# Joint Antenna Placement and Power Allocation for Target Detection in a Distributed MIMO Radar Network

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background and Related Studies

#### 1.2. Contributions

- We formulate the joint power allocation and antenna placement problem as an optimization model subject to the resource budget and area priority level. In the proposed JAPPA model, binary composite hypothesis testing is established to design the Neyman–Pearson (NP) detector for the whole surveillance area with the targets Radar Cross Section (RCS) obeying Rayleigh distribution. Compared to previous oversimplified works [19,20], the weighted NP-based logarithm likelihood ratio test (LRT) function is specified as the utility function that combines all optimization factors into a single objective function. In addition, we evaluate the average utility function values after changing the single stationary target’s position in the whole region to describe the global target detection performance of the radar network.
- We propose an efficient JAPPA strategy that incorporates antenna placement with power allocation to optimize the target detection performance in the radar network. Different from the 0–1 programming in [21,31], we choose suitable antenna deployment positions in the regional grid points to establish the non-linear mixed integer programming problem. To devise computationally feasible methods for practical applications, a two-stage local-search-based algorithm is proposed to split the coupled joint optimization. Herein, it isolates integer programming from continuous variable optimization and provides equivalent performance while requiring less computing effort.
- We develop a joint optimization closed-loop scheme for the joint antenna placement and power allocation optimization [30]. In our scheme, the optimal position results in the current cycle are used for guiding the power allocation. The detection performance will be further improved through this link, which renders a closed-loop scheme to repeat the iteration of transmitting and receiving antenna placement using the optimal power allocation scheme.

#### 1.3. Organization

## 2. System Model and Detection Formulation

#### 2.1. Signal Model

#### 2.2. Neyman–Pearson Detection Model

## 3. Antenna Placement and Power Allocation Strategy

#### 3.1. Optimization Formulation

#### 3.2. Solution Technique

#### 3.2.1. Local Search for Antenna Placement

Algorithm 1: Two-stage greedy dropping heuristic local search for antenna placement |

Input:$M$$,\text{}N$$,\text{}\mathcal{N}\mathcal{E}\mathcal{T}$$,\text{}{\zeta}_{q}$ |

Output: optimal ${L}_{T,k}={[{L}_{T,k}^{1},\dots ,{L}_{T,k}^{m},\dots ,{L}_{T,k}^{M}]}^{T}$ and ${L}_{R,k}={[{L}_{R,k}^{1},\dots ,{L}_{R,k}^{n},\dots ,{L}_{R,k}^{N}]}^{T}$ |

1 Initialize $k=1$, $q=1$, ${L}_{T,1}^{1},\dots ,{L}_{T,1}^{M}$ and ${L}_{R,1}^{1},\dots ,{L}_{R,1}^{N}$; |

2 double $Tem{p}_{0,1}=Tem{p}_{1,0}=0$$,\epsilon =1e-6$; |

3 while$\text{}\mathbb{F}({L}_{T,\mathrm{k}},{L}_{R,q},{P}_{T,average})-Tem{p}_{k-1,q}\ge \epsilon $ do |

4 Replaces $Tem{p}_{k,q}=\mathbb{F}({L}_{T,k},{L}_{R,q},{P}_{T,average})$ |

5 Recombines ${{L}^{\prime}}_{T,k}$: substitute ${L}_{T,k}^{m}$ with a new coordinate $L$$,\text{}L\notin {L}_{T,k}\cup \cdots \cup {L}_{T,0}$ |

6 if $\text{}\mathbb{F}({{L}^{\prime}}_{T,k},{L}_{R,q},{P}_{T,average})\ge Tem{p}_{q,k}$ then |

7 ${L}_{T,k+1}={{L}^{\prime}}_{T,k}$ |

8 Removes the $L$ from the set of candidate points |

9 end if |

10 $k=k+1$ |

11 end while |

12 while $\mathbb{F}({L}_{T,k},{L}_{R,q},{P}_{T,average})-Tem{p}_{k,q-1}\ge \epsilon $ do |

13 Replaces $Tem{p}_{k,q}=\mathbb{F}({L}_{T,k},{L}_{R,q},{P}_{T,average})$ |

14 Recombines ${{L}^{\prime}}_{R,q}$: substitute ${L}_{R,k}^{n}$ with a new coordinate $L$$,\text{}L\notin {L}_{R,k}\cup \cdots \cup {L}_{R,0}$ |

15 if $\text{}\mathbb{F}({L}_{T,k},{{L}^{\prime}}_{R,q},{P}_{T,average})\ge Tem{p}_{k,q}$ then |

16 ${L}_{R,q}={{L}^{\prime}}_{R,q}$ |

17 Removes the $L$ from the set of candidate points |

18 end if |

19 $q=q+1$ |

20 end while |

21 return ${L}_{T,k+1}$, ${L}_{R,k+1}$ |

#### 3.2.2. Local-Search-Based Lagrange-KKT for Power Allocation

#### 3.3. Closed-Loop Resource Allocation Scheme

**Lemma 1.**

**Proof.**

**Lemma 2.**

#### 3.4. Computational Complexity Analysis

- (1)
- The local search for antenna placement;
- (2)
- The Lagrange-KKT for Power Allocation.

## 4. Simulation Results and Discussion

#### 4.1. Factors That Affect the Results of Iterative Optimization

#### 4.1.1. Case 1: Optimization Parameters’ Degree of Freedom

- Single-parameter optimization:
- (1)
- Transmitting antenna position optimization (TAO): The approach simply adjusts the location of the transmitting antenna to enhance the radar network’s target detection capacity.
- (2)
- Receiving antenna position optimization (RAO): The radar network transmitter sites remain constant with evenly distributed transmitting power, and only the placements of the receiving antennas are modified to improve the radar network’s detecting capability.
- (3)
- Transmit power allocation optimization (TPO): The transmitting power is efficiently allocated to increase target detection capabilities based on the random deployment of the radar network antenna space architecture.

- Two-element optimization:
- (1)
- Joint antenna placement optimization (JAP): In this experiment, the transmitting power is uniformly distributed and the antenna deployment positions are optimized using a two-stage method, in which transmitting and receiving antenna positions are iteratively optimized. The iteration completes when the increment reaches the given error limit.
- (2)
- Joint transmitter placement and power allocation optimization (JTPPA): We jointly design the transmit power allocation and transmitters’ placement to optimize radar performance criteria.

#### 4.1.2. Case 2: Iterative Loop Containment Range

#### 4.1.3. Case 3: Numbers of Antennas

#### 4.2. The Efficiency of the Iterative Two-Stage Closed-Loop Solver

#### 4.3. Effectiveness of the Proposed Method

- Random antenna placement with uniform power allocation (RAP-UPA): This method randomly selects the positions of the antennas with the uniform transmitting power resource allocation.
- Random antenna placement withnon-uniform power allocation (RAP-UUPA): The realistic scenario of non-uniform power allocation and random antenna placement will be considered in this simulation.
- Optimal antenna placement with uniform power allocation (OAP-UPA) [13]: In this scenario, the transmitters and receivers are optimally placed sequentially with the power consumption uniformly allocated to the transmitting antennas.
- Three-stage water-filling-type antenna placement and power allocation strategy (TWANPA) [19]: This method improves the radar detection performance through successive optimization under the total power constraint and designs a suboptimal method to effectively locate transmitting and receiving antennas under the water-filling power allocation. It completes the transmitting antenna placement using the power density criterion and positioning the receiving antenna with the SNR criterion. Subsequently, a closed-loop optimization mechanism is established by repeating the antennas’ positions using the allocated powers.

## 5. Conclusions

- The log-LRT function unifies the antenna position variables and transmitter power allocation variables into a single collaborative objective function, which is suitable for guiding the joint resource allocation optimization.
- The joint antenna and power allocation optimization can more effectively improve the target detection performance of the radar system.
- The proposed strategy is effective and efficient in solving the JAPPA problem.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 8.**Simulation results comparison (

**a**) optimum iterative process; (

**b**) average ROC curves for IOAP and JAPPA.

**Figure 9.**Optimal scheme based on IOAP. (

**a**) IOAP Antennas’ geometry distribution; (

**b**) IOAP power assigned to the transmitters.

**Figure 11.**Performance comparison of TWANPA, the exhaustive search, and the JAPPA: (

**a**) average ROC curves; (

**b**) runtime.

Algorithm | Exhaustive Search | TWANPA Method | JAPPA Strategy |
---|---|---|---|

Computational complexity | $\mathcal{O}\left({Q}^{M+N}MN\right)$ | $\mathcal{O}\left(M{Q}^{2}\right(NQ+1)+{N}^{2})$ | $\mathcal{O}\left(\right(M+N){Q}^{3}+MN)$ |

Names | Symbols | Settings |
---|---|---|

Transmitting Antenna Gain | ${\mathrm{G}}_{m}$ | 30 dB |

Receiving Antenna Gain | ${\mathrm{G}}_{n}$ | 30 dB |

Processing Gain At Receiver | ${I}_{p}$ | 20 dB |

Carrier Frequency | ${f}_{c}$ | 10 GHz |

Target Average RCS | ${\sigma}_{0}^{2}$ | 2 m^{2} |

Radar Network Scattering Loss | ${L}_{c}$ | 0 dB |

Radar Network Receiving Loss | ${L}_{r}$ | 0 dB |

Noise Factor | ${\mathrm{F}}_{n}$ | 4 dB |

Bandwidth | $\mathrm{B}$ | 5 MHz |

Transmit Power | ${P}_{TOTAL}$ | 30 KW |

Minimum Transmit Power | ${P}_{T\mathrm{min}}$ | 0 KW |

Maximum Transmit Power | ${P}_{T\mathrm{max}}$ | 30 KW |

Transmitter | x (km) | y (km) |
---|---|---|

#1 | 14 | 10 |

#2 | 2 | 14 |

#3 | 4 | 18 |

Transmitter | x (km) | y (km) |
---|---|---|

#1 | 4 | 2 |

#2 | 10 | 6 |

#3 | 14 | 10 |

Transmitter | x (km) | y (km) |
---|---|---|

#1 | 16 | 8 |

#2 | 12 | 12 |

#3 | 8 | 4 |

Transmitter | #1 | #2 | #3 |
---|---|---|---|

Power Allocation (KW) | 0.133 | 5.744 | 24.123 |

Transmitter | x (km) | y (km) |
---|---|---|

#1 | 18 | 14 |

#2 | 10 | 6 |

#3 | 14 | 10 |

Transmitter | x (km) | y (km) |
---|---|---|

#1 | 16 | 12 |

#2 | 12 | 8 |

#3 | 8 | 4 |

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**MDPI and ACS Style**

Qi, C.; Xie, J.; Zhang, H.
Joint Antenna Placement and Power Allocation for Target Detection in a Distributed MIMO Radar Network. *Remote Sens.* **2022**, *14*, 2650.
https://doi.org/10.3390/rs14112650

**AMA Style**

Qi C, Xie J, Zhang H.
Joint Antenna Placement and Power Allocation for Target Detection in a Distributed MIMO Radar Network. *Remote Sensing*. 2022; 14(11):2650.
https://doi.org/10.3390/rs14112650

**Chicago/Turabian Style**

Qi, Cheng, Junwei Xie, and Haowei Zhang.
2022. "Joint Antenna Placement and Power Allocation for Target Detection in a Distributed MIMO Radar Network" *Remote Sensing* 14, no. 11: 2650.
https://doi.org/10.3390/rs14112650