A new mathematical analysis of the probability of detection in cognitive radio over fading channels
 Omar Altrad^{1}Email author and
 Sami Muhaidat^{2, 3}
https://doi.org/10.1186/168714992013159
© Altrad and Muhaidat; licensee Springer. 2013
Received: 10 December 2012
Accepted: 5 June 2013
Published: 11 June 2013
Abstract
Cognitive radio (CR) enriches wireless technology systems by harnessing spectrum white spaces. Such systems require continuous and reliable sensing to provide highquality service to their users and to minimize the interference they may cause to legacy networks. As the simplicity of implementation of energy detectors and their incoherent requirements make them an ideal candidate for this type of application, this work provides a further mathematical analysis to the probability of detection over different fading channels. We derive a tight closedform expression for the probability of detection over Nakagami channels. In addition, we introduce an accurate recursive algorithm to compute the probability of detection for an odd degree of freedom over additive white Gaussian noise channels, which has been overlooked in the literature so far. Finally, we present the simulation results which concur with the numerical results and which are also a perfect match with the mathematical expressions derived.
Keywords
1 Introduction
The rapid increase of wireless systems and applications raises spectrum demand. However, not all bands of the spectrum are fully utilized at specific times or at specific geographic locations. The Federal Communication Commission (FCC) reported that some of these bands (e.g., unlicensed bands at 2.4 and 5 Ghz) are overcrowded, while others (e.g., licensed bands such as the ultrahigh frequency (UHF) band) are inefficiently used. Therefore, technology is emerging to reduce the spectrum scarcity issue by fully utilizing the unused portion of the spectrum. For example, IEEE 802.22 [1] proposes reusing the television (TV) UHF band without causing any interference to TV receivers. Another considers a cellular communication system that utilizes the wireless local area network system (cf. [2]). Researchers often refer to this technology as cognitive radio (CR) systems.
As the main objective of CR systems is the spectrum efficient utilization, an accurate design for a cognitive radio network (CRN) working under a licensed primary network (PN) needs to be considered. While the CRN may have its own frequency band of operation, it can also utilize the white spaces or spectrum holes in frequency bands of the PN to increase its performance and to provide a higher quality of service to its users. As a result, it is considered to be a secondary network relative to the primary network. One of the major challenge of CRN is spectrum sensing, i.e., a highly reliable sensing function must be implemented in the CRN’s terminals. This arises from the fact that the CR receiver sensitivity must be as high as possible to detect the presence or absence of a primary user (PU) signal and to invoke other functions in the CR device which also depend mainly on sensing functionality. For example, in order to detect a primary signal, the CR system must have a sensitivity as much as 20 to 30 dB higher than that of the primary system [3]. Therefore, the core of CR systems is the spectrum sensing algorithm which determines the validity of a transmission opportunity.
In this paper, our study is limited to the energy sensing method [4–7]. In particular, for a local spectrum sensing scenario, i.e., the sensing is accomplished by a single cognitive radio. This detection method can be applied to any signal type with fewer computational requirements and a simpler implementation. Although several research papers have investigated the detection process using energy detector over a variety of fading channels (cf. [8–13]), the expressions derived for the probability of detection and the probability of false alarms were mainly evaluated for even degrees of freedom (e.g., [14, Eq.]10]). Therefore, we provide an algorithm to compute the detection probability in the case of odd degrees of freedom based on the suboptimal energy detector. Moreover, as spectrum sensing must detect a very low signaltonoise ratio (SNR), which in turn requires a high degree of precision, the previously derived expressions mainly depend on the number of terms in the summation to get highly accurate results. In addition, they are numerically difficult and depend on other functions while their implementation is also susceptible to truncation errors. Therefore, closedform expressions for the detection probability are derived. We summarize our contributions as follows:

We provide a highly accurate recursive algorithm to compute the probability of detection for odd degrees of freedom. It should be noted that the mathematical derivation shows the steps of the algorithm when evaluating the detection probability in case of odd degrees of freedom, i.e., it is an algorithm rather than a mathematical derivation. An example of the algorithm importance is the Marcum function in Matlab which accepts only integer values in its third argument. Therefore, when the number of degrees of freedom is odd, the third argument is no longer accepted and the Marcum function cannot be used to evaluate the detection probability in this case. However, our algorithm solves this problem.

We derive a closedform expression over a Nakagamim fading channel. Here, we use closed form in the sense that no summation and no integration are required. The accuracy of the closed form is very close to the previously reported expressions in which summation and integration are used to get highly accurate results. Our new expressions show how the ratio of the Nakagami parameter m and the average signaltonoise ratio which affects the receiver operation characteristics (ROC) curves.

We compare the derived expressions to the reported expressions in [14, 15] in which summation and integration are used. We also compare our derived expressions to other recently reported expressions, e.g., [16, 17], and we show that our new derived expressions can be used with no limitations. Moreover, the derived expressions are more accurate than the recently reported ones with less or almost the same computational complexity.
We also compare our simulation results with the analytical evaluation of the derived expressions.
The rest of the paper is organized as follows. In Section 2, the energy detector, system model, and the derivation of the recursive algorithm are introduced. We derive closedform expressions for Nakagami channels in Section 3. Simulation and numerical results are introduced in Section 4, and we conclude our paper in Section 5.
2 Energy sensing model
2.1 Probability of detection and false alarm under AWGN channels
which is the same result as in [5].
 (15)can also be reduced to$\begin{array}{l}g({\lambda}^{\prime},l)=g({\lambda}^{\prime},l2)\frac{{\lambda}^{\prime}}{l2}\end{array}$(16)
where the initialization starts with ${G}_{{\chi}_{l2}^{2}}\left({\lambda}^{\prime}\right)=g({\lambda}^{\prime},3)=\sqrt{\frac{2{\lambda}^{\prime}}{\pi}}\text{exp}({\lambda}^{\prime}/2)$.
3 Probability of detection and false alarms under Nakagami fading channels
where in the second step, we substitute $x=\sqrt{\gamma}$ and ${\eta}^{2}=\frac{2m}{\stackrel{\u0304}{\gamma}}$, and in the last step, we substitute $\frac{2}{\Gamma \left(m\right)}{\left(\frac{m}{\stackrel{\u0304}{\gamma}}\right)}^{m}$ with α. Different combinations of m and N/2 lead to different results for the integration defined in the last step. In the following, the probability of detection is evaluated over both Rayleigh and Nakagami fading channels.
3.1 Special case: Rayleigh fading
where u = N/2.
3.2 Nakagami fading
It is clear from (27) how changing various parameters affect the detection process. The new derived expression reveals the fact that the ratio of parameter m to parameter $\stackrel{\u0304}{\gamma}$ is an important consideration when evaluating the probability of detection over Nakagami fading channels. For example, at low $\stackrel{\u0304}{\gamma}<2$ dB and when the degree of freedom u is fixed, the Nakagami parameter m has only a minor effect on the detection process. That means no matter how much m increases, the probability of detection stays almost the same. However, at high $\stackrel{\u0304}{\gamma}>15$ dB, increasing m will greatly improve the probability of detection. This will be discussed further in the simulation section.
where $\beta =\left[2m/\left(2m+\stackrel{\u0304}{\gamma}\right)\right]$.
4 Simulation and numerical results
To evaluate the closedform expressions derived for Nakagami channels, an extensive simulation has been performed using the ROC. The derived expressions are evaluated and compared with the numerical integration of (18) and with the expressions reported by [14–17].^{c}
4.1 Comparison of the derived expressions with Equation 18
4.1.1 Low value of $\stackrel{\u0304}{\gamma}$
At a low value of $\stackrel{\u0304}{\gamma}$, i.e.,$\stackrel{\u0304}{\gamma}=10$ dB, it can be seen that increasing the value of m, (m = 1,2,3), does not improve the misdetection probability for both derived expressions which concurs with the numerical integration of (18). We also note that (29) exactly matches (18); on the other hand, there is a minor discrepancy between (27) and (18).
4.1.2 High value of $\stackrel{\u0304}{\gamma}$
At a high value of $\stackrel{\u0304}{\gamma}$, i.e.,$\stackrel{\u0304}{\gamma}=10$ dB, increasing m will greatly improve the misdetection probability for both derived expressions (27 and 29), which also concurs with the numerical integration of (18) as can be seen in Figure 4. Further, we notice that at a very low false alarm probability, (29) is less accurate compared to (27). However, as the false alarm probability increases, the results for both expressions match that of the numerical integration of (18).
4.2 Comparison of the derived expressions with related works
In this subsection, we compare the new derived expressions with the previously reported expressions for high and low values of $\stackrel{\u0304}{\gamma}$ and m = 1,2,3.
4.2.1 Low value of $\stackrel{\u0304}{\gamma}$
4.2.2 High value of $\stackrel{\u0304}{\gamma}$
4.3 Computational complexity
The derived expressions can also be compared in terms of the number of multiplications. For example, the derived expression in ([14], Eq. 20) is based on the summation of confluent hypergeometric functions, and such functions have a computational complexity of order $O\left[{log}^{2}\left(n\right)\stackrel{\u0304}{M}\left(n\right)\right]$ for ndigit precision [27], where n means computing n digits, and $\stackrel{\u0304}{M}\left(n\right)$ is the bit complexity of multiplication. However, the reported expression of ([15],12) is based on an infinite series of gamma functions, and such functions have a computational complexity of $O\left[\sqrt{n}\stackrel{\u0304}{M}\left(n\right)\right]$. Using this notation, the derived expressions of (27) and (29) have a computational complexity of $O\left(\stackrel{\u0304}{M}\right(n\left)\right)$. Since $O\left[{log}^{2}\left(n\right)\stackrel{\u0304}{M}\left(n\right)\right]>O\left[\sqrt{n}\stackrel{\u0304}{M}\left(n\right)\right]>O\left(\stackrel{\u0304}{M}\right(n\left)\right)$, which is also consistent with the simulation results of Figure 7, therefore, the derived expressions have a lower complexity than ([14], Eq. 20) and ([15], Eq. 12) and have the same computational complexity as ([16], Eq. 13) and ([17], Eq. 13).
5 Conclusions
Spectrum sensing using energy detectors under different fading channels was investigated. We derived tight closedform expressions for the probability of detection in Nakagami channels. The closedform expressions can easily be used for Rayleigh fading channels by setting m = 1. The results of the closedform formula as compared with other expressions based on summation and integration terms are very close. Furthermore, the derived expression of (27) can be used for all $\stackrel{\u0304}{\gamma}$; however, there is a minor limitation of using (29) specifically at high values of $\stackrel{\u0304}{\gamma}$. Moreover, the derived expressions have a lower computational complexity compared to other expressions with only a very small loss of accuracy. In addition, we introduced an accurate recursive algorithm to compute the probability of detection for an odd number of degrees of freedom under AWGN channels. Our simulation shows that the detection process for a binary phase shift keying signal using the recursive formula perfectly coincides with the recursive algorithm.
Endnotes
^{a} In AWGN channels, there is no fading, i.e., h^{2} = 1.
^{b} Different modulation schemes could be used in the simulation since the derived expression is independent of the modulation used.
^{c} For comparison purposes, we have used a value of n = 1 in the expression of [17], Eq. 13), where n represents the number of nodes cooperating in the sensing process according to [17] notations.
^{d} ‘Equation (29) comes from another way of calculating the probability of detection over Nakagami fading channels in order to compare with results from the state of the art, which seemed rather optimistic for low false alarm probability, and that for high SNR cases,’ one of the anonymous reviewer’s comment.
^{e} The conducted simulations show that the computed time will only be scaled by a constant factor and that the calculated computation time will not be affected if we do or do not clear the processor cache of any background application processes.
^{f} The number of terms used to calculate the summation of [15], Eq. 12) was 20.
Declarations
Authors’ Affiliations
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