# Robust Spectrum Sensing Detector Based on MIMO Cognitive Radios with Non-Perfect Channel Gain

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## Abstract

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## 1. Introduction

- (1)
- Observed data vectors in CR with multicell multiple groups at the secondary users are generated while maintaining the SNR levels with range values at the primary users. Then, the Bayesian method is used to assume priors for the unknown probabilistic parameters to extract a posterior probability distribution vector for the observation data samples of the CR system.
- (2)
- We involve the MAP method to determine the posterior probability distribution expression for the unknown probabilistic parameter of the observation data to extract unknown matrices for the distribution parameters.
- (3)
- We present two approaches to address the channel uncertainty and the noise covariance matrix that complicate the resultant optimization problem. The solution for this problem is examined under different approaches; this problem is solved by a sub-optimal solution in the first approach while a robust solution is used in the second approach.
- (4)
- We prove that our approaches in the spectrum sensing problem based on the assumptions are effective methods to address this the uncertainty.

**Notation**

**1.**

## 2. Spectrum Sensing Detector

## 3. System Descriptions

## 4. Methodology

#### 4.1. B-GLRT Detector for Unknown Noise Covariance Matrix with Perfect CSI (B-GLRT1)

#### 4.2. B-GLRT Detector for Unknown Noise Covariance Matrix with Non-Perfect CSI

#### 4.2.1. Sub-Optimal Solution, B-GLT2

#### 4.2.2. Robust Solution, B-GLT3

- Solving for ${\mathrm{\Gamma}}_{0}$: in this case, we assume that ${\mathrm{\Gamma}}_{0}$ is unknown and $\Delta {\mathbf{R}}_{h}$ is known and then solve for ${\mathrm{\Gamma}}_{0}$ as shown in $\mathit{P}6$.$$\begin{array}{ccc}\hfill P6& :& \underset{{\mathrm{\Gamma}}_{0}}{\mathrm{argmax}}(\mathit{L}log|\underset{{\mathbf{R}}_{h}}{\underbrace{({\widehat{\mathbf{R}}}_{h}+\Delta {\mathbf{R}}_{h})}}+{\mathrm{\Gamma}}_{0}{|}^{-1}\hfill \\ & & -tr\left[\mathbf{R}{(\underset{{\mathbf{R}}_{h}}{\underbrace{({\widehat{\mathbf{R}}}_{h}+\Delta {\mathbf{R}}_{h})}}+{\mathrm{\Gamma}}_{0})}^{-1}\right]\hfill \\ & & +(\rho +1)log\left|{\mathrm{\Gamma}}_{0}^{-1}\right|-tr\left[\mathbf{K}{\mathrm{\Gamma}}_{0}^{-1}\right]).\hfill \end{array}$$$${\mathrm{\Gamma}}_{\mathbf{0}}\ge 0,$$For additional details see Appendix A.2.
- Solving for $\Delta {\mathbf{R}}_{h}$: now we can assume that ${\mathrm{\Gamma}}_{0}$ is known and solve for $\Delta {\mathbf{R}}_{h}$. We also define that $\Re ={\widehat{\mathbf{R}}}_{h}+{\mathrm{\Gamma}}_{0}$, then the problem becomes:$$\begin{array}{ccc}\hfill P7& :& \underset{\Delta {\mathbf{R}}_{h}}{\mathrm{argmax}}(\mathit{L}log{\left|(\Re +\Delta {\mathbf{R}}_{h})\right|}^{-1}\hfill \\ & & -tr\left[\mathbf{R}{\left(\Re +\Delta {\mathbf{R}}_{h}\right)}^{-1}\right]\hfill \\ & & +(\rho +1)log\left|{\mathrm{\Gamma}}_{0}^{-1}\right|-tr\left[\mathbf{K}{\mathrm{\Gamma}}_{0}^{-1}\right]).\hfill \end{array}$$$${\u2225\Delta {\mathbf{R}}_{h}\u2225}_{F}\le \u03f5,$$Appendix A.2 provides more detail on this derivation.

## 5. Numerical Evaluation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1

- Equation (24) can be written equivalently as$$\begin{array}{ccc}\hfill {\mathbf{R}}_{\eta}& =& \underset{\underset{\underset{\mathrm{\Psi}\ge 0}{\mathbf{M}:Diagonal}}{{\mathrm{\Gamma}}_{0}:Diagonal}}{\mathrm{argmax}}(\mathit{L}\left|\mathrm{\Psi}\right|-tr\left\{\mathbf{R}\mathrm{\Psi}\right\}\hfill \\ & & +(\alpha +1)log\left|\mathbf{M}\right|-etr\left[\mathbf{B}\mathbf{M}\right])\hfill \end{array}$$$${\mathrm{\Gamma}}_{1}^{-1}\le \mathrm{\Psi}$$$${\mathrm{\Gamma}}_{0}^{-1}\le \mathbf{M}$$$$\mathbf{M}\ge \mathrm{\Psi}.$$This problem can be solved by using Schur complement as shown in our previous work in [32]:$$\left[\begin{array}{cc}\mathrm{\Psi}& \mathbf{I}\\ \mathbf{I}& {\mathbf{R}}_{\eta}+\underset{{\mathbf{R}}_{h}}{\underbrace{({\widehat{\mathbf{R}}}_{h}+\Delta {\mathbf{R}}_{h})}}\end{array}\right]\ge 0$$Similarly,$$\left[\begin{array}{cc}\mathbf{M}& \mathbf{I}\\ \mathbf{I}& {\mathbf{R}}_{\eta}\end{array}\right]\ge 0$$
- In sub-optimal method (B-GLRT2), $\Delta {\mathbf{R}}_{h}$ is equal to $+\u03f5{\mathbf{I}}_{Nr}$ then problem reduces to$$\begin{array}{ccc}\hfill {\mathbf{R}}_{\eta}& =& \underset{\underset{\underset{\mathrm{\Psi}\ge 0}{\mathbf{M}:Diagonal}}{{\mathrm{\Gamma}}_{0}:Diagonal}}{\mathrm{argmax}}(\mathit{L}\left|\mathrm{\Psi}\right|-tr\left\{\mathbf{R}\mathrm{\Psi}\right\}\hfill \\ & & +(\alpha +1)log\left|\mathbf{M}\right|-etr\left[\mathbf{B}\mathbf{M}\right])\hfill \end{array}$$$$\left[\begin{array}{cc}\mathrm{\Psi}& \mathbf{I}\\ \mathbf{I}& {\mathbf{R}}_{\eta}+\underset{{\mathbf{R}}_{h}}{\underbrace{({\widehat{\mathbf{R}}}_{h}+\u03f5{\mathbf{I}}_{Nr})}}\end{array}\right]\ge 0$$$$\left[\begin{array}{cc}\mathbf{M}& \mathbf{I}\\ \mathbf{I}& {\mathbf{R}}_{\eta}\end{array}\right]\ge 0$$$$\mathbf{M}\ge \mathrm{\Psi}$$

#### Appendix A.2

- Solving for ${\mathrm{\Gamma}}_{0}$, the problem in Equation (30) reduced to $\mathit{P}8$:$$\begin{array}{ccc}\hfill P8& :& \underset{{\mathrm{\Gamma}}_{0}:Diagonal}{\mathrm{argmax}}(\mathit{L}log{\left|{\mathbf{R}}_{h}+{\mathrm{\Gamma}}_{0}\right|}^{-1}\hfill \\ & & -tr\left[\mathbf{R}{\left({\mathbf{R}}_{h}+{\mathrm{\Gamma}}_{0}\right)}_{1}^{-1}\right]\hfill \\ & & +(\rho +1)log\left|{\mathrm{\Gamma}}_{0}^{-1}\right|-tr\left[\mathbf{B}{\mathrm{\Gamma}}_{0}^{-1}\right])\hfill \end{array}$$$${({\mathbf{R}}_{h}+{\mathrm{\Gamma}}_{0})}^{-1}\le \mathrm{\Psi}$$$${\mathrm{\Gamma}}_{0}^{-1}\le \mathbf{M}$$$$\mathbf{M}\ge \mathrm{\Psi}$$This problem can be solved according to the solution that is mentioned in Appendix A.1.
- Back to Equation (30) to solve for $\Delta {\mathbf{R}}_{h}$, the problem reduce to:$$\begin{array}{ccc}\hfill P9& :& \underset{\Delta {\mathbf{R}}_{h}}{\mathrm{argmax}}(\mathit{L}log{\left|(\Re +\Delta {\mathbf{R}}_{h})\right|}^{-1}\hfill \\ & & -tr\left[\mathbf{R}{\left(\Re +\Delta {\mathbf{R}}_{h}\right)}^{-1}\right]\hfill \\ & & +(\rho +1)log\left|{\mathrm{\Gamma}}_{0}^{-1}\right|-tr\left[\mathbf{K}{\mathrm{\Gamma}}_{0}^{-1}\right])\hfill \end{array}$$$${(\Re +\Delta {\mathbf{R}}_{h})}^{-1}\le \mathrm{\Psi}{\u2225\Delta {\mathbf{R}}_{h}\u2225}_{F}\le \u03f5.$$This equation can also be solved using Schur complement; the problem will becomes:$$\left[\begin{array}{cc}\mathrm{\Psi}& \mathbf{I}\\ \mathbf{I}& (\Re +\Delta {\mathbf{R}}_{h})\end{array}\right]\ge 0$$

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**Figure 1.**MIMO cognitive radio system model consisting of single PU with ${N}_{t}$ transmitting antennas and single secondary user with ${N}_{r}$ receiving antennas.

**Figure 3.**Probability of missed detection versus the SNR for $Pfa=5\times {10}^{-1}$, ${\mathit{N}}_{\mathit{r}}$ = 3 and $\mathit{L}$ = 20.

**Figure 4.**The ROC performance of the proposed B-GLRT where a SNR = –3 dB, ${\mathit{N}}_{\mathit{r}}$ = 3 and $\mathit{L}$ = 10.

**Figure 5.**Probability of missed detection of the proposed approach versus $\mathit{L}$ where a pfa = $5\times {10}^{-1}$, an average SNR = –3 dB.

**Figure 6.**Probability of detection versus probability of false alarm at SNR = −3 dB, ${\mathit{N}}_{\mathit{r}}$ = 3, ${\mathit{N}}_{\mathit{t}}$ = 2 and $\mathit{L}$ = 20.

**Figure 7.**Probability of detection against probability of false alarm at SNR = – 3 dB, ${\mathit{N}}_{\mathit{r}}$ = 5, ${\mathit{N}}_{\mathit{t}}$ = 3 and $\mathit{L}$ = 25.

**Figure 8.**Probability of missed detection for the detectors versus a SNR with $Pfa=5\times {10}^{-1}$, ${\mathit{N}}_{\mathit{r}}$ = 3, ${\mathit{N}}_{\mathit{t}}$ = 2.

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## Share and Cite

**MDPI and ACS Style**

Al-Amidie, M.; Al-Asadi, A.; Humaidi, A.J.; Al-Dujaili, A.; Alzubaidi, L.; Farhan, L.; Fadhel, M.A.; McGarvey, R.G.; Islam, N.E.
Robust Spectrum Sensing Detector Based on MIMO Cognitive Radios with Non-Perfect Channel Gain. *Electronics* **2021**, *10*, 529.
https://doi.org/10.3390/electronics10050529

**AMA Style**

Al-Amidie M, Al-Asadi A, Humaidi AJ, Al-Dujaili A, Alzubaidi L, Farhan L, Fadhel MA, McGarvey RG, Islam NE.
Robust Spectrum Sensing Detector Based on MIMO Cognitive Radios with Non-Perfect Channel Gain. *Electronics*. 2021; 10(5):529.
https://doi.org/10.3390/electronics10050529

**Chicago/Turabian Style**

Al-Amidie, Muthana, Ahmed Al-Asadi, Amjad J. Humaidi, Ayad Al-Dujaili, Laith Alzubaidi, Laith Farhan, Mohammed A. Fadhel, Ronald G. McGarvey, and Naz E. Islam.
2021. "Robust Spectrum Sensing Detector Based on MIMO Cognitive Radios with Non-Perfect Channel Gain" *Electronics* 10, no. 5: 529.
https://doi.org/10.3390/electronics10050529