# Passive MIMO Radar Detection with Unknown Colored Gaussian Noise

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## Abstract

**:**

## 1. Introduction

## 2. Signal Model and Assumption

## 3. GLRT for Passive MIMO Radar

#### 3.1. Probability Distribution of Received Signal

#### 3.2. GLRT Derivation

_{fa}) is fixed. Nevertheless, for the problem considered in this paper, the likelihood ratio test can not be obtained because the PDF of the received signals depends on the unknown channel coefficients, transmitted signals, and covariance matrix of colored noise. To solve this problem, a typical method is to use the maximum likelihood estimates of unknown parameters as the true values. This method is called GLRT. The GLRT for our problem will be investigated in this subsection.

_{fa}in practice.

**Theorem**

**1.**

#### 3.3. Performance Analysis and Discussion

## 4. Simulation

#### 4.1. Variation of P_{d} with P_{fa}

_{d}) on P

_{fa}is illustrated in Figure 2. In Figure 2a, the TNR is −15 dB and the DNR is −30 dB. We choose the number of training samples to be $K=3$, $K=9$, and $K=15$, respectively. In Figure 2b, the TNRs are set as −18 dB, −15 dB, and −12 dB, respectively. The DNR is −30 dB and $K=15$ training samples are used.

_{d}on P

_{fa}is illustrated in Figure 3. In Figure 3, the TNR is chosen to be −15 dB, and there are $K=15$ training samples. The DNRs are −30 dB, −10 dB, and 10 dB, respectively. When the DNR is improved, both the performance of the proposed GLRT and that of the GLRT in [23] are improved, since more accurate estimates of unknown parameters are obtained with high DNR. Compared with the proposed GLRT, the GLRT in [23] can achieve greater performance gain when the DNR is increased, since the proposed GLRT needs to estimate more unknown parameters.

#### 4.2. Variation of P_{d} with TNR

_{d}on TNR is illustrated in Figure 4. In Figure 4a, we consider that there are $K=6$, $K=9$, $K=15$, and $K=21$ training samples, respectively. The P

_{fa}is chosen to be ${10}^{-3}$, and the DNR is set as −30 dB. In Figure 4b, the DNR is −30 dB, and we consider that there are $K=18$ training samples. The P

_{fa}s are ${10}^{-3}$, ${10}^{-2}$, and ${10}^{-1}$, respectively.

_{d}of both the GLRT in [23] and the proposed GLRT is increased with the increase of TNR, because the estimates of the unknown parameters become more accurate when TNR is increased. Moreover, as the number of training samples increases, the proposed GLRT gradually approaches the GLRT in [23]. Increasing the P

_{fa}, the detection threshold will be reduced, and hence the P

_{d}will be improved.

_{d}with TNR is illustrated with different DNRs. In Figure 5a, the DNRs are considered to be −30 dB, −10 dB, and 10 dB, respectively, and there are $K=21$ training samples. The P

_{fa}is set as ${10}^{-3}$.

_{d}with TNR for different numbers of receivers and transmitters is shown. In Figure 5b, there are $K=20$ training samples, the DNR is −30 dB, and the P

_{fa}is chosen to be ${10}^{-3}$. There are (${N}_{t}=1,{N}_{r}=2$), (${N}_{t}=2,{N}_{r}=2$), (${N}_{t}=2,{N}_{r}=3$), and (${N}_{t}=3,{N}_{r}=3$) transmitters and receivers, respectively.

_{d}is improved when either the number of transmitters or the number of receivers is improved. Obviously, with more transmitters and receivers, we can obtain more observations that will contribute to obtaining more accurate estimates of unknown parameters. Hence, we can achieve higher P

_{d}with more transmitters and receivers.

#### 4.3. P_{d} Loss

_{d}loss is utilized, which is defined as

_{d}of the proposed GLRT, and ${P}_{{d}_{\mathrm{K}}}$ denotes the P

_{d}of the GLRT in [23].

_{d}loss of the proposed method is shown. In Figure 6a, the P

_{fa}is ${10}^{-3}$, the DNR is −30 dB, and the TNRs are, respectively, set as −14 dB, −12 dB, −10 dB, and −8 dB. In Figure 6b, the DNR is considered to be −30 dB, the TNR is chosen to be −12 dB, and the P

_{fa}s are ${10}^{-3}$, ${10}^{-2}$, and ${10}^{-1}$, respectively.

_{d}loss decreases as the number of training samples increases. This is due to that we can achieve more accurate estimates of the unknown covariance matrix of colored noise with the increase of the number of training samples. As shown in Figure 6, when there are more than 25 training samples, the P

_{d}loss will be close to 0. For low TNR, the P

_{d}loss is smaller as shown in Figure 6a. For low TNR, both the GLRT in [23] and the proposed GLRT have small P

_{d}, which will result in a small P

_{d}loss. Moreover, the proposed GLRT needs to estimate more unknown parameters than the GLRT in [23]. Using the same training samples, the GLRT in [23] will achieve more performance improvement when the TNR is increased. Hence, the Pd loss will be increased when the TNR is increased. In Figure 6b, for large P

_{fa}, the P

_{d}loss is larger. Under large P

_{fa}, both the P

_{d}of the GLRT in [23] and that of the proposed GLRT are large. For large P

_{fa}, to achieve the same P

_{d}loss, it needs more training samples for the proposed GLRT.

_{d}loss on the number of training samples is illustrated. In Figure 7, the TNR is −12 dB, the P

_{fa}is ${10}^{-3}$, and the DNRs are −30 dB, −10 dB, and 10 dB, respectively. As the number of training samples increases, the P

_{d}loss is decreased. Moreover, the proposed GLRT will need more training samples to obtain the same P

_{d}loss when the DNR is improved.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

## Appendix B. Maximum Likelihood Estimate $\widehat{\mathbf{U}}$

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**Figure 1.**The notional passive MIMO radar consists of two multichannel receiver and two transmitters.

**Figure 4.**The variation of P

_{d}with TNR. (

**a**) DNR = −30 dB, P

_{fa}= ${10}^{-3}$. (

**b**) DNR = −30 dB, $K=18$.

**Figure 5.**The variation of P

_{d}with TNR. (

**a**) P

_{fa}= ${10}^{-3}$, $K=21$. (

**b**) P

_{fa}= ${10}^{-3}$, $K=20$, DNR = −30 dB.

**Figure 6.**The P

_{d}loss of the proposed method. (

**a**) DNR = −30 dB, P

_{fa}= ${10}^{-3}$. (

**b**) DNR = −30 dB, TNR = −12 dB.

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**MDPI and ACS Style**

Liu, Y.; Liao, G.; Li, H.; Zhu, S.; Li, Y.; Yin, Y.
Passive MIMO Radar Detection with Unknown Colored Gaussian Noise. *Remote Sens.* **2021**, *13*, 2708.
https://doi.org/10.3390/rs13142708

**AMA Style**

Liu Y, Liao G, Li H, Zhu S, Li Y, Yin Y.
Passive MIMO Radar Detection with Unknown Colored Gaussian Noise. *Remote Sensing*. 2021; 13(14):2708.
https://doi.org/10.3390/rs13142708

**Chicago/Turabian Style**

Liu, Yongjun, Guisheng Liao, Haichuan Li, Shengqi Zhu, Yachao Li, and Yingzeng Yin.
2021. "Passive MIMO Radar Detection with Unknown Colored Gaussian Noise" *Remote Sensing* 13, no. 14: 2708.
https://doi.org/10.3390/rs13142708