0 Introduction

The channel resource assigned to a wireless transmitter becomes a waste if it stays idle, while an active transmission of data could be eavesdropped. Spectrum sharing overcomes the first problem by having the primary user (PU) of the channel resource to share it with a secondary user. The latter can then help the former in defending against eavesdropping. The secondary user is expected to be a cognitive radio (CR) equipped with spectrum sensing function that identifies spectral opportunity or unoccupied channel. CR can make use of an unused channel to send its own data. On the other hand, when the PU has data to send, CR relinquishes the channel. CR returns the favor by transmitting artificial noise to jam the eavesdropper (EA), thus protecting the secrecy of primary data. This technique is named as physical layer security^[1-2], which originates from information theory. It utilizes the physics of radio signal decaying with propagation distance. As long as the artificial noise that reaches EA is strong enough, it could garble the information that EA attempts to steal.

PU consists of two nodes, i.e., the primary transmitter (PT) and primary receiver (PR). CR is taken as the secondary user (SU). Conventionally, PU does not like sharing the spectral opportunity with CR, because CR's transmission could harmfully interfere with the data reception at PR. Now there is an incentive for PU, because CR helps defending against eavesdropping when PT transmits. CR sends out friendly jamming signal that is decodable by the PR, which harms only the EA but not PR. In return, CR has the benefit of sending its own data over the channel when it is unused. Such spectrum sharing scheme benefits both PU and CR. Fig. 1 illustrates the proposed model.

Fig. 2 shows a state transition diagram that depicts CR's operation. At first, CR senses the spectrum to identify channel occupancy. If a channel is found to be occupied (by PU), CR transits to the state of jamming EA. Otherwise, it transits to the state of transmitting its data over the unoccupied channel. Although an EA's exact location may not be known, it is assumed to be close to the PR, and that it targets only the data sent out from PT but not CR.

CR's energy and time are divided between the three states depicted in Fig. 2. Suppose the duration of a data packet is T seconds. CR spends t seconds for sensing, and T-t seconds in any of the other two states (transmitting data or jamming EA). Fig. 3 illustrates the time allocation. Let τ=t/T denote the portion of time allocated for spectrum sensing.

The scheme encourages PU to share its spectral channel with CR, since CR can protect PU from being eavesdropped. Here the benefit that the CR gets is a question. CR's throughput increases because it can transmit data when PU becomes inactive. However, jamming EA consumes energy. This may be risky for CR that has limited amount of energy. Whether it is possible to attain high throughput at low energy consumption is the topic for investigation.

Energy efficiency μ is defined as the ratio of channel throughput to energy consumption. Being highly energy efficient is critical for CR. Determining the changes of μ with respect to τ and other parameters is the way to find out how μ varies with the sensing duration and other operation parameters. Intuitively, increasing τ will increase sensing accuracy, thus the throughput should increase too, which results in higher μ. However, increasing τ also reduces the duration for sending CR's data and its throughput, implying reduced μ. These conflicting tendencies result in a maximal μ value. Let μ^* denote the optimal energy efficiency, and τ^* denote the value of τ that optimizes μ. This study seeks to understand how the values of μ^* and τ^* change under different circumstances. Therefore, a novel theoretical expression is contributed, which reveals that the (τ^*, μ^*) pair depends only on the following system parameters:

1) Ratio of the CR's jamming power to transmission power, P_J/P_T;

2) Ratio of the CR's sensing power to transmission power, P_S/P_T;

3) Probability of PU being present and active (Pr₁);

4) Probability of detection (Pr_D), which is the requirement imposed on CR by PU, with the purpose of guaranteeing sufficient protection from CR transmission;

5) Ratio of received PT signal power to noise power (SNR_P);

6) Sampling rate used for sensing (f_s).

In the following sections, how these parameters affect the value of (τ^*, μ^*) will be demonstrated. The remaining part of this paper is organized in the following manner. Section 1 describes the related works. The spectrum sensing technique is described and the problem of finding energy efficiency is formulated in Section 2. Section 3 presents the results and discussions, and Section 4 concludes this paper.

1 Related Works

There are different ways of sharing the spectrum^[3-7]. It could be shared following the use of game theory^[3-5]. The users could bid for the available spectrum in an auction^[3], or enforced among themselves certain rules in order to share the spectrum more efficiently^[4]. In Ref. [5], the PUs compete among themselves in order to provide CRs some spectral opportunities with competitive pricing. In Ref. [6], the gain of utilizing channel fading in favor of CR transmission was analyzed, and the performance gain of using multi-antennas for spectrum sharing was presented in Ref. [7]. In contrast to these works, this study provides an incentive to PU for sharing its spectrum.

The secrecy of PU's data is an important issue^[8-13]. Artificial noise was introduced in Ref. [8] to jam the EA. The idea was then expanded in Refs. [9-10]. The noise pattern was selected from the null space of the receiver's channel but not that of the EA's channel. As a result, the noise causes adverse effect to the EA's reception but not that of the legitimate receiver. Since an eavesdropper is passive, the secrecy scheme proposed in Refs. [8-10] does not assume knowing its presence or even its location. Without the knowledge of the location of EA, there is still a guaranteed minimum rate of transmission secrecy^[9] and minimum secrecy capacity^[10]. A 2-stage protocol is proposed in Refs. [8, 10] to get CR's help in transmitting artificial noise to jam the EA:

Stage 1: Both the PT and PR send out artificial noises. Different weighted versions of them are received by the CR and EA. Let a_XY denote the channel gain between sender X and receiver Y, r_A the signal received by node A. Subscripts T, R, C, and E denote PT, PR, CR, and EA, respectively.

PT transmits signal a_TRx, whereas PR transmits y. CR receives the signal

$ r_{\mathrm{C}}=\alpha_{\mathrm{TC}} \alpha_{\mathrm{TR}} x+\alpha_{\mathrm{RC}} y+n_{\mathrm{C}} $

EA receives the signal

$ r_{\mathrm{E}, 1}=\alpha_{\mathrm{TE}} \alpha_{\mathrm{TR}} x+\alpha_{\mathrm{RE}} y+e_{1} $

Both n_C and e₁ are independent and identically distributed additive white Gaussian noise (AWGN) random variables.

Stage 2: CR retransmits the weighted version of the received noises while PT transmits data that will cancel out the artificial noise at the PR.

CR transmits βr_C, where β is a random weight. PT transmits the signal

$ -\beta \alpha_{\mathrm{TC}} \alpha_{\mathrm{CR}} x+z $

where z is the information signal to be sent by PT. PR receives signal

$ r_{\mathrm{R}}=\alpha_{\mathrm{TR}} z+\beta \alpha_{\mathrm{CR}} \alpha_{\mathrm{RC}} y+\beta \alpha_{\mathrm{CR}} n_{\mathrm{C}}+n_{\mathrm{R}} $

Note that the term associated with x has vanished, i.e., PR is not affected by the artificial noise.

EA receives signal

$ \begin{array}{c} r_{\mathrm{E}, 2}=\alpha_{\mathrm{TE}} z+\beta\left(\alpha_{\mathrm{TR}} \alpha_{\mathrm{TC}} \alpha_{\mathrm{CE}}-\beta \alpha_{\mathrm{TE}} \alpha_{\mathrm{TC}} \alpha_{\mathrm{CR}}\right) x+ \\ \beta \alpha_{\mathrm{RC}} \alpha_{\mathrm{CE}} y+\beta \alpha_{\mathrm{CE}} n_{\mathrm{C}}+e_{2} \end{array} $

n_R and e₂ are i.i.d. AWGN random variables. Note that the term associated with x is artificial noise and non-zero. The proposed scheme is effective in spite of EA's location being unknown.

In the model proposed by Lu et al.^[11], the jamming signal was sent out by PT instead of CR. The latter only relayed the former's message to PR. In Refs. [12-13], the authors aimed to achieve higher secrecy performance for PU by optimizing CR's schedule and power allocation. Our objective is to achieve higher energy efficiency for CR. Xu and Li^[14] have considered optimal scheduling and power allocation among multiple CRs, but not CR's energy efficiency. Yan et al.^[15] have considered CR as the victim of eavesdropping. In this work, PR is the victim of eavesdropping.

Although energy efficiency has been a concern in CR networks^[16-19], the issue has not been considered in the context of CR consuming energy in order to jam the EA. To attain higher energy efficiency, Pei et al.^[16] optimized the times of transmitting and sensing, and the order of sensing the channels. To minimize energy consumption, Deng et al.^[17] proposed an optimal schedule for spectrum sensing. Xie et al.^[18] proposed an energy-efficient resource allocation method in a heterogeneous CR network architecture. The CR network structure considered in Ref. [19] is also different. It consists of a cognitive base station that allocates frequency resources to the CRs. Scheduling of such allocation was analyzed, with the objective of reaching maximum energy efficiency. Liu et al.^[20] has considered the joint issue of spectrum sharing and energy efficiency of PU but not that of CR.

2 Spectrum Sensing and Energy Efficiency

In this section, the relationship between energy efficiency and the performance of spectrum sensing is formulated. For tractable analysis, energy detector is chosen to serve the purpose of spectrum sensing. Urkowitz was the first to present the theory of energy detection^[21], then Digham et al.^[22] extended the theory. Ghasemi and Sousa^[23] applied the theory in spectrum sensing. Now in this study, a T-second frame is considered. CR spends t seconds for sensing and the sampling rate is f_s. The total number of collected energy samples N=f_st. The central limit theorem applies when there is a large number of energy samples^[24]. At the CR receiver, the ratio of PT signal power to noise power is denoted by SNR_P. Given a target probability of detection Pr_D, the probability of false alarm Pr_F can be expressed as follows^[24]:

$ \operatorname{Pr}_{\mathrm{F}}=Q\left(\left[Q^{-1}\left(\operatorname{Pr}_{\mathrm{D}}\right) \sqrt{1+2 \mathrm{SNR}_{\mathrm{P}}}+\mathrm{SNR}_{\mathrm{P}} \sqrt{\frac{f_{s} T \tau}{2}}\right]\right) $

(1)

Q(·) is the standard Gaussian complementary cumulative distribution function (CDF), whereas Q^-1(·) is its inverse.

Let C denote the channel capacity (normalized by channel bandwidth) of CR when PT relinquishes the channel. CR's channel capacity is assumed negligible when PT acquires the channel and is actively transmitting. At the CR receiver, the CR's transmitted signal to noise power ratio is denoted by SNR_CR. C is given by

$ C=\log _{2}\left(1+\mathrm{SNR}_{\mathrm{CR}}\right) $

(2)

Let Pr₁ and Pr₀ denote the respective probabilities of PT actively transmitting and being idle. When PT is idle, CR achieves the channel throughput^[24] that is given by

$ R=(1-\tau) \operatorname{Pr}_{0}\left(1-\operatorname{Pr}_{\mathrm{F}}\right) C $

(3)

where τ=t/T.

The probability of CR perceiving PT to be idle is denoted by Pr₍₀₎. The bracket in the subscript emphasizes that it is only the CR's perception, which may not be the truth. Two conditions contribute towards Pr₍₀₎: 1) PT is idle and there is no false alarm, 2) PT actively transmits but CR does not detect it^[25]. Hence,

$ \operatorname{Pr}_{(0)}=\operatorname{Pr}_{0}\left(1-\operatorname{Pr}_{\mathrm{F}}\right)+\operatorname{Pr}_{1}\left(1-\operatorname{Pr}_{\mathrm{D}}\right) $

(4)

Suppose CR allocates P_S watts for spectrum sensing and P_T watts for transmitting its data. Sensing usually consumes lower power than data transmission. Intuitively, P_S increases with f_s. To relate the magnitudes between P_S and P_T, the following simple model is proposed:

$ P_{\mathrm{S}}=k \cdot\left(f_{s}\right) \cdot P_{\mathrm{T}} $

(5)

where 0 < k·(f_s) < 1, k is a constant, f_s is in the unit of MHz.

Here P_S is assumed to increase linearly with f_s in MHz.

Let E_T and E_S denote the respective energies CR consumes in transmitting data and sensing. Then (as in Ref. [25]),

$ E_{\mathrm{S}}=P_{\mathrm{S}} t $

(6)

$ E_{\mathrm{T}}=P_{\mathrm{T}}(T-t) \operatorname{Pr}_{(0)} $

(7)

Once CR senses an actively transmitting PT, the former relinquishes the channel. CR then sends out artificial noise to EA to garble its reception. It is assumed that EA is always active. Pr₍₁₎ = 1-Pr₍₀₎ denotes the probability of friendly jamming. Let P_J and E_J denote the respective power and energy consumed for jamming.

$ E_{\mathrm{J}}=P_{\mathrm{J}}(T-t) \operatorname{Pr}_{(1)} $

(8)

The total energy that the CR consumes is E= E_S +E_T + E_J.

Being defined as the ratio of channel throughput to consumed energy^{[16, 25]}, energy efficiency is given by

$ \mu=\frac{R}{E} $

(9)

After some steps, it is found that

$ \mu=\frac{C / T}{\frac{P_{\mathrm{S}}}{\tau^{-1}-1}+\frac{\left(P_{\mathrm{T}}-P_{\mathrm{J}}\right) \operatorname{Pr}_{1}\left(1-\operatorname{Pr}_{\mathrm{D}}\right)+P_{\mathrm{J}}}{\operatorname{Pr}_{0}\left(1-\operatorname{Pr}_{\mathrm{F}}\right)}+P_{\mathrm{T}}-P_{\mathrm{J}}} $

(10)

For maximum μ, $ \frac{\mathrm{d} \mu}{\mathrm{d} \tau}=0$.The following expression is obtained, which cannot be further simplified:

$ \begin{array}{l} \frac{{\left[ {{{\Pr }_1}\left( {1 - {{\Pr }_{\rm{D}}}} \right)\left( {\frac{{{P_{\rm{J}}}}}{{{P_{\rm{T}}}}} - 1} \right) + \frac{{{P_{\rm{J}}}}}{{{P_{\rm{T}}}}}} \right]}}{{\left( {1 - {{\Pr }_1}} \right){{[1 - Q(A + B\sqrt \tau )]}^2}(2\sqrt {{\rm{ \mathsf{ π} }}\tau } )}} \cdot \\ \frac{B}{{\exp \left[ {{{(A + B\sqrt \tau )}^2}/2} \right]}} - \frac{{\frac{{{P_{\rm{S}}}}}{{{P_{\rm{T}}}}}}}{{{{(1 - \tau )}^2}}} = 0 \end{array} $

(11)

where $ A=Q^{-1}\left(\operatorname{Pr}_{\mathrm{D}}\right) \sqrt{1+2 \mathrm{SNR}_{\mathrm{P}}}, B=\mathrm{SNR}_{\mathrm{P}} \sqrt{\frac{f_{s} T}{2}}$.

Eq. (11) is a novel expression that allows to find the τ value that optimizes μ(i.e., τ^*). It reveals that the optimal value depends only on the following parameters:

$ \frac{P_{\mathrm{J}}}{P_{\mathrm{T}}}, \frac{P_{\mathrm{S}}}{P_{\mathrm{T}}}, \operatorname{Pr}_{1}, \operatorname{Pr}_{\mathrm{D}}, \mathrm{SNR}_{\mathrm{P}}, f_{s} $

Note that Eq. (11) does not depend on SNR_CR, i.e., SNR_CR does not affect the choice of τ, despite the fact that a strong CR signal will result in high μ^*(i.e., the maximum value of μ). By using numerical method, Eq. (11) can be solved. Subsequently, the sets of (τ^*, μ^*) values that vary with these parameters will be determined.

3 Results and Discussions

Here the relationships between the (τ^*, μ^*) pair and the parameters of $ \frac{P_{\mathrm{J}}}{P_{\mathrm{T}}}, \frac{P_{\mathrm{S}}}{P_{\mathrm{T}}}$, Pr₁, Pr_D, SNR_P, f_s are investigated and reported. To ease referencing, the symbols used, their respective definitions, and their respective default values are listed in Table 1. All the results that follow are generated using the values presented in Table 1, unless specified otherwise.

First, the changes of (τ^*, μ^*) with respect to P_J/P_T are investigated, which vary from 0.2 to 0.7, while P_T remains at 1 W.

From Fig. 4, it can be observed that 1) μ^* decreases linearly with the ratio P_J/P_T; as more power being consumed for jamming EA, the optimal energy efficiency naturally reduces; 2) τ^* increases only a little with P_J/P_T, which shows that τ^* is nearly independent of P_J/P_T.

Now the changes of (τ^*, μ^*) are investigated with respect to P_S/P_T, which varies from 0.1 to 1, while P_T remains at 1 W.

From Fig. 5, it can be observed that 1) μ^* decreases with the ratio P_S/P_T; as more power is consumed for sensing, the optimal energy efficiency is reduced; 2) τ^* also decreases with P_S/P_T; as more power is allocated for spectrum sensing, smaller duration of sensing is needed.

Next, the change of (τ^*, μ^*) is investigated with respect to Pr₁, which varies from 0.01 to 0.99.

From Fig. 6, it can be observed that 1) μ^* decreases significantly with Pr₁; when it becomes more likely for PU to be present and active, there is less chance for the CR to transmit its data, resulting in lower optimal energy efficiency; 2) τ^* increases with Pr₁; when it is more probable for PT to be transmitting, CR needs to allocate a larger portion of a frame for accurately sensing the presence and activity of PU.

τ^* and μ^* change with the target Pr_D, which varies from 0.6 to 0.99. Fig. 7 shows the result.

From Fig. 7, it can be observed that 1) μ^* initially increases and later decreases with Pr_D, both in a very small extent, which means that Pr_D does not have any significant impact on μ^* as long as τ^* varies accordingly; 2) τ^* increases with Pr_D; a more demanding Pr_D requires higher τ^* in order to attain the maximal energy efficiency μ^*.

How τ^* and μ^* change with the SNR_P measured at the CR is also assessed. Fig. 8 shows the result.

From Fig. 8, the following trends can be observed:

1) μ^* increases with SNR_P, but the rate of increment slows down for SNR_P > -15 dB. Higher SNR_P makes detection more accurate. CR can make more accurate judgment on PT's activity. This translates into higher throughput and higher μ^*.

2) τ^* decreases with SNR_P. When the signal from PT is weak (e.g., SNR_P = -22.5 dB), the CR requires large sensing duration to achieve the target detection accuracy. Given the same target detection accuracy, only short sensing duration is required when PT's signal is powerful.

Next, how the changes in sampling rate f_s affect τ^* and μ^* are evaluated. Fig. 9 records the result.

The following trends are observed in Fig. 9:

1) μ^* increases little with f_s, whereas τ^* decreases with f_s.

2) The result shows that increasing the sampling rate does not significantly improve the optimal energy efficiency, other than requiring a shorter sensing duration to reach the same detection accuracy. Based on this, this study argues that a high sampling rate may not be necessary.

All the findings under the given circumstances are summarized as follows:

1) τ^* and μ^* are functions of $ \frac{P_{\mathrm{J}}}{P_{\mathrm{T}}}, \frac{P_{\mathrm{S}}}{P_{\mathrm{T}}}$, Pr₁, Pr_D, SNR_P, f_s, but not SNR_CR;

2) μ^* decreases with P_J/P_T, P_S/P_T, Pr₁, but increases with SNR_P. On the other hand, Pr_D and f_s have little effect on μ^*;

3) τ^* decreases with P_S/P_T, SNR_P, f_s, but increases with Pr₁ and Pr_D. On the other hand, P_J/P_T has little effect on τ^*.

4 Conclusions

This study proposes a spectrum sharing model that counters eavesdropping. The CR jams an eavesdropper in return for the opportunity of spectrum access. CR's energy consumption in friendly jamming and data transmission becomes a concern. Attaining the optimal energy efficiency of CR (μ^*) is the main objective of the study here. The study's primary contribution is a novel mathematical expression that allows us to find the (τ^*, μ^*) pair of values in the given scenario. Following that, the relationships between the (τ^*, μ^*) pair and six different important system parameters have been analyzed. It has been found that a strong PT's signal allows a short sensing duration and it improves μ^*, but there is a limit on the amount of improvement. Increasing the sampling rate of CR may not be worthy since it does not improve μ^* significantly. A more demanding target probability of detection requires a larger sensing duration, but it has insignificant impact on μ^*. In this work, the security threat of eavesdropping is considered. Our future work will consider the security threat of hostile jamming. A possible solution is to require CR to relay PT's data to PR, creating a diversity of information.