# Computationally Efficient Direction-of-Arrival Estimation Algorithms for a Cubic Coprime Array

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## Abstract

**:**

## 1. Introduction

- (a)
- We propose a TA-MUSIC algorithm with CCA geometry to enable massive MIMOs to fully employ DOFs and achieve superior DOA estimation by employing both the auto-covariance matrix and the mutual covariance matrix of the entire array. In addition, we verify that by using the coprime property, the proposed algorithm can suppress the ambiguity problem.
- (b)
- We propose an E-MUSIC algorithm for 2D DOA estimation, which can effectively decrease the complexity of the classic MUSIC algorithm. After utilizing the ESPRIT algorithm to initialize and obtain a rough estimation, we then conduct a fine search within a smaller sector to achieve lower complexity.
- (c)
- Our numerical simulation results confirm that the proposed algorithms outperform the classical ESPRIT algorithm and the PM algorithm in DOA estimation.

## 2. Signal Model

## 3. Proposed Method for DOA Estimation

#### 3.1. Review

#### 3.2. TA-MUSIC Algorithm

**Lemma**

**1.**

**Proof.**

_{1}, M

_{2}, ${T}_{1}$,${T}_{2}$ and ${P}_{1},{P}_{2}$ are coprime integers pairs, respectively, Equation (20) can hold only in the case of ${k}_{u1}={k}_{u2}=0,{k}_{v1}={k}_{v2}=0,{k}_{w1}={k}_{w2}=0$. This indicates that no ambiguity problem has arisen. □

#### 3.3. The E-MUSIC Algorithm

**T**is a nonsingular matrix and ${\widehat{E}}_{s,1}$ is signal subspace which is segmented into ${\widehat{E}}_{x1,1}$ and ${\widehat{E}}_{x2,1}$, denoting ${\widehat{E}}_{s,1}(1:({M}_{1}-1){T}_{1}{P}_{1},:)$ and ${\widehat{E}}_{s,1}({T}_{1}{P}_{1}+1:{T}_{1}{M}_{1}{P}_{1},:)$; and corresponding steering matrices ${A}_{x1,1}$ and ${A}_{x2,1}$, denoting ${A}_{1}(1:({M}_{1}-1){T}_{1}{P}_{1},:)$ and ${A}_{1}({T}_{1}{P}_{1}+1:{M}_{1}{T}_{1}{P}_{1},:)$, respectively. Based on the uniformities of steering matrices, we have:

#### 3.4. Detailed Steps

## 4. Performance Analysis

#### 4.1. Computational Complexity

#### 4.2. Degree of DOF

#### 4.3. Advantages

- The proposed TA-MUSIC performs better DOA estimations by employing all of the array information, including the auto-covariance matrix and the mutual covariance matrix, whereas E-MUSIC only utilizes the auto-covariance matrix information. In addition, TA-MUSIC can fully achieve DOFs of ${M}_{1}{T}_{1}{P}_{1}+{M}_{2}{T}_{2}{P}_{2}-1$, while the algorithms in [34,35,36] only achieve ${M}_{2}{T}_{2}{P}_{2}$ DOFs. Furthermore, E-MUSIC can obtain ${M}_{1}{T}_{1}{P}_{1}$ DOFs, which are larger than ${M}_{2}{T}_{2}{P}_{2}({M}_{1}>{M}_{2})$.
- The proposed TA–MUSIC algorithm can attain paired angles automatically and outperforms the conventional ESPRIT and PM algorithms in DOA estimation performance.

#### 4.4. Cramer–Rao Bound

## 5. Simulation Results

#### 5.1. Comparison of the DOA Estimation Performance of Different Algorithms

#### 5.2. RMSE with a Varying Number of Sensors

#### 5.3. Resolution Performance

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The structure of a CCA configuration (${M}_{1}=4,{T}_{1}=4,{P}_{1}=4$ and ${M}_{2}=3,{T}_{2}=3,{P}_{2}=3$ ).

Algorithm | Complex Multiplication | Running time |
---|---|---|

Classic MUSIC | $1.880416\times {10}^{12}$ | 3406.7221 |

TA-MUSIC | $3.280095\times {10}^{12}$ | 7013.1121 |

E-MUSIC | $2.45380\times {10}^{6}$ | 0.00697637 |

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Gong, P.; Chen, X.
Computationally Efficient Direction-of-Arrival Estimation Algorithms for a Cubic Coprime Array. *Sensors* **2022**, *22*, 136.
https://doi.org/10.3390/s22010136

**AMA Style**

Gong P, Chen X.
Computationally Efficient Direction-of-Arrival Estimation Algorithms for a Cubic Coprime Array. *Sensors*. 2022; 22(1):136.
https://doi.org/10.3390/s22010136

**Chicago/Turabian Style**

Gong, Pan, and Xixin Chen.
2022. "Computationally Efficient Direction-of-Arrival Estimation Algorithms for a Cubic Coprime Array" *Sensors* 22, no. 1: 136.
https://doi.org/10.3390/s22010136