Threshold optimization of a finite sample-based cognitive radio network using energy detector
- Ajay Singh^{1},
- Manav R Bhatnagar^{2}Email author and
- Ranjan K Mallik^{2}
https://doi.org/10.1186/1687-1499-2013-165
© Singh et al.; licensee Springer. 2013
Received: 6 August 2011
Accepted: 22 May 2013
Published: 14 June 2013
Abstract
In this paper, we consider a cognitive radio network containing two cognitive radios (CRs) and one primary user. The CRs utilize finite number of received data samples for estimating the energy of the primary signals and forward these energy estimates to a fusion center (FC). The FC combines the energy estimates and utilizes a global threshold based on the exact knowledge of local thresholds of the CRs for determining the presence or absence of the primary signal. We propose selective and semi-selective soft combining schemes for this set-up. For the proposed schemes, we derive the total probability of error of detecting a spectrum hole. By minimizing the total probability of error in sensing a spectrum hole, we find optimized local and global thresholds. Moreover, we also discuss the optimization of conventional non-selective soft and 1-bit hard combining schemes with multiple (equal to or more than two) collaborative CRs under the total probability of error minimization criterion. It is shown by simulations that the proposed selective soft combination-based scheme significantly outperforms the conventional non-selective schemes based on soft combination and 1-bit hard combination. Further, it is shown by simulation that the proposed selective soft combining scheme along with the total probability of error minimization criterion is able to properly utilize a spectrum hole with interference level less than the standard specified value.
Keywords
Cognitive radio; Collaborative networks; Energy detector; Probability of error; Spectrum hole detection1 Introduction
Due to the utilization of the radio electromagnetic spectrum up to saturation, there is scarcity of free radio spectrum [1, 2]. However, it is seen in practice that even the dedicated radio spectrum is not used efficiently. For example, in television broadcasting, the spectrum allocated is idle for considerable time. This free spectrum can be utilized for some other applications using a cognitive radio network [1, 2]. Hence, effective utilization of the spectrum plays an important role in today’s crowded spectrum environment. Whenever required, a licensed user (primary user) can use its licensed band, otherwise the spectrum can be used for secondary operations. A spectrum hole is said to exist when a particular band of frequencies is not used by the primary user. Searching for spectrum holes is termed as spectrum sensing[1]. Sensing of the spectrum holes is done by the cognitive radios (CRs) which are unlicensed users or secondary users [1, 2]. A CR is an intelligent wireless communication system that periodically monitors the radio spectrum and detects the occupancy of the different parts of the spectrum. The CR opportunistically communicates over spectrum holes without interfering with the primary user’s signals [3].
There are many challenges in detecting a spectrum hole. Few of these are fading, noise uncertainty, and shadowing. These effects may lead to significant interference to a licensed user. It is shown in the literature [4–8] that cooperation among the CRs can be utilized in order to overcome the problems of fading, noise uncertainty, and shadowing. An optimal linear cooperation framework of spectrum sensing for detecting the primary signals is proposed in [4]. The cooperative spectrum sensing in [4] is based on linear combination of local statistics from individual CRs. In [4], each CR uses an energy detector, which estimates the energy of the primary signal from the received data samples. These energy estimates are forwarded over noiseless channels to a fusion center (FC). The performance of the cognitive system is optimized using Neyman-Pearson (NP) criterion by assuming that large number of received data samples are used by each CR for energy estimation in [4]. In [7], a binary decision-based cooperative spectrum sensing scheme is discussed for cognitive networks. Each CR takes a binary decision using energy detector about the presence or absence of the primary signal. The binary decisions are forwarded to the FC which combines them for taking final decision. The sum of probability of missed opportunity and probability of false alarm is minimized for obtaining an optimized threshold for the CRs. Since the hard decision is taken at the CRs in [7], the performance of this scheme is poorer than a soft decision-based scheme [9]. In a soft decision-based scheme, the CRs forward the energy estimates to the FC and FC combines them in order to take the decision about the presence of the primary signal. In [8], the set-up of [7] is generalized to the scenario when the CRs collect very large number of data samples to take binary decisions. The optimized value of the local threshold in the CRs is numerically calculated by minimizing the total probability of error.
Energy detector is proposed for cognitive spectrum sensing because it needs no information about the primary signal and has lower complexity in real-time detection of spectrum hole. For the detection of unknown deterministic signals corrupted by the additive white Gaussian noise (AWGN), an energy detector is derived in [10]. Performance analysis of the energy detector for random signals is studied in [11]. The performance of the energy detector for unknown transmit signal in AWGN and fading environment is discussed in [12]. In [13], optimal soft combination scheme for received energy data is explained. The detail study regarding the performance of spectrum sensing in different scenario is done in [14–24].
In this paper, we consider a collaborative cognitive radio network which utilizes a finite number of samples for deciding the presence or absence of the primary signals. The cognitive network contains two secondary users and one FC. We consider a cognitive set-up, where the CRs do not take decision about the spectrum hole based on their local thresholds. Nevertheless, they can update the local thresholds as per the channel between the primary user and the CRs. These local thresholds are conveyed to the FC and the FC updates its global threshold accordingly. The CRs forward their energy estimates to the FC over noiseless reporting channels and the FC takes decision about the spectrum hole based on the normalized global and local thresholds. In selective soft combining, the FC combines the energy estimates of both CRs for making a decision of the primary signal only if their energy estimates are above the local threshold.
Expressions of the probability of false alarm and the probability of missed opportunity for this collaborative system are derived for the proposed selective combing scheme. The global and local thresholds are optimized by minimizing the total probability of error in detection of a spectrum hole. The proposed scheme of selective soft combination significantly outperforms the conventional soft and 1-bit hard combining schemes. It is also shown that the total probability of error minimization criterion is able to utilize a spectrum hole more efficiently as compared to the NP criterion while keeping the interference to the primary user (PU) within the desired limits for the signal-to-noise ratio (SNR) of the PU-CR link considered in simulations.
The rest of the paper is organized as follows. In Section 2, the system model of the cognitive radio network is explained. Performance analysis of the proposed scheme of selective soft combining and semi-selective soft combining is discussed in Section 3. Optimization of the normalized global and local thresholds is also performed in this section.
Section 4 discusses optimization of non-selective soft combining and 1-bit hard combining schemes for arbitrary number of CRs under the total probability of error minimization criterion. Numerical results are discussed in Section 5. In Section 6, some conclusions are drawn.
2 System model
respectively, where Γ (a) is the Gamma function [26, Eq. (6.1.1)].
3 Performance analysis of the combining schemes
- 1.
C _{00} corresponds to the cost of deciding hypothesis H _{0} when hypothesis H _{0} is true,
- 2.
C _{10} corresponds to the cost of deciding hypothesis H _{1} when hypothesis H _{0} is true,
- 3.
C _{11} corresponds to the cost of deciding hypothesis H _{1} when hypothesis H _{1} is true,
- 4.
C _{01} corresponds to the cost of deciding hypothesis H _{0} when hypothesis H _{1} is true.
It is assumed that C_{00} = C_{11} = 0 which implies that no cost is assigned when correct hypothesis is chosen. This assumption is valid because no error is made if the decision of the presence or absence of the primary signal is correct. However, in order to reduce the probability of error in decision, we have to set C_{01} = C_{10} = 1 so that maximum cost is assigned for a wrong decision.
In CR spectrum sensing, when the primary signal is present and the CR decides that it is not present, then it causes interference to the PU and leads to overutilization of the spectrum. Similarly, when the CR decides a the PU signal is present and actually it is not, then spectrum hole is underutilized. Therefore, we consider that cost assigned in both types of error is set as unity.
It can be deduced that Eq. (10) refers to the total probability of error or average probability of error in deciding the presence or absence of the primary signal. In our analysis, we have chosen this type of error criterion specifically in order to minimize the total probability error. It will be shown in Section 5 that the total probability of error criterion enables the cognitive system to utilize free spectrum more efficiently than the NP criterion, for certain SNR values considered in simulations.
- 1.
u _{1} > λ, u _{2} > λ, and u _{1} + u _{2} > λ _{0},
- 2.
u _{1} > λ, u _{2} < λ, and u _{1} + u _{2} > λ _{0},
- 3.
u _{1} < λ, u _{2} > λ, and u _{1} + u _{2} > λ _{0},
- 4.
u _{1} < λ, u _{2} < λ, and u _{1} + u _{2} > λ _{0}.
The decision regions corresponding to non-detection of the PU, i.e., Z_{0} will be complimentary to the decision regions of Z_{1} in all above conditions. The detection region corresponding to u_{1} < λ, u_{2}<λ, and u_{1} + u_{2} > λ_{0} is non-zero for λ_{0} < 2λ only.
3.1 Proposed selective combining scheme
It is beneficial to check the received signal at each CR and include those signals, which are above a local threshold λ, in making the final decision in the FC. Let us consider a soft combining-based scheme, where the FC takes decision of the presence or absence of the spectrum hole when u_{1} > λ, u_{2} > λ, and u_{1} + u_{2} > 2λ. Hence, selective combining in the FC concerns with the fact that both CRs contribute in the decision about the presence or absence of the primary signal provided that energy estimates of both CRs are above the local threshold λ. Intuitively, the selective combining scheme guarantees the minimum total probability of error in the detection of spectrum hole. Motivated by this fact, we consider a proposed scheme for soft combination of received data in which decision region is u_{1} > λ, u_{2} > λ, and u_{1} + u_{2} > λ_{0}. We will evaluate the performance of the proposed selective combining scheme by deriving the expressions of the probability of false alarm and the probability of missed opportunity for this collaborative system.
3.2 Total probability of error of the selective combining scheme
For finding performance analysis of the proposed soft combining scheme, we split our analysis into three parts λ_{0} > 2λ, λ_{0} = 2λ, and λ_{0} < 2λ.
3.2.1 Total probability of error of the selective combining scheme for λ _{ 0 } > 2λ
where $\gamma ={E}_{s}{\sigma}_{h}^{2}/{\sigma}_{n}^{2}$ is the a verage signal-to-noise ratio (SNR) of the PU-CR link.
3.2.2 Total probability of error of the selective combining scheme for λ_{ 0 } = 2λ
3.2.3 Total probability of error for proposed scheme when λ _{ 0 } < 2λ
3.3 Optimization of the proposed selective soft combining scheme
In this subsection, we will discuss the optimization of the local and global thresholds of the proposed soft combining scheme. The optimization is divided into three different parts, i.e., λ_{0} > 2λ, λ_{0} = 2λ, and λ_{0} < 2λ.
3.3.1 Optimization of thresholds for λ_{ 0 } > 2λ
For finding an optimal value of λ_{ n }, we need to numerically solve Eq. (28). We can also numerically find the joint values of λ_{0n} and λ_{ n } such that total probability of error is minimized [28].
where ${\lambda}_{0n}^{\ast}$ denotes the optimal value of the normalized global threshold in the proposed scheme. It will be shown by simulations in Section 5 that the minimum probability of error in detection of a spectrum hole is achieved for λ_{ n } = 0.
3.3.2 Optimization of thresholds for λ _{ 0 } ≤ 2λ
We can numerically solve Eq. (32) to find the optimized value of λ_{ n }.
3.4 Performance analysis of semi-selective combining scheme
In the semi-selective combining scheme, the FC combines the energy estimates of the CRs corresponding to detection region Z_{1} for the condition that u_{1} > λ, u_{2} < λ, and u_{1} + u_{2} > λ_{0}, u_{1} < λ, u_{2} > λ, and u_{1} + u_{2} > λ_{0}, and u_{1} < λ, u_{2} < λ, and u_{1} + u_{2} > λ_{0} also in addition to u_{1} > λ, u_{2} > λ, and u_{1} + u_{2} > λ_{0} considered in previous subsections. By following the procedure given in Section 3.1, we can split the analysis into three parts λ_{0} > 2λ, λ_{0} = 2λ, and λ_{0} < 2λ and obtain the probability of false alarm and missed detection for these three conditions as follows^{1}.
3.4.1 Probability of false alarm and missed detection for λ _{ 0 } > 2λ
In case of u_{1} < λ, u_{2} < λ, u_{1} + u_{2} > λ_{0}, and λ_{0} > 2λ, the detection region will be zero and probability of miss will be one.
3.4.2 Probability of false alarm and missed detection for λ _{ 0 } = 2λ
When u_{1} < λ, u_{2} < λ, u_{1} + u_{2} > λ_{0}, and λ_{0} > 2λ, the detection region will be zero and probability of miss will be one.
3.4.3 Probability of false alarm and missed detection for λ _{ 0 } < 2λ
The semi-selective soft combining scheme can be optimized by the procedure given in Section 3.2, and optimized values of the local and global thresholds can be obtained by the total probability of error minimization criterion.
4 Soft and hard combination schemes for more than two CRs
It can be seen from the discussion above that analysis of the proposed soft combining scheme is very complex in the case of more than two users. Hence, it is very difficult to find the optimum local and global thresholds for a general collaborative soft combining scheme with more than two CRs. Therefore, in the case of more than two CRs, we can assume that the FC does not have information about the local threshold λ. Therefore, the FC cannot use selective combining discussed in Section 3.2.
4.1 Non-selective soft combination scheme for more than two CRs
In the non-selective soft combining scheme for more than two users, the FC always combines the energy estimates of all CRs for taking a decision. When the FC does not use the local threshold for decision making, the non-selective soft combining scheme is similar to the scheme discussed in [4, 13] for arbitrary number of CRs. However, in [4, 13], the NP criterion is used for finding the optimized value of the global threshold. Moreover, the existing schemes [4, 13] consider a slowly varying Rayleigh fading channel for analysis. It can be seen from Eq. (1) that we consider a fast fading Rayleigh channel between the PU and CRs in our analysis. In addition, it can be seen from Section 3.3 that we use the total probability of error for finding the optimized value of the threshold. Therefore, we need to derive the expression of the total probability of error for the conventional non-selective soft combining scheme of [4, 13], and based on that, we can find a closed form expression of the optimal value of the global threshold.
In the non-selective soft combining scheme, each CR forwards the energy estimate over noiseless channel to the FC, and the FC takes the decision about the spectrum hole based on the global threshold λ_{0}.
which is compared with a predefined threshold λ_{0} to make a decision about the presence of the PU. It can be noted from Eq. (48) that the process of combining summary statistics is similar to equal-gain combining.
4.2 One-bit hard combination scheme
The total error rate of the hard combining-based cooperative scheme will be the sum of P_{ f } and P_{ m }. The optimal number of CRs and optimized value of the local threshold can be calculated by minimizing the total error rate as shown in [7].
5 Numerical results
Optimal values of normalized global threshold for varying values of normalized local threshold
Sample no. | SNR (dB) | Total prob. of error | Normalized local threshold | Normalized global threshold |
---|---|---|---|---|
λ _{ n } | ${\lambda}_{0n}^{\ast}$ | |||
1 | -10 | 0.4630 | 0.0 | 4.4 |
2 | -10 | 0.4672 | 1.0 | 4.4 |
3 | -10 | 0.4778 | 2.0 | 4.4 |
4 | -10 | 0.4901 | 3.0 | 4.4 |
5 | -10 | 0.4967 | 4.0 | 4.4 |
6 | 0 | 0.2493 | 0.0 | 5.6 |
7 | 0 | 0.2647 | 1.0 | 5.6 |
8 | 0 | 0.3057 | 2.0 | 5.6 |
9 | 0 | 0.3642 | 3.0 | 5.6 |
10 | 0 | 0.4218 | 4.0 | 5.6 |
11 | 10 | 0.0118 | 0.0 | 10.4 |
12 | 10 | 0.1462 | 1.0 | 10.4 |
13 | 10 | 0.2286 | 2.0 | 10.4 |
14 | 10 | 0.0359 | 3.0 | 10.4 |
15 | 10 | 0.0534 | 4.0 | 10.4 |
Effect of L on the performance of cooperative spectrum sensing
L | P_{ f }at SNR = -20 dB | P_{ f } at SNR = -2 dB | P_{ m } at SNR -20 dB | P_{ m } at SNR = -2 dB |
---|---|---|---|---|
1 | 0.4532 | 0.2513 | 0.5390 | 0.3812 |
2 | 0.4812 | 0.2027 | 0.5077 | 0.2917 |
3 | 0.4963 | 0.1657 | 0.4901 | 0.2354 |
4 | 0.5069 | 0.1369 | 0.4775 | 0.1946 |
5 | 0.5151 | 0.1140 | 0.4673 | 0.1631 |
6 | 0.5220 | 0.0956 | 0.4588 | 0.1380 |
7 | 0.5280 | 0.0805 | 0.4513 | 0.1176 |
8 | 0.5333 | 0.0681 | 0.4447 | 0.1007 |
Non-selective soft combination scheme with total error probability criterion and NP criterion at SNR = 10 dB and L = 1
Number of CRs ( K) | P _{ f } | P _{ f } | P _{ m } | P _{ m } |
---|---|---|---|---|
(NP criteria) | (Total probability of error) | (NP criteria) | (Total probability of error) | |
5 | 0.02925 | 0.003265 | 0.002447 | 0.007718 |
10 | 0.4579 | 8.833e-005 | 4.663e-008 | 0.0002002 |
15 | 0.9165 | 2.633e-006 | 7.816e-014 | 5.872e-006 |
6 Conclusions
In this paper, we have demonstrated that it is possible for a collaborative cognitive radio network to detect the spectrum hole in an optimal manner by minimizing the total probability of error in decision making with a finite number of the received data samples. It is shown by simulations that the proposed selective scheme for soft combination significantly outperforms the non-selective soft and hard combining schemes. Moreover, the total probability of error minimization criterion performs better than the NP criterion while keeping the probability of false alarm within the desired limits for specified values of SNR of the PU-CR link.
Endnote
^{a} We are skipping analytical details here and writing the final expressions to avoid repetition.
Declarations
Acknowledgements
This work was partially supported by the IDRC Research Grant RP02253.
Authors’ Affiliations
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