the power control form an HPPP Φρ with intensity λρ=pρλ~it=pρλST/M. Any working ST will transmit at power Pρ and the remaining energy is ignored.In the following, for convenience, when we analyze STs and PRs under power control, we still use notations which exist in normal settings, but with a slight difference when the notations involve STs in IT mode. For example,Iit under power control equals ∑Xi∈Φρ∖{X0}PρXi-Z0-αgi.


### 3.2. Energy Storage and Reusing

As described above, an ST has a probability to get enough energy for transmitting in IT mode. If it does not get enough, in our previous settings, the unused energy is ignored and cannot be used in other time slots. Here we consider the setting that the energy can be stored, for a potential reusing. And we assume an ST has an energy storage component with capacityEM. Besides, energy storage follows channel inversion power control, and the remaining energy when an ST can perform active transmission is also stored.If the energy demand for active transmission isED, herein ED=ρdα/M+EC, where EC is the circuit power consumption, the stored energy in a time slot is(17)ES=EH-ED,EH>ED,EH,EH≤ED.Note that we assume an ST can detect whether the harvested energy is enough for active transmission before trying to transmit. If not enough, the ST will not work and cost no energy. Besides, the energy consumption for detection is ignored, so the harvest energy is EH and we do not consider the case that EH<EC. From (8) we know the sensitivity satisfaction probability is G(ED); hence the expectation of stored energy is(18)E¯S=EES=GED·EEH-ED+1-GED·EEH=EEH-GED·ED.For ease of analysis, we assume once an ST has stored enough energy for active transmission before a time slot, it will use that energy in the slot. But the unused part and harvested energy in that slot are ignored. Therefore, the expectation of stored energy afterNS=⌈ED/E¯S⌉ time slots is enough for another active transmission. So in the next time slot, an ST certainly has enough energy for active transmission, and the sensitivity satisfaction probability of the NS+1 slots increases to NSG(ED)+1/NS+1. For simplicity we consider only the next time slot, so the increase of sensitivity satisfaction probability is(19)pρΔ=NSGED+1NS+1-GED1-GED=GED-12NS+1,where 1  −  G(ED) is the dissatisfaction probability, and this part means the improvement works only when harvested energy is not enough.Although the reusing mechanism of stored energy is rough, it provides a view of reusing stored energy and increases the chance of active transmission.Some point processes (p.p.) described in the paper, along with their descriptions, intensities (inten.), and values are listed in Table1 to provide a clear view.Table 1
Some point processes and their descriptions.
p.p. Description Inten. Value ΦU Primary users in a generic channel λ U λ ΦST STs in a generic channel λ S T pchλ Φeh STs in EH mode λ e h (DE/M)λST Φbc STs in BC mode λ b c (DB/M)λST Φbs Busy MBSs λ b s (1-1/M)λ Φ~it STs in IT mode λ ~ i t λST/M Φit Working STs in IT mode λ i t psatλ~it Φρ Working STs (power control) λ ρ pρλ~it


## 3.1. Channel Inversion Power Control

In this part, we use channel inversion power control to let STs make less interference to primary users and conserve energy, while keeping their active transmission reliable. To be specific, an ST in IT mode will not use up its energy but transmit at a power to invert the path-loss to make sure that the average received power at its SR is equal to an SR’s sensitivityρ in IT mode [33]. Specifically, if the available energy is enough to support channel inversion power control, the ST will transmit at power ρdα, where d is the distance between a secondary pair. So ρ will highly impact the performance of STs in IT mode. Besides, if the available energy cannot support the power control to achieve the SR’s sensitivity, the ST will not transmit. So an ST’s transmit power using channel inversion power control is(14)Pρ=ρdα,EH-EC1/M>ρdα,0,EH-EC1/M≤ρdα,which can be rewritten as(15)Pρ=ρdα,EH>ρdαM+EC,0,EH≤ρdαM+EC.The probability that an ST transmits at power Pρ, which takes circuit power constraint into account, is(16)pρ=PEH≥ρdαM+EC=GρdαM+EC,where G(·) is given in (8). For convenience, EH>ρdα/M+EC is termedpower constraint with sensitivity and pρ is termedsensitivity satisfaction probability. Since STs are independent from each other, working STs in IT mode adopting the power control form an HPPP Φρ with intensity λρ=pρλ~it=pρλST/M. Any working ST will transmit at power Pρ and the remaining energy is ignored.In the following, for convenience, when we analyze STs and PRs under power control, we still use notations which exist in normal settings, but with a slight difference when the notations involve STs in IT mode. For example,Iit under power control equals ∑Xi∈Φρ∖{X0}PρXi-Z0-αgi.


## 3.2. Energy Storage and Reusing

As described above, an ST has a probability to get enough energy for transmitting in IT mode. If it does not get enough, in our previous settings, the unused energy is ignored and cannot be used in other time slots. Here we consider the setting that the energy can be stored, for a potential reusing. And we assume an ST has an energy storage component with capacityEM. Besides, energy storage follows channel inversion power control, and the remaining energy when an ST can perform active transmission is also stored.If the energy demand for active transmission isED, herein ED=ρdα/M+EC, where EC is the circuit power consumption, the stored energy in a time slot is(17)ES=EH-ED,EH>ED,EH,EH≤ED.Note that we assume an ST can detect whether the harvested energy is enough for active transmission before trying to transmit. If not enough, the ST will not work and cost no energy. Besides, the energy consumption for detection is ignored, so the harvest energy is EH and we do not consider the case that EH<EC. From (8) we know the sensitivity satisfaction probability is G(ED); hence the expectation of stored energy is(18)E¯S=EES=GED·EEH-ED+1-GED·EEH=EEH-GED·ED.For ease of analysis, we assume once an ST has stored enough energy for active transmission before a time slot, it will use that energy in the slot. But the unused part and harvested energy in that slot are ignored. Therefore, the expectation of stored energy afterNS=⌈ED/E¯S⌉ time slots is enough for another active transmission. So in the next time slot, an ST certainly has enough energy for active transmission, and the sensitivity satisfaction probability of the NS+1 slots increases to NSG(ED)+1/NS+1. For simplicity we consider only the next time slot, so the increase of sensitivity satisfaction probability is(19)pρΔ=NSGED+1NS+1-GED1-GED=GED-12NS+1,where 1  −  G(ED) is the dissatisfaction probability, and this part means the improvement works only when harvested energy is not enough.Although the reusing mechanism of stored energy is rough, it provides a view of reusing stored energy and increases the chance of active transmission.Some point processes (p.p.) described in the paper, along with their descriptions, intensities (inten.), and values are listed in Table1 to provide a clear view.Table 1
Some point processes and their descriptions.
p.p. Description Inten. Value ΦU Primary users in a generic channel λ U λ ΦST STs in a generic channel λ S T pchλ Φeh STs in EH mode λ e h (DE/M)λST Φbc STs in BC mode λ b c (DB/M)λST Φbs Busy MBSs λ b s (1-1/M)λ Φ~it STs in IT mode λ ~ i t λST/M Φit Working STs in IT mode λ i t psatλ~it Φρ Working STs (power control) λ ρ pρλ~it


## 4. Coverage Probability and Achievable Rate

We analyze the coverage probabilities and average achievable rates of an SR in different communication modes and a PR, by using Shannon formula. The average rate is(20)Elog2⁡1+SINR=Eln⁡1+SINRln⁡2.Besides, since SINR≥0, it is easy to derive(21)Eln⁡1+SINR=∫0∞11+TPSINR>TdT,where T is the SINR threshold. So in the following, we will derive the coverage probability of the typical SR or PR in form(22)PSINR>T=PSI+W>T,where S is the desired signal, I is the interference, and W is the noise. After deriving the coverage probabilities, average rates can be easily obtained.Firstly, we show some properties of the derivation processes which will be used for the following analysis and theorems.Property 1.
If the pdf of distancer from an MBS to an ST is f≥Rm(r) in (2), the following expectation of r can be derived easily:(23)Er-2=∫r≥RmRMr-22rRM2-Rm2dr=2ln⁡RM/RmRM2-Rm2.Property 2.
If fadingh~exp⁡(μh), the following expectation of h can be derived easily:(24)Eh2/α=∫0∞μhe-μhhh2/αdh=μh-2/αΓ2α+1,where α is the path-loss exponent and Γ(z)=∫0∞tz-1e-tdt is the Gamma function.Property 3.
Ify is a random variable, the following integral of y can be represented in nonintegral form:(25)Kα=∫0∞y1+yαdy=παsin⁡2π/α,where α is the path-loss exponent. Be similar to [26], we denoted the result in (25) as K(α) with some differences.Property 4.
Given the Laplace transformLX(s) of a continuous random variable X, the pdf fX(x) of X can be recovered by inverse Laplace transform as [21](26)fXx=L-1LXsx.Note that, in this paper, we focus on a generic channel, a typical primary receiver, and a typical secondary receiver. Unless otherwise stated, the following analyzed coverage probabilities and average rates are analyzed under the setting that the receivers exist already.


### 4.1. Coverage Probability of an SR in BC Mode

Here we analyze the coverage probability of the typical secondary receiver working in backscatter mode; that is, its secondary transmitter backscatters signals for data transmission. The desired signal power is given in (4) and the interference is analyzed in Section 2.4.Theorem 5.
The coverage probability of an SR in BC mode located at the origin is(27)pcbc=∫r≥RmRM∫q>0∞μqexp⁡Abc-BbcCdqf≥Rmrdr,where(28)Abc=-μqq-μhβηbqTWdαrα,Bbc=λbcμhTμgq2/αd2,C=2πr2ER-2Eh2/αKα,and E[R-2], E[h2/α], and K(α) are given in (23), (24), and (25), respectively, where R is the distance from a generic interfering ST to its MBS having the same pdf given in (2).Proof.
See Appendix.


### 4.2. Coverage Probability of an SR in IT Mode

Here we analyze the coverage probability of the typical secondary receiver working in information transmission mode, that is, receiving signals transmitted by its secondary transmitter. The desired signal power is given in (11) and the interference is analyzed in Section 2.4.Note that the typical ST in IT mode also has a probability of satisfying circuit power constraint, so the average coverage probability of the typical SR is(29)p¯cit=PSINRit>T,EH≥EC,0,EH<EC,which can be further written as(30)p¯cit=psatPSINRit>T.Since psat has been analyzed, in the following, we mainly derive pcit=P[SINRit>T] under the assumption that the typical ST satisfies the circuit power constraint, andcoverage probability refers to pcit for convenience.Theorem 6.
The coverage probability of an SR in IT mode located at the origin is(31)pcit=∫r≥RmRM∫q>0∞μqexp⁡Ait-BitC-Ditdqf≥Rmrdr,where(32)Ait=-μqq-μhTWξDebqdαrα,Bit=λitμhTμgq2/αd2,Dit=2πλbsμhTμgξDebq2/αd2r2Kα,and C is given in (28).Proof.
See Appendix.


### 4.3. Coverage Probability of a PR

Here we analyze the coverage probability of the typical primary receiver with regard to interference from STs in IT mode. The desired signal comes from its MBS, and the interference is analyzed in Section2.4.Theorem 7.
The coverage probability of a PR located at the origin is(33)pcpr=∫r>0∞exp⁡Apr-BprC-Dprfrdr,where(34)Apr=-μhTWrα,Bpr=λitξDebμhTμg2/α,Dpr=2πλbsμhTμg2/αr2∫μg/μhT1/α∞y1+yαdy,and C is given in (28).Proof.
See Appendix.


### 4.4. Coverage Probability of a PR without Interference from STs

Here we analyze the coverage probability of the typical primary receiver without interference from STs in IT mode. The desired signal still comes from the MBS and the primary receiver suffers only interference from other MBSs.Theorem 8.
The coverage probability of a PR considering no interference from STs is(35)pcpr′=∫r>0∞exp⁡Apr-Dprfrdr,where Apr and Dpr are given in (34).Proof.
See Appendix.


### 4.5. Coverage Probability of an SR in IT Mode under Power Control

Here we analyze the coverage probability of the typical secondary receiver working in information transmission mode and adopting power control. The main differences are the transmit power of STs and their constraint satisfaction probability frompsat to pρ.Similarly to Section4.2, coverage probability averaged over constraint satisfaction probability is(36)p¯c,ρit=pρPSINRitρ>T,and in the following,coverage probability refers to pc,ρit=P[SINRitρ>T] under the assumption that the typical ST satisfies the power constraint with sensitivity.Theorem 9.
The coverage probability of an SR in IT mode under power control is(37)pc,ρit=exp⁡Aitρ-λρDitρ-λbsPρ-2/αDitρ,where(38)Aitρ=-μqTWPρdα,Ditρ=2πμqTμg2/αd2Kα.Proof.
See Appendix.


### 4.6. Coverage Probability of a PR under Power Control

Theorem 10.
The coverage probability of a PR suffering interference from STs in IT mode under power control is(39)pc,ρpr=∫r>0∞exp⁡Apr-Bprρ-Dprfrdr,where Apr, Dpr are given in (34), and(40)Bprρ=2πλρμhTμg2/αr2Pρ2/αKα.Proof.
See Appendix.


## 4.1. Coverage Probability of an SR in BC Mode

Here we analyze the coverage probability of the typical secondary receiver working in backscatter mode; that is, its secondary transmitter backscatters signals for data transmission. The desired signal power is given in (4) and the interference is analyzed in Section 2.4.Theorem 5.
The coverage probability of an SR in BC mode located at the origin is(27)pcbc=∫r≥RmRM∫q>0∞μqexp⁡Abc-BbcCdqf≥Rmrdr,where(28)Abc=-μqq-μhβηbqTWdαrα,Bbc=λbcμhTμgq2/αd2,C=2πr2ER-2Eh2/αKα,and E[R-2], E[h2/α], and K(α) are given in (23), (24), and (25), respectively, where R is the distance from a generic interfering ST to its MBS having the same pdf given in (2).Proof.
See Appendix.


## 4.2. Coverage Probability of an SR in IT Mode

Here we analyze the coverage probability of the typical secondary receiver working in information transmission mode, that is, receiving signals transmitted by its secondary transmitter. The desired signal power is given in (11) and the interference is analyzed in Section 2.4.Note that the typical ST in IT mode also has a probability of satisfying circuit power constraint, so the average coverage probability of the typical SR is(29)p¯cit=PSINRit>T,EH≥EC,0,EH<EC,which can be further written as(30)p¯cit=psatPSINRit>T.Since psat has been analyzed, in the following, we mainly derive pcit=P[SINRit>T] under the assumption that the typical ST satisfies the circuit power constraint, andcoverage probability refers to pcit for convenience.Theorem 6.
The coverage probability of an SR in IT mode located at the origin is(31)pcit=∫r≥RmRM∫q>0∞μqexp⁡Ait-BitC-Ditdqf≥Rmrdr,where(32)Ait=-μqq-μhTWξDebqdαrα,Bit=λitμhTμgq2/αd2,Dit=2πλbsμhTμgξDebq2/αd2r2Kα,and C is given in (28).Proof.
See Appendix.


## 4.3. Coverage Probability of a PR

Here we analyze the coverage probability of the typical primary receiver with regard to interference from STs in IT mode. The desired signal comes from its MBS, and the interference is analyzed in Section2.4.Theorem 7.
The coverage probability of a PR located at the origin is(33)pcpr=∫r>0∞exp⁡Apr-BprC-Dprfrdr,where(34)Apr=-μhTWrα,Bpr=λitξDebμhTμg2/α,Dpr=2πλbsμhTμg2/αr2∫μg/μhT1/α∞y1+yαdy,and C is given in (28).Proof.
See Appendix.


## 4.4. Coverage Probability of a PR without Interference from STs

Here we analyze the coverage probability of the typical primary receiver without interference from STs in IT mode. The desired signal still comes from the MBS and the primary receiver suffers only interference from other MBSs.Theorem 8.
The coverage probability of a PR considering no interference from STs is(35)pcpr′=∫r>0∞exp⁡Apr-Dprfrdr,where Apr and Dpr are given in (34).Proof.
See Appendix.


## 4.5. Coverage Probability of an SR in IT Mode under Power Control

Here we analyze the coverage probability of the typical secondary receiver working in information transmission mode and adopting power control. The main differences are the transmit power of STs and their constraint satisfaction probability frompsat to pρ.Similarly to Section4.2, coverage probability averaged over constraint satisfaction probability is(36)p¯c,ρit=pρPSINRitρ>T,and in the following,coverage probability refers to pc,ρit=P[SINRitρ>T] under the assumption that the typical ST satisfies the power constraint with sensitivity.Theorem 9.
The coverage probability of an SR in IT mode under power control is(37)pc,ρit=exp⁡Aitρ-λρDitρ-λbsPρ-2/αDitρ,where(38)Aitρ=-μqTWPρdα,Ditρ=2πμqTμg2/αd2Kα.Proof.
See Appendix.


## 4.6. Coverage Probability of a PR under Power Control

Theorem 10.
The coverage probability of a PR suffering interference from STs in IT mode under power control is(39)pc,ρpr=∫r>0∞exp⁡Apr-Bprρ-Dprfrdr,where Apr, Dpr are given in (34), and(40)Bprρ=2πλρμhTμg2/αr2Pρ2/αKα.Proof.
See Appendix.


## 5. Numerical Results

We evaluate our analytical results by simulations. The simulation region is a square with side length of 10 km. The simulation results are obtained by averaging over 1000 runs. Unless otherwise stated, parameter values are listed in Table2. To generate uniformly distributed points in an annular region, we use a native method of generating a random point within a circle with radius RM, and if the random value is smaller than the inner radius Rm, the point will be generated repeatedly until it is in the annular region. A mathematical method is to useinverse transform sampling.Table 2
Parameter value list.
Parameter Value Inner radius of annular regionsRm 5 Constant distance from an SR to its STd 10 MBS densityλ 5/km2 Poisson distribution parameterΛ 3 Number of channelsNch 4 Minislot numberM 7 BC minislot numberDB 3 (i.e.,DE=3) Path-loss exponentα 4 Fading parametersμh, μg, μq 1 NoiseW −90 dBm EH efficiencyη 0.6 BC portionβ 0.3 BC efficiencyηb 0.6 IT portionξ 0.5 Circuit energy consumptionEC −30 dBm SR’s sensitivity in IT mode −60 dBm


### 5.1. Channel in Use Probability

Figure4 shows the channel in use probability pch given in (3). The simulation of channel selection is performed by random selection from all possible combinations. STs clustering around an MBS select idle channels using the method in Section 2.2; that is, they randomly select idle channels. We focus on the typical channel (numbered 1) and count how many typical channels are selected by STs clustering around all MBSs. The analytical and simulation curves overlap since the channel selection is simple, and the results are almost identical.Figure 4
Channel in use probability versus Poisson distribution parameter (i.e., average number of STs clusters around an MBS).


### 5.2. Power Satisfaction Probability and Sensitivity Satisfaction Probability

Figure5 shows the power satisfaction probability psat given in (10), of an ST. As circuit power consumption EC increases from −40 dBm (0.1 μW) to −10 dBm (0.1 mW), the probability decreases, but not much. The reason is that a larger EC makes STs closer to MBSs, and STs then can harvest more energy to support a high satisfaction probability.Figure 5
Power satisfaction probability of an ST.Figure6 shows the sensitivity satisfaction probability pρ given in (16), of an ST under power control. As SRs’ sensitivity ρ increases from −70 dBm to −50 dBm, pρ decreases sharply. The reason is that ρ does not change RM, but requires more energy. And when ρ decreases further, the probability is already very low, so the change of satisfaction probability is small. The difference between these two figures also reveals the importance of RM. Moreover, pρ is affected by circuit power constraint as well, which further affects the energy demand to transmit at Pρ.Figure 6
Sensitivity satisfaction probabilities of an ST under power control.


### 5.3. Coverage Probabilities

Figure7 shows the analytical and simulated coverage probabilities of an ST in BC mode, IT mode, and IT mode under power control. Since distance from an ST to its MBS and distance from an SR to its ST are both very short, signals decay not much. Besides, as described in Section 2.4, transmitted signals from MBSs and STs in IT mode make no interference to an SR in BC mode, so coverage probability of BC mode is much higher, while coverage probability of IT mode (under power control) is lower. Moreover, there is an intersection of curves of probabilities of IT mode and IT mode under power control. This happens since an SR in normal IT mode receives varied-power signals related to the ST’s harvested energy. This means an ST may transmit at a high power, resulting in a high SINR, and may transmit at a low power, resulting in a low SINR. So the SINRs of SRs in normal IT mode range widely. On the other hand, an SR in IT mode under power control always receives constant-power (equals its sensitivity) signals from its ST, if fading is not considered. This means, there are few SRs having (extremely) low SINRs or high SINRs, which makes the curve shrink horizontally.Figure 7
Analytical coverage probability of an ST in BC mode (bc), IT mode (it), and IT mode under power control (itρ).Note that these probabilities are not averaged overpsat or pρ; that is, we assume the typical ST has already satisfied the constraints.Figure8 shows the analytical and simulated coverage probabilities of a PR. The probabilities of a PR interfered or not by STs in IT mode change too little to be observed, so the figure shows that only two curves represent analytical and simulated results. Since an ST transmits by using harvested energy, the transmit power is relatively lower to the power of MBSs. Besides, only STs whose MBSs are idle can transmit, and the idle ratio is low (1/7 in our settings), so the number of interfering STs to a PR is small. These are the two main reasons why STs’ interference to a PR is so low. As for an SR in IT mode, since it is very close to its ST, it still gets high SINR even when the ST transmits at low power.Figure 8
Analytical and simulated coverage probabilities of a PR.


### 5.4. Distributions of Interference Caused by STs in IT Mode and under Power Control

As interference caused by STs in IT mode (and under power control) to a PR has quite limited effect on the PR’s SINR, here we give the distributions of interference caused by STs. Following (26) in Property 4, (B.3) in Appendix, and (E.3) in Appendix, the analytical interference distributions caused by STs in IT mode and STs under power control are drawn in Figures 9 and 10, respectively.Figure 9
Analytical distribution of interference caused by STs in IT mode.Figure 10
Analytical distribution of interference caused by STs in IT mode (under power control).


### 5.5. Effect of Minislots Assignment on Coverage Probabilities

The different assignments of minislots, that is, differentDB and DE, result into different numbers of STs in backscatter mode and energy harvesting mode, that is, intensities λbc and λeh of Φbc and Φeh, respectively. The intensity λbc does not affect backscatter power of an ST, because backscattering is instantaneous. But λbc affects interference power since it changes number of STs in backscatter mode. However, as described in previous subsection, SINR of an SR in BC mode is much higher, which means change of interference power makes little effect on the SINR, where noise W affects. So in both analytical and simulated results, when DB changes from 1 to 5, pcbc varies quite little (less than 3%) when 5 dB is selected as the SINR threshold.As for coverage probabilities of an SR in other modes, they change quite little, too (less than1% and 4% in analytical and simulated results, resp.). The reason is that when DB decreases, all STs harvest more energy (on average), so their transmit powers increase together. On the other hand, since we assume each MBS turns into idle in only one minislot, the minislots assignment does not affect number of STs in IT mode. Therefore, DB makes little change on their SINRs. As for PR, since STs’ impact of it is very limited as shown in the previous subsection, its SINR changes little (less than 1% and 3% in analytical and simulated results, resp.) when DB varies.


### 5.6. Coverage Probabilities When Channels Are Fully Used

When considering special casefully used channels, pch=1 holds and other settings remained. The analytical and simulated results of coverage probabilities in this case are very close to the above probabilities under normal settings. The analytical average changes over different SINR thresholds are 0.11%, 0.01%, 0.01%, 0%, 0%, and 0.01% corresponding to BC mode, IT mode, IT mode under power control, a PR, a PR without STs, and a PR (STs under power control), respectively. And the simulated average changes are 0.90%, 1.81%, 0.83%, 0.83%, 0.47%, and 0.83%, respectively. So when Nch=4 and Λ changes from 3 to a high enough value, the coverage probabilities change little.


### 5.7. Average Rates of STs in IT Mode

Figure11 shows average rates of an ST in IT mode versus circuit power consumption EC, with different Λ. As EC increases, the rate becomes higher, too. This reason is that EC affects RM given in (7). A smaller EC means a smaller RM since STs have to be distributed closer to MBSs to get enough energy. However, since we assume a portion of the harvested energy can be used to perform information transmission, the power of the desired signal increases as well, resulting in a higher rate. Figure 12 shows average rates of an ST in IT mode under power control versus SRs’ sensitivity ρ, with different Λ. As ρ increases, an SR receives a higher SINR because the desired signal is stronger, and there are less interfering STs.Figure 11
Average rates of an ST in IT mode versus circuit power consumption.Figure 12
Average rates of an ST in IT mode under power control versus sensitivity.Note that the rates are averaged overpch which represents the utility ratio of a channel by STs. Besides, since it is observed that STs in IT mode have limited interference to PRs, rates of a PR are omitted.


### 5.8. Average Stored Energy and Reusing

A larger circuit power consumptionEC or SRs’ sensitivity ρ makes energy demand ED higher, G(ED) lower, RM smaller, and EH higher. So the impact on average stored energy in a time slot E¯S cannot be observed directly from (18). Figures 13 and 14 show the analytical and simulated results of E¯S versus EC and ρ, respectively.Figure 13
Average stored energy of an ST versus circuit power consumption.Figure 14
Average stored energy of an ST versus sensitivity.Figures15 and 16 show the sensitivity satisfaction probability increase pρΔ given in (19). These results are computed using (19), and the simulation data needed are obtained from simulations. Besides, analytical results are computed in small granularity (about 100 data points in both figures), while a spot of simulated data points is drawn in circles. Under our parameter settings, when EC increases, pρΔ decreases sharply with a slow increase after that. But when ρ increases, pρΔ varies like a staircase function and decreases overall after the beginning increase. The staircase follows NS which contains a ceiling function. Once the energy demand is large enough or averaged stored energy is less enough, NS increases by 1, and the satisfaction probability drops down.Figure 15
Average stored energy of an ST versus circuit power consumption.Figure 16
Average stored energy of an ST versus sensitivity.


## 5.1. Channel in Use Probability

Figure4 shows the channel in use probability pch given in (3). The simulation of channel selection is performed by random selection from all possible combinations. STs clustering around an MBS select idle channels using the method in Section 2.2; that is, they randomly select idle channels. We focus on the typical channel (numbered 1) and count how many typical channels are selected by STs clustering around all MBSs. The analytical and simulation curves overlap since the channel selection is simple, and the results are almost identical.Figure 4
Channel in use probability versus Poisson distribution parameter (i.e., average number of STs clusters around an MBS).


## 5.2. Power Satisfaction Probability and Sensitivity Satisfaction Probability

Figure5 shows the power satisfaction probability psat given in (10), of an ST. As circuit power consumption EC increases from −40 dBm (0.1 μW) to −10 dBm (0.1 mW), the probability decreases, but not much. The reason is that a larger EC makes STs closer to MBSs, and STs then can harvest more energy to support a high satisfaction probability.Figure 5
Power satisfaction probability of an ST.Figure6 shows the sensitivity satisfaction probability pρ given in (16), of an ST under power control. As SRs’ sensitivity ρ increases from −70 dBm to −50 dBm, pρ decreases sharply. The reason is that ρ does not change RM, but requires more energy. And when ρ decreases further, the probability is already very low, so the change of satisfaction probability is small. The difference between these two figures also reveals the importance of RM. Moreover, pρ is affected by circuit power constraint as well, which further affects the energy demand to transmit at Pρ.Figure 6
Sensitivity satisfaction probabilities of an ST under power control.


## 5.3. Coverage Probabilities

Figure7 shows the analytical and simulated coverage probabilities of an ST in BC mode, IT mode, and IT mode under power control. Since distance from an ST to its MBS and distance from an SR to its ST are both very short, signals decay not much. Besides, as described in Section 2.4, transmitted signals from MBSs and STs in IT mode make no interference to an SR in BC mode, so coverage probability of BC mode is much higher, while coverage probability of IT mode (under power control) is lower. Moreover, there is an intersection of curves of probabilities of IT mode and IT mode under power control. This happens since an SR in normal IT mode receives varied-power signals related to the ST’s harvested energy. This means an ST may transmit at a high power, resulting in a high SINR, and may transmit at a low power, resulting in a low SINR. So the SINRs of SRs in normal IT mode range widely. On the other hand, an SR in IT mode under power control always receives constant-power (equals its sensitivity) signals from its ST, if fading is not considered. This means, there are few SRs having (extremely) low SINRs or high SINRs, which makes the curve shrink horizontally.Figure 7
Analytical coverage probability of an ST in BC mode (bc), IT mode (it), and IT mode under power control (itρ).Note that these probabilities are not averaged overpsat or pρ; that is, we assume the typical ST has already satisfied the constraints.Figure8 shows the analytical and simulated coverage probabilities of a PR. The probabilities of a PR interfered or not by STs in IT mode change too little to be observed, so the figure shows that only two curves represent analytical and simulated results. Since an ST transmits by using harvested energy, the transmit power is relatively lower to the power of MBSs. Besides, only STs whose MBSs are idle can transmit, and the idle ratio is low (1/7 in our settings), so the number of interfering STs to a PR is small. These are the two main reasons why STs’ interference to a PR is so low. As for an SR in IT mode, since it is very close to its ST, it still gets high SINR even when the ST transmits at low power.Figure 8
Analytical and simulated coverage probabilities of a PR.


## 5.4. Distributions of Interference Caused by STs in IT Mode and under Power Control

As interference caused by STs in IT mode (and under power control) to a PR has quite limited effect on the PR’s SINR, here we give the distributions of interference caused by STs. Following (26) in Property 4, (B.3) in Appendix, and (E.3) in Appendix, the analytical interference distributions caused by STs in IT mode and STs under power control are drawn in Figures 9 and 10, respectively.Figure 9
Analytical distribution of interference caused by STs in IT mode.Figure 10
Analytical distribution of interference caused by STs in IT mode (under power control).


## 5.5. Effect of Minislots Assignment on Coverage Probabilities

The different assignments of minislots, that is, differentDB and DE, result into different numbers of STs in backscatter mode and energy harvesting mode, that is, intensities λbc and λeh of Φbc and Φeh, respectively. The intensity λbc does not affect backscatter power of an ST, because backscattering is instantaneous. But λbc affects interference power since it changes number of STs in backscatter mode. However, as described in previous subsection, SINR of an SR in BC mode is much higher, which means change of interference power makes little effect on the SINR, where noise W affects. So in both analytical and simulated results, when DB changes from 1 to 5, pcbc varies quite little (less than 3%) when 5 dB is selected as the SINR threshold.As for coverage probabilities of an SR in other modes, they change quite little, too (less than1% and 4% in analytical and simulated results, resp.). The reason is that when DB decreases, all STs harvest more energy (on average), so their transmit powers increase together. On the other hand, since we assume each MBS turns into idle in only one minislot, the minislots assignment does not affect number of STs in IT mode. Therefore, DB makes little change on their SINRs. As for PR, since STs’ impact of it is very limited as shown in the previous subsection, its SINR changes little (less than 1% and 3% in analytical and simulated results, resp.) when DB varies.


## 5.6. Coverage Probabilities When Channels Are Fully Used

When considering special casefully used channels, pch=1 holds and other settings remained. The analytical and simulated results of coverage probabilities in this case are very close to the above probabilities under normal settings. The analytical average changes over different SINR thresholds are 0.11%, 0.01%, 0.01%, 0%, 0%, and 0.01% corresponding to BC mode, IT mode, IT mode under power control, a PR, a PR without STs, and a PR (STs under power control), respectively. And the simulated average changes are 0.90%, 1.81%, 0.83%, 0.83%, 0.47%, and 0.83%, respectively. So when Nch=4 and Λ changes from 3 to a high enough value, the coverage probabilities change little.


## 5.7. Average Rates of STs in IT Mode

Figure11 shows average rates of an ST in IT mode versus circuit power consumption EC, with different Λ. As EC increases, the rate becomes higher, too. This reason is that EC affects RM given in (7). A smaller EC means a smaller RM since STs have to be distributed closer to MBSs to get enough energy. However, since we assume a portion of the harvested energy can be used to perform information transmission, the power of the desired signal increases as well, resulting in a higher rate. Figure 12 shows average rates of an ST in IT mode under power control versus SRs’ sensitivity ρ, with different Λ. As ρ increases, an SR receives a higher SINR because the desired signal is stronger, and there are less interfering STs.Figure 11
Average rates of an ST in IT mode versus circuit power consumption.Figure 12
Average rates of an ST in IT mode under power control versus sensitivity.Note that the rates are averaged overpch which represents the utility ratio of a channel by STs. Besides, since it is observed that STs in IT mode have limited interference to PRs, rates of a PR are omitted.


## 5.8. Average Stored Energy and Reusing

A larger circuit power consumptionEC or SRs’ sensitivity ρ makes energy demand ED higher, G(ED) lower, RM smaller, and EH higher. So the impact on average stored energy in a time slot E¯S cannot be observed directly from (18). Figures 13 and 14 show the analytical and simulated results of E¯S versus EC and ρ, respectively.Figure 13
Average stored energy of an ST versus circuit power consumption.Figure 14
Average stored energy of an ST versus sensitivity.Figures15 and 16 show the sensitivity satisfaction probability increase pρΔ given in (19). These results are computed using (19), and the simulation data needed are obtained from simulations. Besides, analytical results are computed in small granularity (about 100 data points in both figures), while a spot of simulated data points is drawn in circles. Under our parameter settings, when EC increases, pρΔ decreases sharply with a slow increase after that. But when ρ increases, pρΔ varies like a staircase function and decreases overall after the beginning increase. The staircase follows NS which contains a ceiling function. Once the energy demand is large enough or averaged stored energy is less enough, NS increases by 1, and the satisfaction probability drops down.Figure 15
Average stored energy of an ST versus circuit power consumption.Figure 16
Average stored energy of an ST versus sensitivity.


## 6. Conclusions

In this work, we have analyzed the performance of ambient backscatter communications in RF-powered cognitive radio networks based on stochastic geometry. Besides, we have applied channel inversion power control to active information of secondary users. After that, a simple energy storage and reusing mechanism has been designed and analyzed to improve utilization of harvested energy. Analytical results for constraint satisfaction probabilities and coverage probabilities of secondary users and of primary users considering both communication modes of secondary users have been obtained. Besides, average rates of secondary users have been obtained based on coverage probability. As for energy reusing, we have analyzed average stored energy during a time slot and the increase of constraint satisfaction probability when power control is applied. The numerical results validate our theoretical analysis. Also, the results show performance improvement of secondary systems with only limited impact on the performance of primary systems. The analytical results and simulations demonstrate that integrating ambient backscatter communications into RF-powered cognitive radio network is a promising way to achieve energy and spectrum efficient wireless communications, which is suitable for certain Internet of things (IoT) applications.



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*Source: 8509693-2018-04-02.xml*