Spectrum Sensing Techniques Based on Last Status Change Point Estimation for Dynamic Primary User in Additive Laplacian Noise

A real time scenario of dynamic primary user (PU) is considered in additive Laplacian noise. Two transitions or status changes of PU in the fixed sensing time are considered. The last status change point (LSCP) is estimated with maximum likelihood estimation by using dynamic programming. We consider Cumulative Sum (CuSum) based weighted samples for detection. We consider three detection schemes such as sample mean detector, energy detection and improved absolute value cumulation detection. We derive closed form expressions of detection probability (PD)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_D)$$\end{document} and false alarm probability (PF)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_F)$$\end{document} for all the three schemes. We present our results with receiver operating characteristic (ROC) for the considered schemes. We also present simulation results, which are closely matching with their analytical counterparts. We compare the ROC of the considered system with the ROC of conventional techniques. In the conventional techniques, all the samples in the sensing time are used for detection without LSCP estimation and weight. It is found that the considered system outperforms the conventional schemes.


Introduction
Spectrum sensing techniques in cognitive radio are used to utilize the spectrum effectively by unlicensed user or secondary users (SU) without causing interference to the primary users [1]. In spectrum sensing techniques, the PU is usually assumed to be static [2,3]. It means PU is either present or absent during the entire sensing duration [4]. This holds good for low PU traffic. However, for medium to high traffic, PU is assumed to possess dynamic behaviour [5]. The dynamic behaviour of the PU means

System Model
The null and alternate hypothesis are denoted by H 0 and H 1 respectively. The received symbols at the cognitive terminal under the random arrival and the random departure of the PU can be expressed as where 1 < N 1 < N 2 < N , and 1 < N ′ 1 < N ′ 2 < N . Here, N 1 and N ′ 1 are first transitions of PU, and N 2 and N ′ 2 are the second ones. The N indicates the total number of samples present during the sensing period. The s m is the unknown PU signal and w m denotes sample of Laplacian noise with mean 0 and variance 2b 2 , where b is the scale parameter of Laplacian noise. The average SNR is defined as = (1∕N)

Performance Analysis
In this section, we briefly present LSCP Estimation along with CuSum based weighing scheme [11]. Then, we derive the expressions of P D and P F . The last status change point (LSCP) estimation is found out by maximum likelihood estimation (MLE). The PU (s m ) is assumed to be unknown constant C. The joint MLE of C, N 1 and N 2 under hypothesis H 0 is found out by minimizing the following cost function [11] (1) Similarly, under H 1 , the cost function to be minimized can be expressed as Here, the estimated values of N 2 and N ′ 2 are taken as N 2 and N′ 2 respectively. We use dynamic programming [11] for the same. Now, with N 2 and N′ 2 , the effective hypotheses H ′ 0 and H ′ 1 can be expressed as After LSCP estimation, we use CuSum based weighted samples in the following three detectors.

Sample Mean Detector
The test statistics can be expressed as [11] where L scp denotes the last status change point of the PU, i.e. L scp ∈ {N 2 ,N � 2 } . Now, using the central limit theorem [6], the pdf of Z � ( ) can be expressed as Gaussian under both the hypotheses with the following mean and variance.
where E[⋅] and var[⋅] denote mean and variance, respectively. Derivation of (8) are given in appendix A and B while derivation of (9) are given in Appendix C and D. It should be noted here that similar steps given in the Appendix to calculate mean and variance have been followed in case of ED as well as i-AVCD.

Energy Detection
The test statistics can be expressed as Now, using the central limit theorem, the pdf of Z � ( ) can be expressed as Gaussian under both the hypotheses with the following mean and variance.

improved-Absolute Value Cumulation Detection
The test statistics can be expressed as Now, using the central limit theorem, the pdf of Z � ( ) can be expressed as Gaussian under both the hypotheses with the following mean and variance. (10) The P F and the P D in each case can be expressed, by taking respective mean and variance, as Similarly, here, is the detection threshold. .

Simulation Results
In this section, we present performance of the considered system with receiver operating characteristics (ROC). After estimating the LSCP, we apply CuSum based weighted samples for the considered detection schemes such as SMD, ED and i-AVCD. Using Neyman Pearson test, we determine threshold using (18) and subsequently P D using (19). Then, we plot the ROC for the considered schemes i-AVCD for P = 0.1 , SMD and ED. Fig. 1 shows the ROC at = −5 dB, N 1 = 30 , N 2 = 40 , N � 1 = 10 , N � 2 = 40 and N = 50.
It can be seen that the i-AVCD outperforms the remaining two schemes. We have also presented simulation results for all the three schemes. The close matching of the simulation results with analytical counterparts, validates our analysis. Figure 2 shows the ROC for the considered schemes at = −10 dB, N 1 = 30 , N 2 = 40 , N � 1 = 10 , N � 2 = 50 and N = 50 . We have also shown the performance of the three schemes without the LSCP and CuSum based test statistics. We refer to them as Conventional schemes. It can be seen that the considered schemes outperform the conventional scheme. Figure 3 shows the ROC for the considered SMD scheme as 'two PU change' at = −20 dB, N 1 = 10 , N 2 = 30 , N � 1 = 10 , N � 2 = 30 and N = 100 . We have also presented the performance of the 'One PU change', in which only one transition of PU is there at N 1 and N ′ 1 in H 0 and H 1 respectively with total samples of N in the sensing time. After applying LSCP estimation using dynamic programming, we get N 1 and N ′

Conclusion
In this letter, the effect of dynamic behaviour of PU was observed on the ROC in the additive Laplacian noise channel. The dynamic behaviour of the PU was assumed by two status changes of PU in the fixed sensing time. We estimated the last status change point (LSCP) by dynamic programming and then detected the PU using the samples available from the  LSCP to the end of sensing time by ignoring the samples before LSCP and boosting the samples after LSCP. Then, we used Cumulative Summation (CuSum) based weighted samples in the detection schemes such as improved absolute value cumulation detection (i-AVCD), Sample Mean Detector (SMD) and Energy Detection (ED). We derived the expressions of P D and P F for all the three schemes. We plot the ROC for all the three schemes and found that the i-AVCD outperforms the remaining two schemes. We also presented simulation results and found close matching between their analytical counterparts. We compared the considered schemes with the conventional schemes, where no LSCP or CuSum based weighted samples were used. In this case, the considered schemes outperform the conventional schemes. Finally, we compared the considered two PU status changes schemes with one PU status change. We observed that one PU status change outperforms the two PU status changes due to less estimation error in LSCP. (7), we have Expanding the above expression, we get

Appendix
can be further simplified as can be further simplified as As L scp =N � 2 , the final expression becomes (25)