He Huang, Chaowei Yuan＊

School of Information and Communication Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China

I. INTRODUCTION

Cognitive radios (CRs) have been proposed to make full use of limited spectrum bands as wireless services develop rapidly [1-3]. There are mainly three categories spectrum sensing techniques, which include matched filter detection, cyclostationary feature detection and energy detection (ED). ED is a non-coherent signal detection algorithm and it has the advantages in low complexity, implementation simplicity and detection without a priori knowledge [4, 5], it employs ED radiometer at the receiver to compare the received energy value with fixed threshold, then determine the state of PU is absent or present instantaneously.

Urkowitz firstly adopted binary hypothesis testing signal detection over a flat band-limited Gaussian noise channel by deriving the probability of detection Pdand false alarm Pf[6].Next, Kostylev and Alouini et al. considered ED model over multipath fading conditions such as Rayleigh, Nakagami-q and Nakagami-m [7-9]. Since then, ED algorithm has been widely used in corresponding fading scenarios with diversity combining, such as maximal ratio combining (MRC), selection combining(SC) and equal gain combining (EGC) etc [9-11]. In Ref. [12] unified close-form expressions for average detection probability of ED with cooperative spectrum sensing (CSS) are derived over generalized multipath fading channels. In Ref. [13] novel expressions have been derived over extended generalized K composite fading channels to provide unified models for fading channel statistics. Likewise,exact closed-form expressions of detective pattern over N*Rayleigh channels are derived with Meijer G-function, then they are extended to the case of square-law selection (SLS) to obtain better detective performance.

On the other hand, as wireless radio propagation is affected by multipath and shadowing simultaneously, the adequate fading expressions are needed to represent the fundamental characteristics for fading statistical patterns[14]. The generalized fading channels κ-μ and η-μ which describe the line-of-sight (LOS)and non-line-of-sight (NLOS) communication conditions are proposed to provide accurate representation for radio propagation [15, 16].Furthermore, the general complex fading model α-η-κ-μ has been put forward to account for short-term propagation phenomena by employing joint phase-envelope method, which has been represented in the device to device communications, vehicle to vehicle communications and indoor to outdoor propagation for 5G [17]. Moreover, severe fading channels and composite fading/shadowed models have been developed to measure the practical communication channels in the complex conditions [18-20].

The α-κ-μ distribution is a general flexible fading model which contains more fading parameters to describe the nonlinear LOS small-scale transmission scenario. Specially,it contains α-μ when κ is approached to 0 and κ-μ distribution when α is 2. It obviously shows that the α-κ-μ fading model is more effective and practical than α-μ, κ-μ, Rayleigh,Rice and Nakagami-m distributions. However,although a large number of papers have been devoted to study the SS performance over different generalized fading channels, no studies are related to ED over generalized non-linear LOS fading channels α-κ-μ in the open technical literatures, in addition how to improve the performance evaluation of detection algorithm is a key problem to be solved.

This paper investigates the SS based on ED that is implemented over α-κ-μ fading channels to reveal the relationship between the performance of ED and nonlinear LOS fading channels.

Motivated by above, in this paper we consider ED algorithm over α-κ-μ generalized fading channels and improve detective performance. The contributions are summarized as follows:

1) The α-κ-μ fading distributions have been derived under instantaneous SNR condition to represent the non-linear variation of fading signals, which could be used for investigating the real-time and short-range sensing features for SS in severe fading communication scenarios.

2) The novel exact close-form detection models are derived over α-κ-μ generalized fading channels with moment generating function (MGF) method and probability density function-based (PDF-based) approach firstly for point-to-point communication.

3) The novel close-form expressions for area under the receiver operating characteristics curve (AUC) and average AUC () are derived over α-κ-μ fading channels for revealing and quantifying the relationship between the behavior of ED with the variations of the involved fading parameters algebraically.

4) CSS with diversity reception cases have been considered to mitigate the shadowed fading features and improve detection probability practically. The tight closed-form upper bound of detection probability with MRC is derived to evaluate the optimized detection performance theoretically.

The rest of this paper is organized as follows. In section II the ED system models are presented. In section III the α-κ-μ fading models are derived under instantaneous SNR condition. In Section IV the close-form detection expressions have been derived over general-ized fading channels. In Section V theperformance of ED has been analyzed over α-κ-μ fading models. The derivations of CSS with diversity techniques are given in Section VI. Simulation results are presented in Section VII and conclusions are provided in Section VIII.

II. SYSTEM MODEL

The ED model is assumed to be the binary hypothesis-testing problem in Eq. (1) (H0: signal is absent; H1: signal is present) [5],

where y(t) is the received signal, n(t) is additive white Gaussian noise (AWGN), h is the channel gain, s(t) is the transmitted primary signal. From figure 1 the ED model can be obtained as

where m is the sampling number of received signal,is the AWGN for received signal yi(t). De fining the time bandwidth product as u=TW, T is the time interval and W is the single-sided signal bandwidth. The probability of detection Pdand the probability of false alarm Pfcan be expressed as [5]

where Qu(a,b) is the u-th order generalized Marcum-Q function, Г(.) is the Gamma function and Г(.,.) is the incomplete Gamma function. γ is the instantaneous signal-to-ratio(SNR) and λ is the ED threshold. The cumulative density function (c.d.f.) are obtained as

From Eq. (2) the probability density function (p.d.f.) of y(t) is given by

Fig. 1. ED system model.

where Iu-1(.) is the first kind modified Bessel function with the order u-1.

III. THE α-κ-μ FADING MODELS

The α-κ-μ fading distribution is a general fading distribution that represents the small-scale fading characteristics in the non-linear LOS condition. The envelope p.d.f. can be shown as[14]

where α is the nonlinear characteristics of the propagation medium, κ is the ratio between the total power of the dominant components and the total power of the scattered waves, μ is related to the number of multipath waves. For the fading signal with normalized power is w/E(w), the power p.d.f. can be expressed as

The SNR p.d.f. can be derived from [Eq.(2), Eq. (6), Eq. (7) and Eq. (8), 15],

where

IV. SPECTRUM SENSING OVER α-κ-μ FADING CHANNELS

Evaluating detection probability Pdwith Eq.(3) and Marcum-Q function from Ref. [6] for single sensing node in CRs,

The average probability of detection over α-κ-μ fading channels can be obtained as

wheredenotes the probability of detection,is the p.d.f. of the α-κ-μ distribution under instantaneous SNR condition.Substituting Eq. (9) and Eq. (11) into Eq. (12),the average probability of detection can be expressed as

The p.d.f. of the moment generating function (MGF) under the instantaneous SNR over α-κ-μ fading channels is shown as [21],

where E(.) denotes the expectation. Deducing the n-th derivative of Eq. (14) as

Likewise, the average probability of detection can be obtained in another way with Eq.(15),

Moreover, the closed-form expression over α-κ-μ fading channels can be evaluated as

The modified Bessel function of the first kind with the order v in Eq. (17) can be simplified with [Eq. (8.445), 22] as

Then Eq. (17) can be derived by using infinite series representation

Evaluating the integral in Eq. (19) with extended incomplete Gamma function,

Eq. (19) can be simplified as

Furthermore, the Eq. (21) can be evaluated with [Theorem 3.1, 23] as

where

The FOX-H function can be expressed as,

V. AVERAGE AREA UNDER THE ROC CURVE (AUC) OVER α-κ-μ FADING CHANNELS

5.1 AUC under instantaneous SNR condition

The AUC measurement is usually used for characterizing the performance of ED [24],health care field tests [25] and machine learning algorithms [26] for plotting Pdversus Pf(ROC) or missed detection probability Pm(Pm=1-Pd) versus Pf(complementary ROC).Here we introduce the ED threshold λ varies from 0 to ∞ to analyze the capability of energy detector. When the instantaneous SNR value denotes γ, the AUC can be shown as [27, 28]

Taking the derivative with Pf(u, λ), Eq. (25)can be written as

With Eq. (3) and Eq. (4), Eq. (26) can be derived as

5.2 over α-κ-μ fading channels

The average AUC () can be investigated with the p.d.f. of generalized fading models to indicate the properties of the fading channels,therefore,can be shown as

where

Eq. (28) is derived as with Eq. (25)-Eq. (27)

Depending on Eq. (9) and Eq. (30), Eq. (30)can be derived as

By means of [Eq. (8.445), 22], eq. (31) can be derived as

With Eq. (10), eq. (32) can be simplified as,

Simplifying the integral B1in Eq. (33)based on Taylor series [22],

Expanding Eq. (34) with Binomial theorem,

The integral in Eq. (35) can be simplified as

Lastly, the exact close-form expression can be obtained by substituting Eq. (36) into Eq.(33), Eq. (33) can be simplified as

where

VI. ED WITH RECEIVER DIVERSITY OVER α-κ-μ FADING CHANNELS

6.1 Upper bound of detection performance with MRC

It makes sense that implementing ED with diversity over fading channels at the receiver can improve detection probability in CRs. [12,21].

Although MRC requires prior channel knowledge of the signal, it estimates the upper bound on the performance of ED in practice[4]. The total instantaneous SNR of MRC is given by

where L is the number of antennas for each SU, the SNR of i-th receiver branch is defined as γi.

When the number of diversity branches is L, the average detection probability with MRC can be computed by substituting Eq. (39) and Eq. (22) into Eq. (13),

where the Φ(·) denotes the MGF with MRC.Using [Eq. (24), 10], the MGF is given by

It is worth noting that when the number of product terms of functions is two the Leibniz`s rule can be obtained as [Eq. (0.42), 22],

Deriving the n-order Leibniz`s rule with the aid of [Eq. (26), 10], the n-th derivative of (41) can be deduced with Eq. (14) and Eq.(15) as

Furthermore, from Eq. (22), Eq. (24), Eq.(40) and Eq. (43), the close-form detective expression of ED over α-κ-μ fading channels is given by

where

6.2 Cooperative spectrum sensing with diversity reception SLC and SLS

Square Law Combining (SLC): the SLC scheme requires the received signals are integrated and squared, then summed together[28]. The degrees of freedom is 2Lu and the total received SNR γSLCis equal to combined instantaneous SNR γMRC, besides the time bandwidth product u is replaced by Lu for Eq.(44) to represent the detection capacity over α-κ-μ fading channels. When the number of diversity branches is L, the detection probability can be shown as

where

Square Law selection (SLS) [29]: In SLS scheme the maximum decision statistics ySLSis selected to calculate the average detection probability as

where

Fig. 2. Simulation for average probability of detection versus average SNR (dB)with u=2 and Pf =0.01.

Cooperative spectrum sensing (CSS): In CSS SUs send own decisions to fusion center(FC) respectively, then FC makes the global decision by combining the received information to determine the absence or presence of PU [24]. As N is the number of collaborative users, the detection probability Pd-CSSwith SLC and SLS for the OR-rule are given by

VII. NUMERICAL SIMULATION AND ANALYSIS

Numerical simulation and analysis for the behavior of ED have been provided to reveal the crucial impact over α-κ-μ fading channels with MATHEMATICA [30]. The corresponding performances of Section IV to Section VI are quantified in the following figures to show the various numerical features of severe shadowing conditions. Figure 2 illustrates the average detection probability versus average SNR with different parameters α, κ and μ, when u=2 and Pf=0.01. Although α is low, the raise of κ and μ can improve the detection probability because higher ratio between the total power of the dominant components and the total power of the scattered waves and higher related variable of multipath clusters will lead to more received power of dominant components.Besides, the average detection probability also improves substantially, and if κ and μ are constant, higher α will lead to better detection.

Figure 3 and figure 4 showagainstto present performance characteristics for ED like Ref. [14, 19, 20]. Figure 3 depictsversusfor low non linearity parameter. It can be seen that under low values of α small variations of μ will evidently improve the sensing performance of ED, as κ changes,it has been demonstrated that μ is more important than κ when α is low.

Figure 4 illustratesagainst average SNR for higher α to Figure 3. Although under high values of α, it can be seen that higher α will lead to better detection, the variations of κ and μ could not significantly alteras parameter α increases. Besides, from Figure 3 and Figure 4, it shows higher α corresponds to better simulation results when average SNR is greater than 3 dB.

In Figure 5 the upper bound of detection probability with MRC have been analyzed if u=2 and Pf=0.01. It shows the performance of detection is proportional to average SNR and number of diversity branch. As the diversity branch increases, the average detection probability can be obviously raised. For example,despite of low Pf, Pd≈0.835 for L=3 and it far exceeds the detection probability (Pd≈0.617)for L=2 when average SNR is 4 dB.

Figure 6 and figure 7 show CSS with diversity combining jointly improve sensing performance over α-κ-μ fading channels. The average probability of detection versus average SNR with different collaborative users numbers are analyzed respectively. It indicates under low fixed α, diversity techniques and CSS both im-prove detection performance, although the Pffor SLC can be expressed as Г(Lu, λ/2)/Г(Lu)and the Pffor SLS is 1-(1-Г(u, λ/2)/Г(u))^N.Moreover, it can be inferred SLC has better effects and diversity combining is almost similar to CSS for detection probability improvement.

Fig. 3. Average AUC versus average SNR (dB) over α-κ-μ fading channels with u=2 for low values of α.

Fig. 5. The average probability of detection versus average SNR(dB) for upper bound of detection probability with MRC when u=2,α=1.35, κ=1.0, μ=1.0 and Pf =0.01.

Fig. 4. Average AUC versus average SNR (dB) over α-κ-μ fading channels with u=2 for high values of α.

Fig. 6. The average probability of detection versus average SNR(dB) for CSS with SLC and SLS when u=2, N=1, 2, α=1.35, κ=1.0,μ=1.0 and Pf =0.01.

Fig. 7. The average probability of detection versus average SNR (dB) for CSS with SLC and SLS when u=2, N=2, 3, α=1.35, κ=1.0, μ=1.0 and Pf =0.01.

VIII. CONCLUSIONS

This paper investigates the SS based on ED that is implemented over α-κ-μ fading channels to reveal the relationship between the performance of ED and nonlinear LOS fading channels. The novel unified close-form expressions of ED over α-κ-μ fading channels have been deduced to show essential sensing probability. In addition, exact close-form expressions of average AUC have been derived to quantify the behavior of ED with different values of nonlinear coefficient. Besides, it is demonstrated that diversity techniques and CSS can jointly improve the detection performance and the upper bound with MRC have been inferred to evaluate the sensing performance theoretically. Generally speaking, the sufficient results that are derived above can be completely used to quantify the performance of SS over α-κ-μ nonlinear LOS fading scenarios, and it can radically improve the energy efficiency for CR systems in wireless communications.

ACKNOWLEDGEMENTS

This work is supported by the science and technology project of state grid headquarters of China (SGLNDK00KJJS1700200).

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