An interframe dynamic doublethreshold energy detection for spectrum sensing in cognitive radios
 Caimei Fu^{1},
 Youming Li^{1}Email author,
 Yucheng He^{2},
 Ming Jin^{1},
 Gang Wang^{1} and
 Peng Lei^{1}
https://doi.org/10.1186/s1363801708970
© The Author(s) 2017
Received: 30 November 2016
Accepted: 3 June 2017
Published: 28 June 2017
Abstract
This paper proposes an interframe dynamic double threshold (IFDDT) spectrum sensing algorithm in order to improve the sensing performance based on energy detection (ED) in cognitive radios (CRs). Based on both the activity model of the primary user (PU) and the sensing mechanism of the secondary user (SU), the proposed algorithm exploits the relationship between two adjacent sensing frames and designs dynamic double thresholds for each sensing frame to enhance spectrum sensing performance when the received energy cannot give a reliable local decision. The detection probability and false alarm probability of the proposed sensing scheme are analyzed, and an algorithm for searching the optimal dynamic double thresholds is derived with very low complexity according to the NeymanPearson (NP) test criterion. Theoretical analysis and simulation results show that the proposed algorithm outperforms the ED algorithm.
Keywords
1 Introduction
Nowadays, wireless communications have been experiencing an increasing tension of the frequency resource which, however, is far from fully utilized under the static spectrum assignment policy [1]. Cognitive radio (CR) technology provides a promising solution to the conflict by allowing the second user (SU) to opportunistically access the licensed spectrum band when it is temporarily not occupied by the primary user (PU) [2].
Spectrum sensing is a critical aspect of CR systems that aims to identify the working state of the PU (i.e., ON or OFF, indicating whether the licensed spectrum is occupied or not, respectively) before allowing the SU temporarily access the channel without causing harmful interference to the PU. Typically, the SU operates spectrum sensing frame by frame, and each frame is divided into two parts: the sensing duration and the data duration [3–5]. The SU detects whether the licensed spectrum is accessed within the sensing duration, and then transmits data within the data duration if the sensing result is OFF (which identifies that the PU’s state is OFF); otherwise, the SU keeps silent.
The accuracy of spectrum sensing is measured by two probabilities: the false alarm probability (denoted as P _{ f }) and the detection probability (denoted as P _{ d }). The false alarm probability is defined as the probability that the PU’s state is identified as ON when its real state is OFF, while the detection probability is defined as the probability that the PU’s state is identified as ON when its real state is also ON. Clearly, a low false alarm probability improves the efficiency of the unused spectrum, whereas a high detection probability reduces the resulting interference in the licensed users [4, 6].
A number of spectrum sensing methods have been proposed for CR systems, such as energy detection (ED), cyclostationarity feature detection, secondorder statistics detection, matchedfiltering detection, compressive sensing detection [7, 8], and multiple antenna detection [9]. The energy detection has been preferred due to its feasible applicability and low implementation complexity. The cyclostationarity feature detection differentiates noise from PU signals by exploiting the cyclostationarity features of the received signals. However, it has high computational complexity and requires the knowledge of the cyclic frequencies which is difficult to be obtained. The secondorder statisticsbased detection is subject to the modulations of primary signals. The matchedfiltering detection is the optimum method only when perfect information about the waveforms of the PU is given. The compressive sensing detection makes use of the sparse structure of primary signals. The multiple antenna detection effectively overcomes the noise uncertainty and improves the detection performance based on the spatial correlation in the multiantenna receiver. Nevertheless, it requires the SU equipped with multiantenna receivers.
In order to avoid error detection caused by dramatic energy decrease, the average signal power of several past sensing frames is used in [10] for the ED schemes. However, conventional EDbased spectrum sensing schemes usually employ only a single fixed threshold. When the received energy of the SU nears the threshold, the decision for the PU’s state is subject to a high probability of misjudgment, thus yielding a loss in sensing performance. In order to alleviate this problem, the doublethreshold energy detection method is discussed in [11]. This method can decrease the interference of the SU to the PU. However, how to set the two thresholds and deal with the received energy falling between the two thresholds are not discussed. Doublethreshold energy detection method is also discussed in cooperative spectrum sensing environment [12]. In this method, each SU sends its observed energy to the fusion center (FC) when the energy falls between the two thresholds; otherwise, the SU sends its decision results. The final decision is made based on the soft combination of the received energy and hard combination of the received decision results in the FC. Therefore, the performance improvements of cooperative spectrum sensing schemes are obtained at the cost of extra control messages and increased computation complexities [13].
An alternative way of improving the ED schemes is to take into account the PU’s activity model during the sensing process. A model of twostate Markov chain is used as an adequate mean accurately describing spectrum occupancy in the time domain in [14–17]. On the one hand, the durations of the PU’s states (ON and OFF) are respectively exponentially distributed [18, 19]. On the other hand, the frame length of the SU is much shorter than both the average durations of the PU’s ON and OFF states [20, 21], which implies that the PU’s activities over successive sensing frames are not independent, especially between adjacent frames. Furthermore, the PU stays at the same state with a high probability during the whole sensing frame [22]. These properties can be exploited to improve the sensing performance of the ED schemes.
In [22], the received energy sequence over successive sensing frames is modeled as a continuous hidden Markov chain, and the final decision is made according to the combined observations from all previous sensing frames. However, this scheme needs to perform simultaneously spectrum sensing and data transmission and hence suffers from implementation difficulties due to selfinterference [23]. In [24] and [25], the PU activity is modeled as a Markov chain, and the final decision is also based on history observations but using different combining rules from that of [22]. However, in the schemes in [24] and [25], each sensing frame is further divided into many short slots for either dynamic spectrum sensing or data transmission. This frequent sensingtransmitting alternating strategy may cause synchronization difficulty on the receiving terminal.
In this paper, using a twostate Markov chain model of the PU’s activity, an interframe dynamic double threshold (IFDDT) ED scheme is proposed. Firstly, the relationship between two adjacent sensing frames is analyzed, and the IFDDT scheme for each sensing frame is designed based on dynamic double thresholds. Then, according to the NeymanPearson (NP) test criterion, an optimization problem is formulated to acquire the optimal dynamic double thresholds which maximize the detection probability while maintaining a maximum tolerable false alarm probability. Finally, the optimization problem is transformed into an easily solvable problem and an algorithm for searching the optimal dynamic double thresholds are derived with very low complexity. Theoretical analysis and computer simulations show that the proposed IFDDT scheme can achieve performance improvement due to dynamic double thresholds even when the received energy cannot give a reliable local decision. Compared to reference [11], theoretical analysis and the implementation method are provided for the proposed doublethreshold energy detection scheme.
The remainder of this paper is organized as follows. In Section 2, the system model is presented. In Section 3, the IFDDT scheme is proposed, and the optimal dynamic double thresholds are derived. Section 4 evaluates the proposed scheme for spectrum sensing through simulations. Finally, Section 5 concludes the paper.
2 System model
2.1 The PU’s activity model and the SU’s sensing mechanism
Let T _{1} and T _{0} denote the durations that the PU stays respectively in ON and OFF states as shown in Fig. 1. It is well accepted that the durations are exponentially distributed with expectations \(\overline {T}_{1}~=~\alpha \) and \(\overline {T}_{0}~=~\beta \) [25]. The PU’s average activity period is defined as \(\overline {T}_{PU}~=~\overline {T}_{1}~+~\overline {T}_{0}\), and the PU’s traffic load is defined as \(\eta _{\text {PU}}~=~{\overline {T}_{1}}/{\overline {T}_{\text {PU}}}\).
We assume that the SU allocates each frame k with a dynamic length T F _{ k }, consisting of fixed sensing duration τ and varying data transmission duration T F _{ k }−τ for k=1,2,…. Furthermore, we assume that the lengths satisfy the conditions that \(\tau \ll {TF}_{k},{TF}_{k}\ll \overline {T}_{1}\), and \({TF}_{k}\ll \overline {T}_{0}\).
Let z _{ k }∈{0,1} denote the real state of the PU within the sensing duration of any specific frame k at the SU. On the above assumptions, it is clear that there is a high probability that z _{ k } remains unchanged within a whole frame and that z _{ k } is closely related to the past state z _{ k−1} with respect to the preceding frame k − 1 for k > 1 [18, 20, 26].
where \(p_{ij,k~~1} \triangleq P\{z_{k}=jz_{k1}=i\}\) denotes the state transition probability over two adjacent frames k − 1 and k for i,j∈{0,1}; it is easy to identify that p _{ i0,k − 1} + p _{ i1,k − 1} = 1 for i = 0,1.
Although the frame length T F _{ k } varies with k, it holds that (α + β)T F _{ k − 1}≪1 on the above assumptions. It follows in general that p _{00,k − 1} > p _{01,k − 1} and p _{11,k − 1} > p _{10,k − 1}. This states an inertia property of the PU activity that z _{ k } is more likely to remain the same as z _{ k − 1} between any two adjacent frames k − 1 and k. In the CR system model, the SU is assumed to have full knowledge of parameters α and β [20, 22, 25].
2.2 Statistics for ED
where n = 1,2,…,N; w _{ k }(n) denotes the independent and identically distributed (i.i.d.) AWGN samples with zeromean and normalized unit variance; and s _{ k }(n) denotes the PU’s signal samples with constant transmit power. Hence, the average received signaltonoise ratio (SNR) per sample at the SU is expressed as \(\gamma ~=~\mathbb {E}\left (s_{k}(n)^{2}\right)\).
where \(\mathbb {N}(\mu,\sigma ^{2})\) is the notation of the Gaussian distribution with mean μ and variance σ ^{2}.
2.3 The conventional ED rule
In conventional ED schemes, it is usual to preset a fixed threshold λ _{ E D,k } = λ _{ ED } for all k. Thus, the probability measures \(P_{f,k}^{ED}\) and \(P_{d,k}^{ED}\) are independent of k. It has been shown that the single fixed threshold scheme yields a high probability of erroneous decision when the received energy U _{ k } nears the threshold λ _{ E D,k } [11].
3 The IFDDT scheme
In this section, we propose an IFDDT spectrum sensing scheme by exploiting the inertia property of the PU’s activity described in Section 2.1. Following the presentation of the decision rule, we formulate an optimization problem to acquire the optimal dynamic double thresholds. Furthermore, we finally transform the optimization problem into an easily solvable problem and derive the optimal dynamic double thresholds with very low complexity.
3.1 The decision rule using IFDDT
where λ _{1,k } and λ _{0,k } represent the double dynamic thresholds for frame k at the SU, and λ _{0,k }≤λ _{1,k }.
For k = 1, the two thresholds should be identically initialized as λ _{0,1} = λ _{1,1}. For k > 1, if the received energy U _{ k } lies in the open range (λ _{0,k },λ _{1,k }), the sensing result d _{ k } for frame k retains the same as the sensing result d _{ k − 1} for the preceding frame k − 1.
where i,j∈{0,1}. Then, \(p_{ij,k}^{*}=p_{ij,k~~1}\) is derived.
3.2 Derivation of thresholds
To derive the double thresholds, there exists two hypothesis testing criterions: the NeymanPearson (NP) test [8] and the Bayesian test [28]. We consider only the NP test criterion, which aims at maximizing P _{ d,k } with the constraint P _{ f,k } ≤ P _{ f,target}, or alternatively minimizing P _{ f,k } with the constraint P _{ d,k }≥P _{ d,t a r g e t }, where P _{ f,target} and P _{ d,target} represent the tolerable maximum false alarm probability and minimum detection probability, respectively.
where P _{ d,k } and P _{ f,k } are written as bivariate functions of thresholds.
Theorem 1
where c is the Lagrange multiplier.
So we convert the optimization problem in Eq. (16) into a nonlinear programming problem consisting of Eqs. (20) and (22).
Clearly, the existence of the solution to the optimization problem depends upon the system parameters, including the sample length N, the average SNR value γ, the frame lengths T F _{ k }, the expectation of ON and OFF state period \(\overline {T}_{1}\), \(\overline {T}_{0}\), and the target false alarm probability P _{ f,target}. On the other hand, the solution to the nonlinear programming problem cannot be expressed in a closed form. Therefore, in order to facilitate a heuristic approach for the solution, we derive the range for the above nonlinear programming problem firstly.
3.3 Solution analysis
Obviously, the zero point of g(λ _{0,k }) is the solution of λ _{0,k } to the nonlinear programming problem in Eqs. (20) and (21) meanwhile.
where \(\lambda _{1,k} = \sqrt {2N}\mathbb {Q}^{1}\left (\frac {P_{f,\text {target}}\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)P_{0,k}} {1P_{0,k}} \right)+N\).
 (1)
\( \frac {P_{f,\text {target}}}{P_{0,k}}<{ \mathbb {Q}\left ( \frac {1}{2} \sqrt {\frac {N}{2}} \right)}\). It can be easily obtained that \( \frac {\lambda _{0,k}a}{\lambda _{1,k}a}>0\), and thus, we can get g ^{′}(λ _{0,k })<0.
 (2)\( \frac {P_{f,\text {target}}}{P_{0,k}} \geq { \mathbb {Q}\left ( \frac {1}{2} \sqrt {\frac {N}{2}} \right)}\).

λ _{0,k }∈(a,λ ^{∗}). Similar to (1), it can be obtained that g ^{′}(λ _{0,k }) < 0.

\(\lambda _{0,k}\in (\lambda _{0,k}^{*},a)\), Eq. (20) can be transformed into$$\begin{array}{@{}rcl@{}} &&\left[1\frac{1}{2}er~f~c\left(\frac{N\lambda_{0,k}}{2\sqrt {N}} \right) \right]P_{0,k}\\&&\quad+\frac{1}{2}er~f~c\left(\frac{\lambda_{1,k}N}{2\sqrt {N}} \right)(1P_{0,k})=P_{f,\text{target}} \end{array} $$(33)
where \(er~f~c(x)~=~2\mathbb {Q}(\sqrt {2}x)\).

By taking Eq. (36) into Eq. (32), it can be proved that g ^{′}(λ _{0,k }) < 0 when \(\lambda _{0,k}\in \left (\lambda _{0,k}^{*},a \right)\) ; the nonlinear programming problem is then transformed into a problem of searching for the zero point of a strictly decreasing function, which can be easily solved by the dichotomy or Newton’s method. It should be noticed that no zero point exists when \(g\left (\lambda _{0,k}^{*} \right)~<~0 \) since g(λ ^{∗}) < 0.
4 Simulation results
In this section, numerical results are presented to evaluate the effectiveness of the IFDDT sensing scheme. The proposed algorithm is tested under various average activity periods and traffic loads of the PU.
4.1 Simulation setup
Firstly, the thresholds of the IFDDT scheme are simulated. Secondly, the actual false alarm of the new algorithm is provided to examine the correctness of the thresholds. Thirdly, the performance of the IFDDT algorithm is quantified in different average PU active periods and PU traffic loads, respectively. Finally, the total error probability of the IFDDT scheme is provided to verify the detection performance.
In the simulations, the sample length N is set to be 1000, the sensing frame length T F _{ k } is set to be 10 ms according to the IEEE 802.22 standard, and a Markov chain with 10^{6} frames is randomly generated to model the PU state varying in each simulation. The detection performance of the proposed IFDDT scheme is compared with that of the ED scheme and the doublethreshold (DT) energy scheme in [11].
4.2 Numerical results
From Fig. 4a, b, we can see that the double thresholds quickly converge to stable values and actually keep invariant. It means that the calculation for the thresholds of the subsequent sensing frames can be saved as long as the system parameters remain unchanged. Furthermore, the marked points show that in lots of frames where the received energy cannot give a reliable local decision, the conventional ED algorithm seem powerless but our algorithm tends to make the right judgment. It is also indicated that the gap between the double thresholds is smaller when the SNR is high.
5 Conclusions
In this paper, the PU’s activity model and the SU’s sensing mechanism are utilized and an interframebased dynamic doublethreshold spectrum sensing scheme, which exploits the relationship of two adjacent sensing frames, is proposed to enhance energy detection performance. When the received energy cannot give a reliable local decision to distinguish the ON and OFF states of the PU, the detector retains the previous sensing result. The detection probability and false alarm probability of the proposed sensing scheme are analyzed, and the optimal double thresholds are derived with very low complexity according to the NeymanPearson (NP) test criterion. This sensing scheme is simple but shown to outperform the ED algorithm from both theoretical analysis and simulation results.
In this paper, our algorithm is based on energy detection. However, it can be extended to other traditional spectrum sensing method, such as coherent detection and feature detection, where the PU’s activity model and the SU’s sensing mechanism have not been exploited. The same way can be followed to obtain the corresponding dynamic double thresholds and the detection probability together with the false alarm probability.
6 Appendix 1: Derivation of Eqs. (12) and (13)
7 Appendix 2: Proof of Theorem 1
Consider first k>1. In Eqs. (10) and (11), P _{0,k } and P _{1,k } can be viewed as constants in the open range (0,1). Thus, the probabilities P _{ f,k } and P _{ d,k } are both strictly decreasing functions of λ _{0,k } and λ _{1,k }, respectively.
which contradicts the assumption of the maximum value \(P_{d,k}^{*}\).
On the other hand, if \(\lambda ^{\prime }_{0,k}< \lambda ^{\prime }_{1,k}\), we may also reduce \(\lambda ^{\prime }_{1,k}\) to \(\lambda ^{*}_{1,k} \in [ \lambda ^{\prime }_{0,k},\lambda ^{\prime }_{1,k})\) with \(\lambda ^{\prime }_{0,k}\) given, such that \(P_{f,k}\left (\lambda ^{\prime }_{0,k},\lambda ^{*}_{1,k}\right)=P_{f,\text {target}}\), followed by \(P_{d,k}\left (\lambda ^{*}_{1,k},\lambda ^{\prime }_{0,k}\right)>P_{d,k}^{*}\).
In the special case k = 1, the probabilities P _{ f,1} and P _{ d,1} become strictly decreasing functions of λ _{1} as described underneath Eq. (9). Similarly, it can be proved that P _{ d,1} reaches \(P_{d,1}^{*}\) only when P _{ f,1} = P _{ f,target} using a proper threshold \(\lambda _{1}^{*}\).
Therefore, it is concluded that P _{ d,k } reaches the maximum value \(P_{d,k}^{*}\) only when P _{ f,k } = P _{ f,target} with pair thresholds \(\left (\lambda ^{*}_{0,k},\lambda ^{*}_{1,k} \right)\) for all k > 1.
8 Appendix 3: Proof of Eq. (23)
The proof is composed of two steps:
Firstly, \(\frac {P_{f,\text {target}}}{P_{0,k}} \geq \mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)\) can be derived from Eq. (20). From \(\frac {\lambda _{0,k}N}{\sqrt {2N}}=\sqrt {\frac {N}{2}}+\frac {\lambda _{0,k}}{\sqrt {2N}}\geq \sqrt {\frac {N}{2}}\) and the monotonic decreasing property of \(\mathbb {Q}\) function, we obtain \(\mathbb {Q}\left (\sqrt {\frac {N}{2}}\right) \geq \mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)\).
Therefore \(\min \left [\frac {P_{f,\text {target}}}{P_{0,k}},\mathbb {Q}\left (\sqrt {\frac {N}{2}}\right)\right ]\geq \mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right) \).
Secondly, from λ _{0,k }≤λ _{1,k }, we have \(\frac {\lambda _{0,k}N}{\sqrt {2N}} \leq \frac {\lambda _{1,k}N}{\sqrt {2N}}\), then \(\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right) \geq \mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)\).
On the one hand, \(\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)P_{0,k}+\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)(1P_{0,k}) = \mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)\). On the other hand, \(\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)P_{0,k}+\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)(1P_{0,k}) \geq \mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right)P_{0,k}+\mathbb {Q}\left (\frac {\lambda _{1,k}N}{\sqrt {2N}}\right)(1P_{0,k})=P_{f,\text {target}}\). Therefore, \(\mathbb {Q}\left (\frac {\lambda _{0,k}N}{\sqrt {2N}}\right) \geq P_{f,\text {target}}\).
Declarations
Acknowledgements
The authors would gratefully acknowledge the grants from the National Natural Science Foundation of China (61571250), the Natural Science Foundation of Ningbo City of China (2015 A610121), and the Natural Science Foundation of Fujian Province of China (2014J01243) and the K. C. Wong Magna Fund in Ningbo University.
Authors’ contributions
PL, YH, and CF conceived and designed the study. CF and PL performed the simulation experiments. CF and PL wrote the paper. YH, YL, MJ, and GW reviewed and edited the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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