Fusion Schemes Based on IRS-Enhanced Cooperative Spectrum Sensing for Cognitive Radio Networks

Abstract

The detection performance of cooperative spectrum sensing (CSS) is poor when there are obstacles blocking. Therefore, fusion schemes based on intelligent reflecting surface (IRS)-enhanced CSS are investigated. Existing fusion schemes can be divided into the soft combination and the hard combination. In the soft combination, each secondary user (SU) uploads the decision statistic to a fusion center (FC). Comparatively, during the hard combination, each SU reports the local 0/1 decision result to FC instead of decision statistic. In this paper, the equal gain combination (EGC) and the selection combination (SC) schemes are studied for the soft combination. The weighted hard combination (WHC) scheme is studied for the hard combination, and an optimal set combination (OSC) scheme is proposed based on signal-to-noise ratio (SNR). By using the definition of the non-centrality chi-square distribution and the moment-matching method, the closed-form expressions for the average probability of detection of all the schemes are derived. Simulation results reveal that the detection performance of IRS-enhanced CSS obviously outperforms the decode and forward (DF) relay case and the no IRS case. In addition, our proposed OSC are superior to the K-rank in terms of the detection performance.

1. Introduction

With the rapid development of wireless devices in recent years, the demand for spectrum resources is increasing. Today, most countries adopt the fixed spectrum assignment policy. Although it ensures the stable development of communication system, the spectrum utilization is poor [1]. Cognitive radio (CR) was proposed to tackle this issue [2]. Without interfering with the primary user (PU), CR can independently recognize and analyze the radio environment, and find and utilize the spectrum holes.

In a CR network (CRN), spectrum sensing is a critical front-end technology. It is utilized to observe the spectrum occupancy status and find the spectrum holes. There are three main ways for a single user to sense the spectrum, namely energy detection, matched filter detection, and cyclostationary feature detection [3]. Energy detection has high study value in the spectrum sensing since it is a type of blind sensing technique and has the lowest complexity. However, one SU cannot sense the PU accurately due to fading, shadowing and other effects. CSS was considered as one of the solutions to these problems.

In the CSS, local sensing information of all SUs are sent to the FC. FC obtains the final decision result by using a certain fusion scheme. The selection of fusion schemes will directly affect the detection performance of the CSS. The existing fusion schemes are mainly divided into the soft combination and the hard combination [4]. In the soft combination, each SU reports the decision statistic to FC. Comparatively, In the hard combination, each SU reports the local 0/1 decision result. The classic soft combination schemes are SC, EGC, and maximum ratio combination (MRC). The performance of the EGC is very close to the MRC, and it is simple to implement [5]. Therefore, this paper will not focus on the MRC scheme. The traditional hard schemes are AND, OR, and K-rank. The K-rank is an extension of the AND and the OR, so it has better applicability and higher research value [6].

In recent years, many works have been completed on the fusion schemes in the CSS. The authors mainly focused on the threshold of the fusion scheme in [7,8,9]. Adaptive threshold or multi-threshold was adopted to alleviate the poor detection performance of single fixed threshold under low SNR, but the cost of these schemes is much greater bandwidth and energy consumption. In [10,11], the authors combined weighting or node filtration with the hard combination to weaken the bad influence of low SNR SUs. In [5,12], the authors summarized the performance of the traditional hard and soft combination. However, the hard combination in their study was too simple, and none of the fusion schemes derived the closed-form expressions for the average probability of detection.

However, when there are obstacles blocking, the receiving SNR of all SUs is low. At this time, any improvements of the fusion scheme are hard to obtain the ideal detection performance. Recently, IRS has been envisioned and drawn significant attention. It can significantly enhance the signal strength for the following reasons: (1) IRS is a planar surface with massive passive reflecting elements. The amplitude and/or phase of the incident signal can be adjusted by controlling different elements [13]; (2) proper deployment of the IRS can help create virtual line-of-sight (LoS) links between the PU and SUs [13]; (3) IRS is able to achieve passive reflection without incurring any noise amplification [14]. Therefore, IRS-assisted can significantly improve the SU’s perception of weak signals.

Currently, the study of IRS in spectrum sensing mainly focuses on the field of single SU system [15,16,17]. The authors in [15] exploited the large aperture and passive beamforming gains of IRS to boost the PU signal strength. The authors in [16] configured the IRS as a primary receiver to improve the detection performance. In [17], the authors have proposed an IRS-enhanced energy detection, but it mainly focuses on the single SU case. To the best of our knowledge, the study on IRS-enhanced CSS is still relatively scarce, and the detection performance of the related systems is unknown, so it is urgent to analyze and provide necessary theoretical guidance for actual design.

Thus, this paper studies fusion schemes based on IRS-enhanced CSS. The main contributions of this paper are summarized as follows.

• The fusion schemes in IRS-enhanced CSS are analyzed. The EGC and the SC schemes for the soft combination are studied. The WHC scheme and the OSC schemes for the hard combination are investigated;
• An OSC scheme is proposed, which forms an optimal set based on each SU’s SNR to weaken the bad influence of low SNR SUs. The adjustable threshold makes the OSC have good applicability;
• Each scheme considers both cases where there are or not direct links between the PU and all SUs. The closed-form expressions for the average probability of detection of all the schemes in both the cases are obtained;
• Simulation results show that the IRS-enhanced CSS is superior to the benchmark schemes in terms of the detection performance. In addition, the detection performance of the OSC outperforms the K-rank.

The remaining parts are organized as follows. The system model of IRS-enhanced CSS is described in Section 2. Section 3 studies the EGC and the SC schemes for the soft combination. Section 4 studies the WHC and the OSC schemes for the hard combination. The simulation results are provided in Section 5. Finally, the conclusion is presented in Section 6.

Notation 1.

${\chi }_{\alpha }^2$ denotes the centrality chi-square distribution with α degree of freedom. ${\chi }_{\alpha }^2\left(v\right)$ denotes the non-centrality chi-square distribution with non-centrality parameter v and α degree of freedom. $N\left(\mu ,{\sigma }^2\right)$ denotes the normal distribution with μ mean and ${\sigma }^2$ variance. $Rayleigh\left(b\right)$ denotes the Rayleigh distribution with scale parameter b. $Gamma\left(a,b\right)$ denotes the Gamma distribution with shape parameter a and scale parameter b. $exponential\left(c\right)$ denotes the Exponential distribution with mean waiting time c. $p\left(\cdot \right)$ denotes the probability density function (PDF). $E\left(\cdot \right)$ denotes the expectation of a random variable. $Var\left(\cdot \right)$ denotes the variance of a random variable.

2. System Model

As shown in Figure 1, an IRS-enhanced CSS is considered in a CRN. The network consists of an IRS with $M\times L$ passive reflective elements, a PU, a FC, and L SUs. The elements in the IRS are divided into L groups and each SU is served by M elements. Two cases are considered with or without direct links between the PU and the L SUs.

The channels from the PU to the k-th element in the i-th group, from the k-th element in the i-th group to the i-th SU (SU_i) and from the PU to the SU_i are denoted by $h_{k,i}$, $g_{k,i}$, and $v_i$, respectively, where $k\in \{1,···,M\}$, $i\in \{1,···,L\}$. Since the channels between different elements and the PU are almost the same, the channel differences between different groups and the PU can be ignored, i.e., $h_{k,i}=h_k$. Rayleigh fading is used to model the channels. Thus, the corresponding channel-envelope $\left|h_k\right|\sim Rayleigh\left(\sqrt{{\beta }_h}\right)$, $\left|g_{k,i}\right|\sim Rayleigh\left(\sqrt{{\beta }_{g,i}}\right)$, $\left|v_i\right|\sim Rayleigh\left(\sqrt{{\beta }_{v,i}}\right)$, where ${\beta }_h$, ${\beta }_{g,i}$ and ${\beta }_{v,i}$ are the path-loss coefficients.

Energy detection is adopted for each SU. Then, the SU_i reports the decision statistic $Y_i$ or the 0/1 decision result $r_i$ to a FC via the control channel. Assume that all control channels are ideal, i.e., the sensing information obtained by FC for each SU is accurate. Finally, FC implements a specific fusion scheme depending on the sensing information received.

In order to study the IRS-enhanced CSS, we require the distribution information of a single SU energy detector. A binary hypothesis can be used to model the single SU system. Thus, the received signal of the SU_i can be described as below

$$H_0:y_i=n,$$

(1a)

$$H_1:\{\begin{array}{cc}{\mathrm{y}}_i=\sqrt{P}{\displaystyle \sum _{k=1}^M}{h}_k\exp \left(j{\phi }_k\right)g_{k,i}x+n\hfill & \left(1\mathrm{b}\right)\\ {\mathrm{y}}_i=\sqrt{P}{\displaystyle \sum _{k=1}^M}{h}_k\exp \left(j{\phi }_k\right)g_{k,i}x+\sqrt{P}{v}_{i}x+n\hfill & \left(1\mathrm{c}\right)\end{array}$$

where x is the primary signal, and $y_i$ is the received signal of the $\mathrm{S}{\mathrm{U}}_i$. The $H_0$ denotes the absence of the PU, and n is the additive white Gaussian noise (AWGN) with zero mean and variance ${\sigma }^2$. The $H_1^{}$ denotes the presence of the PU, and P is its transmit power. The expression (1b) presents the case where there is not a direct link between the PU and the SU_i, and the expression (1c) presents otherwise. The ${\phi }_k\in \left[0,2\pi \right)$ denotes the phase adjustment coefficient of the k-th element. The M denotes the number of elements in each group.

The $\mathrm{S}{\mathrm{U}}_i$ processes the received signal $y_i$ by detecting its energy, and then obtains the decision statistic $Y_i$. Figure 2 shows the energy detection process. The decision statistic $Y_i$ follows

$$Y_i\sim \left\{\begin{array}{cc}{\chi }_{2u}^2,\hfill & H_0^{},\hfill \\ {\chi }_{2u}^2\left(2{\gamma }_i^{}\right),\hfill & H_1^{},\hfill \end{array}\right.$$

(2)

where $u=BT$ is the time-bandwidth product, ${\gamma }_i$ is the instantaneous received SNR of the $\mathrm{S}{\mathrm{U}}_i$. According to [13], the optimal phase shift of the k-th element in IRS ${\phi }_k^{*}$ is given by ${\phi }_k^{*}=-\left({\omega }_k+{\varpi }_{k,i}\right)$, where $k\in \{1,···,M\}$, ${\omega }_k$ and ${\varpi }_{k,i}$ are the channel phases of $h_k$ and $g_{k,i}$, respectively. In both the cases, the expressions of maximum received SNR ${\gamma }_i^{*}$ can be given as

$$\begin{array}{cc}\hfill & {\gamma }_{i,withdirectlinks}^{*}={\gamma }_0^{}{\left|W_i^{}+\left|v_i\right|\right|}_{}^2,\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & {\gamma }_{i,withoutdirectlinks}^{*}={\gamma }_0{W}_i^2,\hfill \end{array}$$

(3b)

where ${\gamma }_0=P/{\sigma }^2$ is the transmit SNR, $W_i={\displaystyle \sum _{k=1}^M}\left|h_k\right|\left|g_{k,i}\right|$. Since there are many elements in the IRS [13], using the central limit theorem [18], it can be obtained that $W_i\sim N\left({\mu }_{W,i},{\sigma }_{W,i}^2\right)$, where ${\mu }_{W,i}=M\pi$$\sqrt{{\beta }_h{\beta }_{g,i}}/2$, ${\sigma }_{W,i}^2=M{\beta }_h{\beta }_{g,i}\left(4-{\pi }^2/4\right)$.

The expressions for the probability of false alarm ($P_{f,i}$), the probability of detection ($P_{d,i}$) and the average probability of detection (${\overline{P}}_{d,i}$) can be given by

$$\begin{array}{cc}\hfill & P_{f,i}^{}=P\left(\left.Y>{\lambda }_i\right|H_0^{}\right)=\frac{\Gamma (u,{\lambda }_i/2)}{\Gamma \left(u\right)},\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill & P_{d,i}^{}=P\left(\left.Y>{\lambda }_i\right|H_1^{}\right)=Q_u^{}\left(\sqrt{2{\gamma }_i},\sqrt{{\lambda }_i}\right),\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,i}=\underset{0}{\overset{\infty }{\int }}{Q}_u^{}\left(\sqrt{2{\gamma }_i},\sqrt{{\lambda }_i}\right)p\left({\gamma }_i\right)d{\gamma }_i,\hfill \end{array}$$

(4c)

where ${\lambda }_i$ is the decision threshold of the $\mathrm{S}{\mathrm{U}}_i$, $\Gamma (·)$ is Gamma function, $\Gamma (·,·)$ is upper incomplete Gamma function, $Q_u^{}(·,·)$ is the u-th order Marcum Q-function.

3. IRS-Enhanced Soft Combination

In the soft combination, the SU_i reports the decision statistic $Y_i$ to the FC via the control channel. For simplicity, Rayleigh channels between the IRS and the SUs, the PU and the SUs are considered to be independent and identically distributed (IID), i.e., ${\beta }_{g,i}={\beta }_g$, ${\beta }_{v,i}={\beta }_v$. FC obtains the final decision statistic Y by a weighted combination of $Y_i$, and then compares it with an appropriate decision threshold $\lambda$. The expression of Y can be written as

$$Y=\sum _{i=0}^L{w}_i^{}{Y}_i^{}\left\{\begin{array}{cc}\ge {\lambda }_{}{{}_{,}}_{}\hfill & \hfill H_1,\\ <{\lambda }_{}{{}_{,}}_{}\hfill & \hfill H_0,\end{array}\right.$$

(5)

where $w_i^{}\ge 0$ is the weight factor of the $\mathrm{S}{\mathrm{U}}_i$. Without loss of generality, we suppose ${\displaystyle \sum _{i=0}^L}∥w∥=1$.

3.1. Equal Gain Combination

In the EGC scheme, the weight factors are evenly distributed, i.e., $w_i=1/\sqrt{M}.$ Thus, the expression of the decision statistic $Y^{EGC}$ can be written as

$$Y^{EGC}=\frac{1}{\sqrt{L}}\sum _{i=0}^L{Y}_i^{}\left\{\begin{array}{cc}\ge {\lambda }_{}{{}_{,}}_{}\hfill & \hfill H_1,\\ <{\lambda }_{}{{}_{,}}_{}\hfill & \hfill H_0.\end{array}\right.$$

(6)

Since $\left\{Y_i\right|i=1,···,L\}$ are IID, it can be deduced that $Y^{EGC}$ follows

$$\sqrt{L}{Y}^{EGC}\sim \left\{\begin{array}{cc}{\chi }_{2Lu}^2,\hfill & \hfill H_0^{},\\ {\chi }_{2Lu}^2\left(2\sum _i^L{\gamma }_i^{}\right),\hfill & \hfill H_1^{}.\end{array}\right.$$

(7)

According to (4a)–(4c), the probability of false alarm ($P_f^{EGC}$) and the average probability of detection (${\overline{P}}_d^{EGC}$) can be obtained similarly. Their expressions can be written as

$$P_f^{EGC}=\frac{\Gamma (Lu,\epsilon /2)}{\Gamma \left(Lu\right)},$$

(8)

$$\begin{array}{cc}\hfill & {\overline{P}}_d^{EGC}=\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}···\underset{0}{\overset{\infty }{\int }}{Q}_{Lu}^{}\left(\sqrt{2\sum _{i=0}^L{\gamma }_i^{}},\sqrt{\epsilon }\right)\prod _{i=0}^{L}p\left({\gamma }_i^{}\right)d{\gamma }_1^{}d{\gamma }_2^{}···d{\gamma }_L^{},\hfill \end{array}$$

(9)

where $\epsilon =\sqrt[]{L}\lambda$. It is clear that $P_f^{EGC}$ is independent of ${\gamma }_i$, while ${\overline{P}}_d^{EGC}$ is related to ${\gamma }_i$. It can be seen from (9) that ${\overline{P}}_d^{EGC}$ is the complex multiple integral of the variable ${\displaystyle \sum _{i=0}^L}{\gamma }_i^{}$. To solve it, it is converted into a definite integral. If $\gamma ={\displaystyle \sum _{i=0}^L}{\gamma }_i^{}$ is defined, the expression (9) can be rewritten as

$${\overline{P}}_d^{EGC}=\underset{0}{\overset{\infty }{\int }}{Q}_{Lu}^{}\left(\sqrt{2\gamma },\sqrt{\epsilon }\right)p_L^{}\left(\gamma \right)d\gamma ,$$

(10)

where $p_L^{}\left(\gamma \right)$ is the PDF of $\gamma$.

3.1.1. EGC without Direct Links

This subsection analyzes the case where there are not direct links. Since ${\beta }_{g,i}={\beta }_g$ and ${\beta }_{v,i}={\beta }_v$, it can be deduced that $W_i=W\sim N\left({\mu }_W,{\sigma }_W^2\right)$, where ${\mu }_W=M\pi \sqrt{{\beta }_h{\beta }_g}/2$, ${\sigma }_W^2=M{\beta }_h{\beta }_g\left(4-{\pi }^2/4\right)$. Thus, it can be seen from (3b) that ${\gamma }_i^{*}={\gamma }_0{W}^2\sim {\chi }_1^2\left({\gamma }_0{\mu }_W^2\right)$. Since $\gamma ={\displaystyle \sum _{i=0}^L}{\gamma }_i$, according to the definition of the non-central chi-square distribution [19], the distribution of $\gamma$ can be expressed as $\gamma \sim {\chi }_L^2\left(L{\gamma }_0{\mu }_W^2\right)$. Thus, its PDF can be given as

$$\begin{array}{cc}\hfill & p_L\left(\gamma \right)=\frac{1}{2{\gamma }_0^{}{\sigma }_W^2}{\left(\frac{\gamma }{L{\gamma }_0^{}{\mu }_W^2}\right)}^{\frac{L-2}{4}}exp\left(-\frac{\gamma +L{\gamma }_0^{}{\mu }_W^2}{2{\gamma }_0^{}{\sigma }_W^2}\right)I_{\frac{L}{2}-1}\left(\frac{{\mu }_W\sqrt{L\gamma }}{\sqrt{{\gamma }_0}{\sigma }_W^2}\right),\hfill \end{array}$$

(11)

where $I_k^{}\left(\cdot \right)$ is the first kind kth-order Bessel function. Then, the alternative series representation of Marcum-Q function can be given as [19]

$$Q_{Lu}^{}\left(\sqrt{2\gamma },\sqrt{\epsilon }\right)={\displaystyle \sum _{n=0}^{\infty }}\frac{{\gamma }^{n}exp\left(-\gamma \right)}{n!}{\displaystyle \sum _{k=0}^{n+Lu-1}}\frac{exp\left(-\frac{\epsilon }{2}\right)}{k!}{\left(\frac{\epsilon }{2}\right)}_{}^k.$$

(12)

Substituting (11) and (12) into (10), the expression for the average probability of detection can be written as

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withoutdirectlinks}^{EGC}=\eta {\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\underset{0}{\overset{\infty }{\int }}{\gamma }_{}^{n+\frac{L-2}{4}}exp\left(-\gamma -\frac{\gamma }{2{\gamma }_0^{}{\sigma }_W^2}\right)\hfill \\ \hfill & \times I_{\frac{L-2}{2}}^{}\left(\frac{{\mu }_W^{}\sqrt{L}}{\sqrt{{\gamma }_0^{}}{\sigma }_W^2}\sqrt{\gamma }\right)d\gamma {\displaystyle \sum _{k=0}^{n+Lu-1}}\frac{1}{k!}{\left(\frac{\epsilon }{2}\right)}_{}^k,\hfill \end{array}$$

(13)

where $\eta =exp\left(-\frac{L{\mu }_W^2}{2{\sigma }_W^2}-\frac{\epsilon }{2}\right){\left(L{\gamma }_0^{}{\mu }_W^2\right)}_{}^{-\frac{L-2}{4}}/2{\gamma }_0^{}{\sigma }_W^2.$ First, use the expression [20]

$$\begin{array}{cc}\hfill & \underset{0}{\overset{\infty }{\int }}{x}_{}^{u-\frac{1}{2}}{e}_{}^{-\alpha x}{I}_{2v}^{}\left(2\beta \sqrt{x}\right)dx=\frac{\Gamma \left(u+v+\frac{1}{2}\right)}{\Gamma \left(2v+1\right)}{\beta }_{}^{-1}{e}_{}^{\frac{{\beta }_{}^2}{2\alpha }}{\alpha }_{}^{-u}{M}_{-u,v}^{}\left(\frac{{\beta }_{}^2}{\alpha }\right),\hfill \end{array}$$

(14)

where $M_{a,b}^{}\left(\cdot \right)$ is the Whittaker function. Then, combine with the relation as follows [20]

$$M_{-u,v}^{}\left(z\right)=z_{}^{v+\frac{1}{2}}{e}_{}^{-\frac{z}{2}}{}_1{F}_1^{}\left(v+u+\frac{1}{2};2v+1;z\right),$$

(15)

where ${}_1{F}_1\left({·}_{}{;}_{}{·}_{}{;}_{}·\right)$ is the first class Confluent Hypergeometric Function. Finally, the closed-form of ${\overline{P}}_{d,withoutdirectlinks}^{EGC}$ can be given by

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withoutdirectlinks}^{EGC}=\varsigma {{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\Gamma \left(n+\frac{L}{2}\right){\left(1-\frac{1}{2{\gamma }_0^{}{\sigma }_W^2}\right)}^n}_{}\hfill \\ \hfill & \times {}_1{F}_1\left(n+\frac{L}{2};\frac{L}{2};\frac{L{\mu }_W^2}{2{\sigma }_W^2\left(2{\gamma }_0^{}{\sigma }_W^2+1\right)}\right){\displaystyle \sum _{\mathrm{k}=0}^{n+Lu-1}}\frac{1}{k!}{\left(\frac{\epsilon }{2}\right)}_{}^k,\hfill \end{array}$$

(16)

where $\varsigma =exp\left(-\left(\frac{\epsilon }{2}+\frac{L{\mu }_W^2}{2{\sigma }_W^2}\right)\right)/\left(\Gamma \left(\frac{L}{2}\right){\left(1+2{\gamma }_0^{}{\sigma }_W^2\right)}_{}^{\frac{L}{2}}\right).$

3.1.2. EGC with Direct Links

This subsection analyzes the case where there are direct links. Since $W_i=W$, the expression (3a) can be rewritten as

$${\gamma }_i^{*}={\gamma }_0^{}{\left(\underset{R}{\underbrace{W+\left|v\right|}}\right)}_{}^2={\gamma }_0^{}\left(\underset{a}{\underbrace{W_{}^2}}+\underset{b}{\underbrace{2W\left|v\right|}}+\underset{c}{\underbrace{{\left|v\right|}_{}^2}}\right),$$

(17)

where $R=W+\left|v\right|$, $a=W_{}^2$, $b=2W\left|v\right|$, $c={\left|v\right|}_{}^2$. The distribution of R can be given as $R\sim Gamma\left({\alpha }_R,{\beta }_R\right)$ [17], the expressions of ${\alpha }_R$ and ${\beta }_R$ are ${\alpha }_R={\left(\sqrt{{\beta }_v\pi }+\frac{\pi \iota }{2}\right)}^2/\left(\left(4-\pi \right){\beta }_v+\frac{o}{4}\right),$${\beta }_R=\left(4-\pi \right)\left({\beta }_v+\right.\left.\frac{M\left(\pi +4\right){\beta }_h{\beta }_g}{4}\right)/\left(\pi \iota +2\sqrt{{\beta }_v\pi }\right)$, where $\iota =M\sqrt{{\beta }_h{\beta }_g}$, $o=M\left(16-{\pi }^2\right){\beta }_h$${\beta }_g$. Thus, the PDF of ${\gamma }_i^{*}$ in this case can be obtained as:

$$\begin{array}{cc}\hfill & p_{{\gamma }_i^{*}}^{}\left(x\right)=p_R^{}\left(\sqrt{\frac{x}{{\gamma }_0}}\right){\frac{1}{2\sqrt{x{\gamma }_0}}}^{}\stackrel{}{}=\frac{{{\beta }_R}^{{\alpha }_R}}{2\Gamma \left({\alpha }_R\right){\left({\gamma }_0\right)}^{{\alpha }_R/2}}{\left(\sqrt{x}\right)}^{{\alpha }_R-2}exp\left(\frac{-{\beta }_R\sqrt{x}}{\sqrt{{\gamma }_0}}\right).\hfill \end{array}$$

(18)

It is obvious that the PDF of ${\gamma }_i^{*}$ is not known, hence the solution of $p_L\left(\gamma \right)$ is extremely difficult. Thus, the Gamma distribution (moment-matching method) can be used as the approximate distribution of ${\gamma }_i^{*}$.

There are three factors to consider: (1) the Gamma distribution belongs to the Pearson-III distribution. It is widely used to approximate the distribution of unknown positive random variables when combined with the moment-matching method [21]; (2) the PDF of b is related to the Gamma function; (3) the Gamma distribution has additivity [22]. If ${\gamma }_i^{*}$ is approximated as a Gamma distribution, the $\gamma$ can also be approximated as a Gamma distribution.

Since $W\sim N\left({\mu }_W,{\sigma }_W^2\right)$, the distribution of a can be expressed as $a\sim {\chi }_1^2\left({\mu }_W^2\right)$. Thus, $E\left(a\right)={\sigma }_W^2+{\mu }_W^2$, $Var\left(a\right)=2{\sigma }_W^4+4{\sigma }_W^2{\mu }_W^2$. Since W and $\left|v\right|$ are independent of each other, and $\left|v\right|\sim Rayleigh\sqrt{{\beta }_v})$, it can be obtained that $E\left(b\right)=2{\mu }_W^{}{\mu }_{\left|v\right|}^{}$, $Var\left(b\right)=4\rho$, where $\rho =2\left({\sigma }_W^2+{\mu }_W^2\right){\beta }_v-{\left({\mu }_W^{}{\mu }_{\left|v\right|}^{}\right)}_{}^2$, ${\mu }_{\left|v\right|}^{}=\sqrt{{\beta }_v\pi /2}$. According to the relation between the Rayleigh distribution and the Exponential distribution [22], the distribution of c can be given by $c\sim exponential\left(2{\beta }_v\right)$. Therefore, $E\left(c\right)=2{\beta }_v$, $Var\left(c\right)=$$4{\beta }_v^2$. In summary, it can be deduced that $E\left({\gamma }_i^{*}\right)={\gamma }_0^{}\left[E\left(a\right)+E\left(b\right)+E\left(c\right)\right]$, $Var\left({\gamma }_i^{*}\right)={\gamma }_0^2\left[Var\left(a\right)+\right.\left.Var\left(b\right)+Var\left(c\right)\right]$.

Then, we use the moment-matching method ($Gamma\left(k,\theta \right)$) to approximate the distribution of ${\gamma }_i^{*}$, where its k and $\theta$ can be written as

$$k=\frac{{\left\{E\left({\gamma }_i^{*}\right)\right\}}^2}{Var\left({{\gamma }^{*}}_i\right)}=\frac{{\left({\sigma }_W^2+{\mu }_W^2+2{\mu }_W^{}{\mu }_{\left|v\right|}^{}+2{\beta }_v\right)}_{}^2}{2{\sigma }_W^4+4{\sigma }_W^2{\mu }_W^2+4\rho +4{\beta }_v^2},$$

(19)

$$\theta =\frac{Var\left({\gamma }_i^{*}\right)}{E\left({\gamma }_i^{*}\right)}={\gamma }_0^{}\left(\frac{2{\sigma }_W^4+4{\sigma }_W^2{\mu }_W^2+4\rho +4{\beta }_v^2}{{\sigma }_W^2+{\mu }_W^2+2{\mu }_W^{}{\mu }_{\left|v\right|}^{}+2{\beta }_v}\right).$$

(20)

The accuracy of this approximation is shown in Figure 3.

According to the additivity of the Gamma distribution [22], the distribution of $\gamma$ can be given by $\gamma \sim Gamma\left(Lk,\theta \right)$. Thus, the PDF of $\gamma$ can be written as

$$p_L\left(\gamma \right)=\frac{{\beta }_{}^{\alpha }}{\Gamma \left(\alpha \right)}{\gamma }^{\alpha -1}exp\left(-\beta \gamma \right),$$

(21)

where $\alpha =Lk,\beta =\theta$. Integrating (12), (21) over (10), the average probability of detection can be expressed as

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withdirectlinks}^{EGC}=\frac{{\beta }_{}^{\alpha }}{\Gamma \left(\alpha \right)}{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\underset{0}{\overset{\infty }{\int }}{x}_{}^{n+\alpha -1}{e}_{}^{-\left(\beta +1\right)x}dx{\displaystyle \sum _{k=0}^{n+\mathrm{Lu}-1}}\frac{e_{}^{-\frac{\epsilon }{2}}}{\mathrm{k}!}\left(\frac{\epsilon }{2}\right){}_{}^k.\hfill \end{array}$$

(22)

Then, by using the expression [20]

$$\underset{0}{\overset{\infty }{\int }}{x}_{}^{q-1}{e}_{}^{-\psi x}dx=\frac{1}{{\psi }_{}^q}\Gamma \left(q\right),$$

(23)

the closed-form expression of ${\overline{P}}_{d,withdirectlinks}^{EGC}$ can be represented as

$${\overline{P}}_{d,withdirectlinks}^{EGC}=\tau {\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\frac{\Gamma \left(n+\alpha \right)}{{\left(\beta +1\right)}_{}^{n+\alpha }}{\displaystyle \sum _{k=0}^{n+Lu-1}}{\frac{1}{k!}\left(\frac{\epsilon }{2}\right)}_{}^k,$$

(24)

where $\tau =exp\left(-\frac{\epsilon }{2}\right)\frac{{\beta }_{}^{\alpha }}{\Gamma \left(\alpha \right)}$.

3.2. Selection Combination

In the SC scheme, FC selects the SU with the maximum SNR for decision. It is assumed that the j-th SU ($\mathrm{S}{\mathrm{U}}_j$) has the maximum SNR, where j is an arbitrary constant and $j\in \{1,···,L\}$. Thus, the weight factors can be given by

$$w_i=\left\{\begin{array}{cc}{1}_{},\hfill & \hfill i=j,\\ 0_{},\hfill & \hfill i\ne j.\end{array}\right.$$

(25)

Since the SC is a soft combination, substituting (25) into (5), the expression of decision statistic $Y^{SC}$ can be written as $Y^{SC}={\displaystyle \sum _{i=0}^L}{w}_i{Y}_i=Y_j$. It can be seen that the distribution of $Y^{SC}$ is similar to the IRS-enhanced single-user case. Therefore, similar to (2), the decision statistic $Y^{SC}$ can be expressed as

$$Y^{SC}\sim \left\{\begin{array}{cc}{\chi }_{2u}^2,\hfill & \hfill H_0^{},\\ {\chi }_{2u}^2\left(2{\gamma }_{max}\right),\hfill & \hfill H_1^{},\end{array}\right.$$

(26)

where ${\gamma }_{max}$ is the instantaneous received SNR of the $\mathrm{S}{\mathrm{U}}_j$.

Similar to (4a) and (4c), the expressions for the probability of false alarm (${\overline{P}}_f^{SC}$) and the average probability of detection (${\overline{P}}_d^{SC}$) can be written as

$$P_f^{SC}=\frac{\Gamma (u,\lambda /2)}{\Gamma \left(u\right)},$$

(27)

$${\overline{P}}_d^{SC}=\underset{0}{\overset{\infty }{\int }}{Q}_u\left(\sqrt{2{\gamma }_{max}},\sqrt{\lambda }\right)p\left({\gamma }_{max}\right)d{\gamma }_{max}.$$

(28)

Since ${\gamma }_{max}={\gamma }_j$, the ${\overline{P}}_d^{SC}$ also needs to be analyzed for the cases with or without direct links. Compared (28) with (4c), it can be found that the derivation of ${\overline{P}}_d^{SC}$ is similar to the IRS-enhanced single-user spectrum sensing. In [17], the closed-form expressions of ${\overline{P}}_d$ and $P_f$ have been obtained, considering two cases where there is or not a direct link between the PU and the SU. Thus, in the SC scheme, the average probability of detection in both the cases can be, respectively, expressed as

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withoutdirectlinks}^{SC}=\vartheta {\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\Gamma \left(n+\frac{1}{2}\right){\left(1+\frac{1}{2{\sigma }_{}^2}\right)}^{-n}\hfill \\ \hfill & \times {}_1{F}_1\left(n+\frac{1}{2};\frac{1}{2};\frac{{\mu }_W^2}{2{\sigma }_W^2\left(2{\sigma }_{}^2+1\right)}\right){\displaystyle \sum _{\mathrm{k}=0}^{n+u-1}}\frac{1}{k!}{\left(\frac{\lambda }{2}\right)}_{}^k,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withdirectlinks}^{SC}=\eta {\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}{\left(2{\gamma }_0^2\right)}^{-\frac{2\mathrm{n}+{\alpha }_R}{2}}\Gamma (2n+{\alpha }_R)exp\left(\frac{{\beta }_R}{8{\gamma }_0}\right)\hfill \\ \hfill & \times D_{-\left(2n+{\alpha }_R\right)}\left(\frac{{\beta }_R}{\sqrt{2{\gamma }_0^2}}\right){{\displaystyle \sum _{k=0}^{n+u-1}}\frac{1}{k!}\left(\frac{\lambda }{2}\right)}^k,\hfill \end{array}$$

(30)

where $\vartheta =exp\left(-\left(\frac{\lambda }{2}+\frac{{\mu }_W^2}{2{\sigma }_W^2}\right)\right)/\left(\Gamma \left(\frac{1}{2}\right){\left(1+2{\gamma }_0^{}{\sigma }_W^2\right)}_{}^{\frac{1}{2}}\right)$, $\eta =exp\left(-\frac{\lambda }{2}\right){{\beta }_R}^{{\alpha }_R}/\Gamma \left({\alpha }_R\right),$$D_{·}\left(·\right)$ is the Parabolic Cylinder Function, ${}_1{F}_1\left({·}_{}{;}_{}{·}_{}{;}_{}·\right)$ has been explained in (15).

3.3. Verification of the Approximation

Figure 3 shows the comparison of the proposed Gamma distribution (moment-matching) and the empirical PDF of ${\gamma }_i^{*}$ with direct links, where ${\gamma }_0=1\underset{}{}\mathrm{dB}$, $L=4$, $M=16$. Their probability density curves are highly coincident, indicating that the approximation is accurate.

4. IRS-Enhanced Hard Combination

In the hard combination, according to the decision rule $Y_i\underset{H_0}{\stackrel{H_1}{\begin{array}{c}\ge \\ <\end{array}}}{\lambda }_i$, the $\mathrm{S}{\mathrm{U}}_i$ makes the local decision and then obtains local 0/1 decision result $r_i$, where ${\lambda }_i$ is the pre-determined decision threshold of the SU_i. The decision strategy of the $r_i$ can be given by

$$r_i=\left\{\begin{array}{cc}0,\hfill & \hfill H_0,\\ 1,\hfill & \hfill H_1,\end{array}\right.$$

(31)

where $r_i=0$ denotes that $\mathrm{S}{\mathrm{U}}_i$’s decision as $H_0$, $r_i=1$ denotes that $\mathrm{S}{\mathrm{U}}_i$’s decision as $H_1$. Then, the $\mathrm{S}{\mathrm{U}}_i$ reports the $r_i$ to FC via the control channel. FC makes the final decision according to the certain hard combination scheme and $\left\{r_i\right|i=1,···,L\}$.

4.1. Weighted Hard Combination

The WHC scheme consists of two steps: (1) FC assigns a weight factor related to the SNR to each SU. The expression of the weight factor $p_i$ can be expressed as $p_i=L{\gamma }_i/{\displaystyle \sum _{i=1}^L}{\gamma }_i$; (2) FC combines the OR-rule with weight factors.

Thus, in the WHC scheme, the probability of false alarm ($P_f^{WHC}$) and the average probability of detection (${\overline{P}}_{\mathrm{d}}^{WHC}$) can be written as

$$P_f^{WHC}=1-\prod _{i=1}^L\left(1-P_{f,i}\right)=1-\prod _{i=1}^L\left(1-\frac{\Gamma (u,{\lambda }_i/2)}{\Gamma \left(u\right)}\right),$$

(32)

$${\overline{P}}_{\mathrm{d}}^{WHC}=1-\prod _{i=1}^L{p}_i\left(1-\overline{P_{d,i}}\right),$$

(33)

where ${\lambda }_i$ is the decision threshold of the $\mathrm{S}{\mathrm{U}}_i$. It is clear that $P_f^{WHC}$ is independent of ${\gamma }_i$ and ${\overline{P}}_{\mathrm{d}}^{WHC}$ is related to $\overline{P_{d,i}}$. Thus, ${\overline{P}}_{\mathrm{d}}^{WHC}$ also needs to be analyzed in both the cases.

Substituting (29) and (30) into (33), respectively. The closed-form expressions for the average probability of detection in two cases can be written as

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withoutdirectlinks}^{WHC}=1-\prod _{i=1}^L{p}_i\left[1-{\xi }_i{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\Gamma (n+\frac{1}{2})\right.\hfill \\ \hfill & \left.\times a_i^{-n}{}_1{F}_1\left(n+\frac{1}{2};\frac{1}{2};b_i\right){{\displaystyle \sum _{k=0}^{n+u-1}}\frac{1}{k!}\left(\frac{{\lambda }_i}{2}\right)}^k\right],\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withdirectlinks}^{WHC}=1-\prod _{i=1}^L{p}_i\left[1-{\psi }_i{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}{\left(2{\gamma }_0^2\right)}^{-\frac{2\mathrm{n}+{\alpha }_{R,i}}{2}}\right.\Gamma (2n+{\alpha }_{R,i})\hfill \\ \hfill & \left.\times exp\left(\frac{{\beta }_{R,i}}{8{\gamma }_0}\right)D_{-\left(2n+{\alpha }_{R,i}\right)}\left(\frac{{\beta }_{R,i}}{\sqrt{2{\gamma }_0^2}}\right){{\displaystyle \sum _{k=0}^{n+u-1}}\frac{1}{k!}\left(\frac{{\lambda }_i}{2}\right)}^k\right],\hfill \end{array}$$

(35)

where ${\xi }_i=exp\left(-\left(\frac{{\lambda }_i}{2}+\frac{{\mu }_{W,i}^2}{2{\sigma }_{W,i}^2}\right)\right)/\left(\Gamma \left(\frac{1}{2}\right){\left(1+2{\gamma }_0^{}{\sigma }_{W,i}^2\right)}_{}^{\frac{1}{2}}\right),$$a_i=\left(2{\gamma }_0{\sigma }_{w,i}^2+1\right)/2{\gamma }_0{\sigma }_{w,i}^2,$$b_i={\mu }_{W,i}^2/\left(2{\sigma }_{w,i}^2\left(2{\gamma }_0{\sigma }_{w,i}^2+1\right)\right),$${\psi }_i=exp\left(-\frac{{\lambda }_i}{2}\right){{\beta }_{R,i}}^{{\alpha }_{R,i}}/\Gamma \left({\alpha }_{R,i}\right).$ The expressions of ${\alpha }_{R,i}$ and ${\beta }_{R,i}$ can be given as ${\alpha }_{R,i}={\left(\sqrt{{\beta }_{v,i}\pi }+\frac{\pi {\iota }_i}{2}\right)}^2/\left(\left(4-\pi \right){\beta }_{v,i}+\frac{o_i}{4}\right),$${\beta }_{R,i}=\frac{\left(4-\pi \right)}{\left(\pi {\iota }_i+2\sqrt{{\beta }_{v,i}\pi }\right)}\times \left(\frac{M\left(\pi +4\right){\beta }_h{\beta }_{g,i}}{4}+{\beta }_{v,i}\right)$, where ${\iota }_i=M\sqrt{{\beta }_h{\beta }_{g,i}}$, $o_i=M\left(16-{\pi }^2\right){\beta }_h{\beta }_{g,i}$.

4.2. Optimal Set Combination

The OSC scheme also consists of two steps: (1) FC selects W ($W\in \{1,···,L\}$) SUs with higher SNR from all SUs to form a optimal set $\varphi$, i.e., ${\gamma }_{i\in \varphi }\ge {\gamma }_{i\notin \varphi }$. The W is defined as the threshold of the optimal set; (2) FC combines the 0/1 decision results of SUs in $\varphi$ with the OR-rule.

Therefore, in the OSC scheme, the probability of false alarm ($P_f^{OSC}$) and the average probability of detection (${\overline{P}}_d^{OSC}$) can be written as:

$$P_f^{OSC}=1-\prod _{i\in \varphi }\left(1-P_{f,i}\right)=1-\prod _{i\in \varphi }\left(1-\frac{\Gamma (u,{\lambda }_i/2)}{\Gamma \left(u\right)}\right),$$

(36)

$${\overline{P}}_d^{OSC}=1-\prod _{i\in \varphi }\left(1-\overline{P_{d,i}}\right).$$

(37)

Similar to (34) and (35), the average probability of detection in both the cases can be given by

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withoutdirectlinks}^{OSC}=1-\prod _{i\in \varphi }^{}\left[1-{\varsigma }_i{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}\Gamma (n+\frac{1}{2})\right.\hfill \\ \hfill & \left.\times a_i^{-n}{}_1{F}_1\left(n+\frac{1}{2};\frac{1}{2};b_i\right){{\displaystyle \sum _{k=0}^{n+u-1}}\frac{1}{k!}\left(\frac{{\lambda }_i}{2}\right)}^k\right],\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\overline{P}}_{d,withdirectlinks}^{OSC}=1-\prod _{i\in \varphi }^{}\left[1-{\zeta }_i{\displaystyle \sum _{n=0}^{\infty }}\frac{1}{n!}{\left(2{\gamma }_0^2\right)}^{-\frac{2\mathrm{n}+{\alpha }_{R,i}}{2}}\right.\Gamma (2n+{\alpha }_{R,i})\hfill \\ \hfill & {\left.\times exp\left(\frac{{\beta }_{R,i}}{8{\gamma }_0}\right)D_{-\left(2n+{\alpha }_{R,i}\right)}\left(\frac{{\beta }_{R,i}}{\sqrt{2{\gamma }_0^2}}\right){{\displaystyle \sum _{k=0}^{n+u-1}}\frac{1}{k!}\left(\frac{{\lambda }_i}{2}\right)}^k\right]}_{},\hfill \end{array}$$

(39)

where ${\xi }_i$, $a_i$, $b_i$, ${\psi }_i$, ${\alpha }_{R,i}$, and ${\beta }_{R,i}$ have been explained in (34) and (35).

5. Simulation Results

In this section, simulation results are given to evaluate the detection performance of all the above schemes and compare them with the benchmark schemes. The adopted parameters during the simulation are shown in the

Table 1. Simulation parameters.

Parameters	Values
Path-loss fading ${\delta }_{\mu }\left(\mu \in \left\{h,g,v\right\}\right)$	${\delta }_0{\left(d_0/d_{\mu }\right)}^k$
Reference path loss ${\delta }_0$	$43{}^{}\mathrm{dB}$
Reference distance $d_0$	$1 \mathrm{m}$
Path-loss exponent k	$2.7$
Distance between the PU and the IRS $d_h$	$110 \mathrm{m}$
Distance between the IRS and SUs $d_g$	$110 \mathrm{m}$
Distance between the PU and SUs $d_v$	$220 \mathrm{m}$
Number of SUs L	4
Number of passive reflective elements $M\times L$	$16\times 4$
Threshold of the optimal set W	2
Time-bandwidth product u	5

below for clarity.

Figure 4 shows the Receiver Operating Characteristics (ROC) curves (the probability of miss detection $P_m^{}=1-P_d^{}$ VS the probability of false alarm $P_f^{}$), which involve the EGC, the SC, the WHC, and the OSC, and compare it to the detection performance of a single SU. The transmit SNR is set as ${\gamma }_0=-2^{}\mathrm{dB}$. Figure 4a denotes the case without direct links, and Figure 4b denotes otherwise. It shows that in both cases, with the same $P_f$, the detection performance of each scheme from good to bad is WHC, OSC, SC, EGC, and single SU. Both the OSC and the SC are unaffected by SUs with low SNR, so their detection performance is better. Because the WHC is essentially a fusion of the high SNR priority criterion and the OR-rule, it outperforms the OSC. However, its applicability is poorer than that of the OSC.

Figure 5a shows three scenarios: (i) there is an IRS between the PU and the SUs; (ii) there is a DF relay between the PU and the SUsl; (iii) there is not an IRS and a relay between the PU and the SUs. The transmit SNR is set as ${\gamma }_0=-1^{}\mathrm{dB}$. Take soft combination as an example. The detection performance is compared when there are direct links. The $P_f^{}$ and $P_d^{}$ of the scenario (ii) and the scenario (iii) are given in [23,24], respectively. It shows that the IRS-enhanced CSS outright outperforms the DF relay scenario and the no IRS scenario. This is because the IRS can construct multiple passive auxiliary reflection links, which can overcome signal’s randomness and do not incur any noise amplification.

Figure 5b shows three configurations: (i) the IRS is placed at closer distances, i.e., $d_h+d_g\approx d_v$ (configuration 1, $d_h=d_h=110\mathrm{m}$, $d_v=200\mathrm{m}$); (ii) the IRS is placed at moderate distances, i.e., $d_h+d_g>d_v$ (configuration 2, $d_h=d_h=220\mathrm{m}$, $d_v=200\mathrm{m}$); (iii) the IRS is placed at far-away distances, i.e., $d_h+d_g\gg d_v$ (configuration 3, $d_h=d_h=450\mathrm{m}$, $d_v=200\mathrm{m}$). The transmit SNR is set as ${\gamma }_0=2^{}\mathrm{dB}$. Take the EGC as an example. The detection performance is compared when there are direct links. The detection performance of the IRS-enhanced CSS is hindered with increasing distance, mainly due to the large increase in distance-dependent path loss effects. It is also worth noting that the curve order for all three configurations are the same because three ROC curves are plotted by varying the distances and keeping other parameters fixed.

Figure 6a shows ROC curves for the average probability of detection (${\overline{P}}_d$) versus the number of elements in one group (M). The probability of false alarm is set as $P_f$ = 0.05. The transmit SNR of PU is set as ${\gamma }_0=-7\mathrm{dB}$. Take the EGC and the OSC as an example. If the target ${\overline{P}}_d$ is given, the minimum M can be obtained through the ROC curves in the Figure 6a. For example, when the target ${\overline{P}}_d$ = 0.78, the minimum positive integer $M=20$ (OSC, with direct links), $M=32$ (OSC, with direct links). When the target ${\overline{P}}_d$ = 0.47, the minimum positive integer $M=24$ (EGC, with direct links), $M=36$ (EGC, no direct links) (upward rounding). It can be seen from the trend of ROC curves that the higher requirement of detection performance (${\overline{P}}_d$) needs the more elements in the IRS when the other parameters are the same. This is because the more elements in the IRS, the more auxiliary reflection links can be constructed, which can better overcome the randomness of the signal.

Figure 6b shows ROC curves for the average probability of detection (${\overline{P}}_d$) versus the number of SUs (L) when given the limitation of the number of elements in the IRS. The probability of false alarm is set as $P_f$ = 0.05. The transmit SNR of PU is set as ${\gamma }_0=-5\mathrm{dB}$. The limitation of the number of elements is set as 64. Additionally, take the EGC and the OSC as an example. If the target ${\overline{P}}_d$ is given, the maximum L can be obtained through the ROC curves in the Figure 6b. For example, when the target ${\overline{P}}_d$ = 0.4, the maximum positive integer $L=5$ (EGC, no direct links), $L=7$ (EGC, with direct links), $L=9$ (OSC, no direct links), and $L=12$ (OSC, with direct links) (downward rounding). It can be seen from the trend of ROC curves that the lower requirement of detection performance (${\overline{P}}_d$) can serve the more SUs when the other parameters are the same. This is because the increase in the number of SUs means a higher detection threshold and a reduction in the number of elements at one group. In addition, it can also be seen from Figure 6a,b that the IRS-enhanced CSS has good scalability. Because it can adapt to different detection requirements by adjusting the number of elements in the IRS and the number of SUs.

Figure 7 shows ROC curves for the average probability of detection (${\overline{P}}_d$) versus the transmit SNR of PU (${\gamma }_0$) in the IRS-enhanced CSS. The probability of a false alarm is set as $P_f$ = 0.05. If the target ${\overline{P}}_d$ is given, the required minimum ${\gamma }_0$ can be obtained through the ROC curves in the Figure 7. For example, when the target ${\overline{P}}_d$ = 0.61, the required minimum ${\gamma }_0=-5.1\mathrm{dB}$ (EGC, no direct links), ${\gamma }_0=-7\mathrm{dB}$ (OSC, no direct links). When the target ${\overline{P}}_d$ = 0.52, the required minimum ${\gamma }_0=-6\mathrm{dB}$ (EGC, with direct links), and ${\gamma }_0=-8\mathrm{dB}$ (OSC, with direct links). It can be seen from the trend of ROC curves that the higher detection performance (${\overline{P}}_d$ ) requires the higher ${\gamma }_0$ when other parameters are the same. This is because both poor transmission environment and weak transmit power (P) can negatively affect the decision of the presence of PU.

Figure 8 shows the detection performance of the OSC scheme (W = 2, 3, 4) and the K-rank scheme (K = 2). The transmit SNR is set as ${\gamma }_0=-3^{}\mathrm{dB}$. The result shows that the OSC scheme can further improve the detection performance comparing to the K-rank scheme. The adjustable threshold W makes the OSC similar to the K-rank in terms of the applicability. When $W=L$, the OSC degenerates into the OR-rule. Furthermore, the detection performance of the OSC is proportionally associated with W. The performance improvement offered by increasing W from 3 to 4 is only 0.167 times the performance improvement achieved by increasing W from 2 to 3, which is cost-ineffective. The reduction in W makes the SUs with low SNR participate in cooperation, so W should not be too large.

Figure 9 shows ROC curves for the probability of detection ($P_d$) versus the received SNR (${\gamma }_i$). Take the EGC as an example (other schemes have similar analysis procedures and the same result). It can be seen from the monotonicity of the ROC curves in Figure 9 that the larger received SNR of SU means the higher probability of detection. This shows that maximizing the probability of detection is equivalent to maximizing the received SNR of SU. Therefore, the optimal phase shift corresponding to the maximum probability of detection probability can still be given as ${\phi }_k^{*}=-\left({\omega }_k+{\varpi }_{k,i}\right)$, where ${\phi }_k^{*}$ is the optimal phase shift of the k-th element in IRS, ${\omega }_k$ and ${\varpi }_{k,i}$ are the channel phases of $h_k$ and $g_{k,i}$, respectively.

6. Conclusions

This paper studied fusion schemes based on IRS-enhanced CSS. The EGC and the SC schemes for the soft combination were investigated. The WHC scheme was studied for the hard combination, and an OSC scheme was proposed based on SNR. Both cases with and without direct links between the PU and all SUs were considered for each scheme. The closed-form expressions for the probability of false alarm and the average probability of detection were derived. Numerical results showed the detection performance of the four schemes ranked from good to bad is WHC, OSC, SC, and EGC. In addition, IRS-enhanced CSS outperforms than the DF relay case and the no IRS case. Finally, the detection performance of the OSC outperforms the K-rank scheme, but its optimal threshold should not be to large.