# Statistical Properties of Energy Detection for Spectrum Sensing by Using Estimated Noise Variance

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## Abstract

**:**

## 1. Introduction

- What is the expectation of ${\widehat{P}}_{fa}$?
- What is the variance of ${\widehat{P}}_{fa}$? (equivalently, the second moment of $E{\widehat{P}}_{fa}^{2}$)
- What is the limitation of $E{\widehat{P}}_{fa}$ for a fixed ${P}_{fa}$ as N, the number of samples used to estimate the noise variance, and M, the number of samples used to perform detection, tend to infinity?

## 2. Model Setting and Hypothesis Testing with Known Noise Variance

## 3. Energy Detection Performance Using Estimated Noise Variance

**Example**

**1.**

**Example**

**2.**

## 4. Calculations of $E{\widehat{P}}_{fa}$ and $E{\widehat{\tilde{P}}}_{fa}$

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Remark**

**1.**

**Proof**

**of**

**Theorem**

**3.**

**Theorem**

**4.**

**Remark**

**2.**

**Proof.**

## 5. Upper Bounds of $\mathit{E}{\widehat{\mathit{P}}}_{\mathit{fa}}^{\mathbf{2}}$ and $\mathit{E}{\widehat{\tilde{\mathit{P}}}}_{\mathit{fa}}^{\mathbf{2}}$

**Proposition**

**1.**

**Proposition**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

**Theorem**

**5.**

**Remark**

**3.**

**Proof**

**of**

**Theorem**

**5.**

**Theorem**

**6.**

**Remark**

**4.**

**Proof**

**of**

**Theorem**

**6.**

## 6. New Thresholds Based on $E{\widehat{P}}_{fa}$ and $E{\widehat{\tilde{P}}}_{fa}$

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**2.**

## References

- Xue, D.; Ekici, E.; Vuran, M.C. Cooperative spectrum sensing in cognitive radio networks using multidimensional correlations. IEEE Trans. Wirel. Commun.
**2014**, 13, 1832–1843. [Google Scholar] [CrossRef] - Arjoune, Y.; Kaabouch, N. A Comprehensive Survey on Spectrum Sensing in Cognitive Radio Networks: Recent Advances, New Challenges, and Future Research Directions. Sensors
**2019**, 19, 126. [Google Scholar] [CrossRef] [PubMed] - Ali, A.; Hamouda, W. Advances on Spectrum Sensing for Cognitive Radio Networks: Theory and Applications. IEEE Commun. Surv. Tutorials
**2016**, 19, 1277–1304. [Google Scholar] [CrossRef] - Plata, D.M.M.; Reatiga, A.G.A. Evaluation of energy detection for spectrum sensing based on the dynamic selection of detection-threshold. Procedia Eng.
**2012**, 35, 135–143. [Google Scholar] [CrossRef] - Hu, X.-L.; Ho, P.-H. Performance Analysis of Maximum Likelihood Estimation for Transmit Power Based on Signal Strength Model. J. Sens. Actuator Netw.
**2018**, 7, 38. [Google Scholar] [CrossRef] - Sahai, A.; Tandra, R.; Mishra, S.M.; Hoven, N. Fundamental design tradeoffs in cognitive radio systems. In Proceedings of the First International Workshop on Technology and Policy for Accessing Spectrum, Boston, MA, USA, 2–5 August 2006. [Google Scholar]
- Oner, M.; Jondral, F. Cyclostationarity based air interface recognition for software radio systems. In Proceedings of the IEEE Radio and Wireless Conference, Atlanta, GA, USA, 22 September 2004; pp. 263–266. [Google Scholar]
- Gardner, W. Exploitation of spectral redundancy in cyclostationary signals. IEEE Signal Process. Mag.
**1991**, 8, 14–36. [Google Scholar] [CrossRef] - Urkowitz, H. Energy detection of unknown deterministic signals. Proc. IEEE
**1967**, 55, 523–531. [Google Scholar] [CrossRef] - Kostylev, V. Energy detection of a signal with random amplitude. In Proceedings of the 2002 IEEE International Conference on Communications. Conference Proceedings, New York, NY, USA, 28 April–2 May 2002; Volume 3, pp. 1606–1610. [Google Scholar]
- Digham, F.F.; Alouini, M.; Simon, M.K. On the energy detection of unknown signals over fading channels. IEEE Trans. Commun.
**2007**, 55, 21–24. [Google Scholar] [CrossRef] - Ghasemi, A.; Sousa, E.S. Collaborative spectrum sensing for opportunistic access in fading environments. In Proceedings of the First IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, MD, USA, USA, 8–11 November 2005; pp. 131–136. [Google Scholar]
- Ye, Z.; Memik, G.; Grosspietsch, J. Energy Detection Using Estimated Noise Variance for Spectrum Sensing in Cognitive Radio Networks. In Proceedings of the Wireless Communications and Networking Conference, Las Vegas, NV, USA, 31 March–3 April 2008; pp. 711–716. [Google Scholar]
- Arjoune, Y.; Mrabet, Z.E.; Ghazi, H.E.; Tamtaoui, A. Spectrum sensing: Enhanced energy detection technique based on noise measurement. In Proceedings of the 2018 IEEE 8th Annual Computing and Communication Workshop and Conference (CCWC), Las Vegas, NV, USA, 8–10 January 2018. [Google Scholar] [CrossRef]
- Kustra, M.; Kosmowski, K.; Suchanski, M. Performance of hybrid sensing method in environment with noise uncertainty. J. Telecommun. Inf. Technol.
**2018**, 1, 51–57. [Google Scholar] [CrossRef] - Yusuf, D.P.; Onwuka, E.; Alenoghena, C.; Agajo, J. Discrete wavelet packet based spectrum sensing in cognitive radio using an improved adaptive threshold. In Proceedings of the International Conference on Industrial Engineering and Operations Management, Bandung, Indonesia, 6–8 March 2018. [Google Scholar]
- Joshi, D.R.; Popescu, D.C.; Dobre, O.A. Adaptive spectrum sensing with noise variance estimation for dynamic cognitive radio systems. In Proceedings of the Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA, 17–19 March 2010; pp. 1–5. [Google Scholar]
- Wu, J.; Luo, T.; Yue, G. An Energy Detection Algorithm Based on Double-Threshold in Cognitive Radio Systems. In Proceedings of the International Conference on Information Science and Engineering, Nanjing, China, 26–28 Decemner 2009; pp. 493–496. [Google Scholar]
- Peh, E.; Liang, Y. Optimization of cooperative sensing in cognitive radio networks. In Proceedings of the WCNC 2007, Kowloon, China, 11–15 March 2007; pp. 27–32. [Google Scholar]
- Shellhammer, S.; Shankar, S.; Dandra, R.; Tomcik, J. Performance of power detector sensors of DVT signals in IEEE 802.22 WRANs. In Proceedings of the First International Workshop on Technology and Policy for Accessing Spectrum, Boston, MA, USA, 5 August 2006. [Google Scholar]
- Kuang, J. Applied Inequalities, 3rd ed.; Shandong Science and Technology Press: Jinan, China, 2004. (In Chinese) [Google Scholar]

**Figure 1.**Energy detector using estimated noise variance by available similar channel. The noise variance is estimated by ${\widehat{\sigma}}_{n}^{2}=\frac{1}{N}{\sum}_{k=1}^{N}{n}_{k}^{2}$ from the other available similar channel. Then the empirical average energy $u=\frac{1}{M}{\sum}_{k=1}^{M}{x}_{k}^{2}$ is detected by principle constant false alarm rate (CFAR) or constant detection rate (CDR) with the threshold $\widehat{\lambda}$ using the estimated noise variance ${\widehat{\sigma}}_{n}^{2}$.

**Figure 2.**Plots of Examples 1 and 2. (

**a**) Average false alarm probability ${\overline{\widehat{P}}}_{fa}>{P}_{fa}$ vs. P

_{fa}= 0, 0.05, …, 1. (

**b**) ${\overline{\widehat{P}}}_{fa}$ vs. M = N = 1, 2, …, 100

**Figure 3.**The plot of ${\overline{\widehat{P}}}_{fa}$ by ’+’, $E{\widehat{P}}_{fa}$ and $E{\widehat{\tilde{P}}}_{fa}$ under setting of P

_{fa}= 0, 0.05, …, 1 and M = N = 1, 2, …, 100. (

**a**) The plot of ${\overline{\widehat{P}}}_{fa}$ by ’+’, $E{\widehat{P}}_{fa}$ and $E{\widehat{\tilde{P}}}_{fa}$ vs P

_{fa}= 0, 0.05, …, 1. (

**b**) The plot of ${\overline{\widehat{P}}}_{fa}$, $E{\widehat{P}}_{fa}$ and $E{\widehat{\tilde{P}}}_{fa}$ vs. $M=N=1,2,\dots ,100$, under P

_{fa}= 0.05.

**Figure 4.**The plot of approximation of $E{\widehat{P}}_{fa}^{2}$ and $E{\widehat{\tilde{P}}}_{fa}^{2}$ and their upper bounds. (

**a**) The plot of ${\overline{{\widehat{P}}^{2}}}_{fa}$ by ’+’, $\mathbf{B}({\widehat{P}}_{fa}^{2})$ and $\mathbf{B}({\widehat{\tilde{P}}}_{fa}^{2})$ vs. ${P}_{fa}$ = 0, 0.05, …, 1. (

**b**) The plot of ${\overline{{\widehat{P}}^{2}}}_{fa}$ by ’+’, $\mathbf{B}({\widehat{P}}_{fa}^{2})$ and $\mathbf{B}({\widehat{\tilde{P}}}_{fa}^{2})$ vs. $M=N=1,\dots ,100$.

**Figure 5.**The plot of preassigned false alarm rate ${P}_{fa}^{0}$ to guarantee $E{\widehat{P}}_{fa}={P}_{fa}$ and $E{\widehat{\tilde{P}}}_{fa}={P}_{fa}$ by Theorems 1 and 3 respectively. (

**a**) Preassigned rate derived by Theorem 1 to guarantee $E{\widehat{P}}_{fa}={P}_{fa}$. (

**b**) Preassigned rate derived by Theorem 1 to guarantee $E{\widehat{\tilde{P}}}_{fa}={P}_{fa}$.

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**MDPI and ACS Style**

Hu, X.-L.; Ho, P.-H.; Peng, L.
Statistical Properties of Energy Detection for Spectrum Sensing by Using Estimated Noise Variance. *J. Sens. Actuator Netw.* **2019**, *8*, 28.
https://doi.org/10.3390/jsan8020028

**AMA Style**

Hu X-L, Ho P-H, Peng L.
Statistical Properties of Energy Detection for Spectrum Sensing by Using Estimated Noise Variance. *Journal of Sensor and Actuator Networks*. 2019; 8(2):28.
https://doi.org/10.3390/jsan8020028

**Chicago/Turabian Style**

Hu, Xiao-Li, Pin-Han Ho, and Limei Peng.
2019. "Statistical Properties of Energy Detection for Spectrum Sensing by Using Estimated Noise Variance" *Journal of Sensor and Actuator Networks* 8, no. 2: 28.
https://doi.org/10.3390/jsan8020028