Mathematical analysis for detection probability in cognitive radio networks over wireless communication channels

: In this study, the authors consider the problem of spectrum sensing based on energy detection method in cognitive radio over wireless communication channels when users experience fading and non-fading effects. The closed-form analytical expressions for the detection probability are derived over non-fading additive white Gaussian noise channel and Rayleigh and log-normal shadowing fading channels. The detection probability involves Marcum-Q function, summations and integrations in the early research papers, which are replaced by closed-form expressions in this study. The probability distribution function of fading channels is used to obtain the expressions for detection probability. The new derived numerical results are simulated under various parameters. The performance of the derived theoretical expressions closely matches with the simulated results.


Introduction
Owing to the increasing popularity of wireless devices in recent years, the radio spectrum has been an extremely scarce resource. However, all bands of the spectrum are not fully utilised at specific times or at particular geographic locations according to the report of Federal Communications Commission (FCC) [1]. The underutilisation of spectrum in many of the frequency bands is because of the conventional fixed spectrum allocation policy. To overcome the problem of spectrum scarcity, the license exempted secondary users (SUs) are allowed to exploit the unused spectrum holes over some frequency ranges by using the cognitive radio (CR) technology [2]. The challenge in implementing the CR technology is that the SUs must accurately monitor and be aware of the presence of the primary users (PUs) over a particular spectrum. Several efficient methods have been proposed to address this challenge [3][4][5][6]. The detection of unknown deterministic signals with energy detection (ED) is the simplest and most popular method since it does not require any a priori knowledge of the PUs signal and has much lower computational and implementation complexity. The performance of ED over non-fading additive white Gaussian noise (AWGN) channel was first addressed by Urkowitz and comprehensively derived analytic expressions for the probability of detection and the probability of false alarm [7]. Recently, the detection problem over fading channels has been revised in [8,9]. The expressions derived for the detection probability over fading channels using energy detector in [9][10][11][12][13][14] depend on the number of terms in the summation and other functions. The implementations of these expressions are susceptible to truncation errors. We summarise our contributions as follows. We provide approximate expressions for the detection probability over non-fading AWGN channel and Rayleigh and log-normal shadowing fading channels. The expressions reported in [9][10][11][12][13][14] for AWGN channel involves the Marcum-Q function, which is replaced by closed-form expression in our derivation (16). It has much less computational complexity. The derived analytical expressions over fading channels in the recent papers consist of summations and integrations with the limitations. These are susceptible to truncation errors. We compare our new derived expressions (16), (22) and (30) with the expressions reported in [9][10][11][12][13][14] and we show that our new derived expressions do not involve summations and integrations and they can be used with no limitations. The simulation results of the considered ED method over both AWGN and fading channels are compared with the theoretical evaluation of the derived expressions. The rest of this paper is organised as follows. In Section 2, the system model is discussed. We derive expressions for the detection probability over fading channels in Section 3. The performance of the derived expressions and simulation results are presented in Section 4. Conclusions are drawn in Section 5.

Model
The energy-sensing model is as shown in Fig. 1. The received signal r(t)i sfiltered to the required bandwidth B to reject the noise and adjacent signals. The filtered signal is sampled and quantised using analog-to-digital converter. Next, a square-law device and an integrator measure the received signal energy. The output of the integrator, represented by the decision statistic Y, is compared with a predetermined threshold γ to determine the presence (H 1 )or absence (H 0 ) of a PU. The presence or absence of a PU signal can be modelled as binary hypotheses problem as originally proposed by Urkowitz [7] and Pridham and Urkowitz [15] and latter followed by most researchers [16,17]. The received signal samples under two hypotheses defined as where n =1,2,…, N. s(n) is the PU signal which is assumed to be an unknown deterministic signal and w(n) is the AWGN with zero mean and variance s 2 n . h is the channel gain between PU and SU. The energy detector is defined as The distribution of the decision variable Y will be central chi-square X 2 N under H 0 and non-central chi-squareX 2 N with N degrees of freedom under H 1 . The distribution can be expressed as [18] Its probability density function can be written as where the non-centrality parameter z = N i=1 m 2 i and μ i is the mean of the ith Gaussian random variable of test Y. I M (·) is the Mth modified Bessel function of the first kind, which has a series representation [19] The test statistic Y is compared with a predetermined threshold γ, to decide on one of the two hypotheses H 0 and H 1 . The probability of detection (P d ) and probability of false alarm (P fa ) can be computed as

Probability of false alarm under AWGN channel
To derive the probability of false alarm over non-fading AWGN channel, the probability density function defined under H 0 in (4) is inserted in (7). We define signal-to-noise ratio (SNR) (ρ)a s r = h 2 z/s 2 Substituting t = y/s 2 n in (8) and further integrating the P fa results in The right-hand side of (9) is the chi-squared density [20, 26.4.2] which can be written as P fa = Q X 2 N g/s 2 n . In the energy detector, the threshold γ can be computed using g = s 2 x e −u 2 /2 du is the standard Gaussian tail probability function.

Probability of detection under AWGN channel
To derive the probability of detection over non-fading AWGN channel, the probability density function defined under H 1 in (4) is inserted in (6). By substituting t = y/s 2 n and using definition of ρ, the probability of detection is given by (10) Substituting t = x 2 , ρ = a 2 and with the further simplification, P d can be written as (11) We can rewrite (11) Since the precise computation of Marcum-Q function is quite difficult, it is represented in the new form as [21, 2.6] where L k n (x) is the generalised Laguerre polynomial of degree n and order k. The absolute convergence of the series in (13) can be easily shown [21,22] Inserting (15) in (13) the P d for AWGN channel is approximated as

Probability of detection under fading channels
We define the average SNR r as r = E[h 2 ]z/s 2 n , where E(·) denotes the expectation operator.

Rayleigh fading channel
The probability distribution function (PDF) of ρ in case of Rayleigh fading channel is given as Using (16), we obtain the probability of detection as (see (18) and (19)).
The first term of (19) is simplified to The second term of (19) is simplified to After substituting (20) and (21) in (19), the approximate probability of detection for Rayleigh channel is

Log-normal shadowing channel
The link quality in terrestrial and satellite land-mobile systems is affected by slow variation of the mean signal level because of the shadowing from terrain, buildings and trees. The performance of a communication system will depend only on the shadowing if the radio receiver is able to average out the fast multipath fading or if an efficient micro-diversity system is used to eliminate the effects of multipath. Empirical measurements reveal that shadowing can be modelled by a log-normal distribution for various outdoor and indoor environments, in which case the probability distribution of ρ is given by [23] f (r) = 4.34 2p √ sr exp − (10 log 10 r − m) 2 2s 2 (23) where μ(dB) and σ(dB) are the mean and the standard deviation of 10 log 10 ρ, respectively. To the best of our knowledge, closed-form expression cannot be obtained for the detection probability using (16) and (23). The log-normal distribution can be closely approximated by the Wald distribution [10]. Its PDF is given by where θ = E(ρ) denotes the expectation of ρ and η is the shape parameter. These parameters are related with μ and σ as given below Using (16) and (24), the approximate probability of detection for log-normal shadowing channel is given as (see (26) and (27)).
Using [24, 3.471-9], the first and second terms of (27) are reduced to (see (28) and (29)) where K v (x) is a modified Bessel function of second kind. By substituting (28) and (29) in (27), the approximate probability of detection over log-normal shadowing channel is given by (see (30) at the bottom of the next page).

Results and discussion
In this section, we present the performance of the derived expressions for AWGN and fading channels through simulation.
Receiver operating characteristic (ROC) has been widely used in the signal detection theory. It is an ideal technique to quantify the tradeoff between the probability of detection and probability of false alarm. In the simulation, we use complementary ROC curves (P md against P fa ) to show the detection performance of ED over non-fading AWGN and fading channels. The noise variance factor s 2 n is considered in each simulation result. In Fig. 2, we generate the complementary ROC curves for AWGN channel for different values of ρ with N = 10 and noise variance s 2 n = 0.5. In this figure, the deviation between (12) which involves generalised Marcum-Q function and the derived closedform expression (16) is shown. The deviation between (12) and (16) is too small. It is clear that increasing SNR decreases the miss-detection probability and improvement in the detection probability diminishes at low SNR. In Fig. 4, we have presented the performance evaluation of ED over Rayleigh channel based on both analytical and simulation for different values of N with r =−5 dB and s 2 n = 0.5. As can be observed, the simulation results well confirm the analytical expression (22). Here, increasing the value of N improves the detection probability. Fig. 5 shows the complementary ROC curves for different values of r with N = 8 and s 2 n = 0.5. We note from the figure that there is an improvement in the performance with several dB's increment in r. In Fig. 6, we plot complementary ROC curves for log-normal shadowing channel for different values of mean (μ) with N =8, σ = 4 and s 2 n = 0.5. There is a great improvement in the detection probability when μ is increased from 6 to 12 dB. Fig. 7 shows the complementary ROC curves for log-normal shadowing channel for different values of N with μ = 5 dB, σ = 3 and s 2 n = 0.6. We note that detection probability is low when N =6 and we can achieve very high detection probability for N = 12.
Here simulation results closely match with analytical results. Finally, we say that all the derived expressions for the detection probability over non-fading and fading channels are simulated in MATLAB under various parameters which concur with the simulation results.

Conclusions
In this study, we derived the closed-form expressions for the probability of detection over non-fading AWGN and fading channels. To the best of our knowledge, (16), (22) and (30) are the new numerical results for the detection probability over considered channels. These theoretical expressions are evaluated under various parameters and verified by the simulation results. Our numerical results in Section 3 provide new mathematical analysis for the detection probability over fading channels.