# Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Arrays Configuration and Signal Model

_{i}, φ

_{i}) (i = 1, 2, …, q) denoting nominal azimuth and nominal elevation of the ith sources impinging into arrays. θ

_{i}$\in \left[0,\mathsf{\pi}\right]$, φ

_{i}$\in \left[0,\mathsf{\pi}\right]$. λ is the wavelength of the impinging signal. The additive noise is considered as Gaussian white with zero mean and uncorrelated with sensors.

**x**-axis defined as subarray

**x**

_{1}can be written as

**n**

_{x}

_{1}(t) denotes the noise vector of subarray. The M × 1 dimensional vector

**α**

_{1}(θ,φ) denotes the steering vector of subarray

**x**

_{1}with respect to point source, which can be expressed as

_{i}(θ,φ,t) is the complex random angular signal density of the distributed source representing the reflection intensity of the source from angle (θ,φ) at the snapshot index t. Unlike a point source, signal of a distributed source exists not only in the direction of (θ

_{i},φ

_{i}) but also in a spatial distribution around (θ

_{i},φ

_{i}). A distributed source is defined as incoherently distributed if s

_{i}(θ,φ,t) from one direction is uncorrelated with other directions, which can be modeled as a random process as

_{i}is the power of the source, f

_{i}(θ,φ;

**u**

_{i}) is its normalized angular power density function (APDF) reflecting geometry and surface property of a distributed source. APDF is determined by parameter set

**u**

_{i}. For a Gaussian ID source,

**u**

_{i}= [${\theta}_{i}$, ${\varphi}_{i}$, ${\sigma}_{\theta i}$, ${\sigma}_{\varphi i}$, $\rho $] denoting nominal azimuth, nominal elevation, azimuth spread, elevation spread, and covariance coefficient respectively, APDF can be expressed as

**u**

_{i}= [${\theta}_{i}$, ${\varphi}_{i}$, ${\sigma}_{\theta i}$, ${\sigma}_{\varphi i}$] denoting nominal azimuth, nominal elevation, azimuth spread and elevation spread respectively. APDF can be expressed as

**x**-axis, which is defined as subarray

**x**

_{k}, can be written as

**α**

_{k}(θ,φ) denotes the steering vector of subarray

**x**

_{k}with respect to point source, which can be written as

**x**-axis can be expressed as

**z**-axis defined as subarray

**z**

_{1}can be written as

**β**

_{1}(θ,φ) denotes the steering vector of subarray

**z**

_{1}with respect to point source,

**n**

_{z}

_{1}(t) denotes the noise vector of subarray

**z**

_{1}.

**β**

_{1}(θ,φ) can be expressed as

**z**-axis defined as subarray

**z**

_{m}can be written as

**β**

_{m}(θ,φ) denotes the steering vector of subarray

**z**

_{m}with respect to point source, which can be written as

**z**-axis can be expressed as

## 3. Proposed Method

#### 3.1. Generalized Steering Matrix

**x**

_{k}and

**x**

_{k−}

_{1}at (θ

_{i}, φ

_{i}), we have

_{i}, φ

_{i}), the following relationship can be obtained from Equation (7)

**x**

_{1}as $[{A}_{11},{A}_{12},{A}_{13}]$.

**A**

_{11},

**A**

_{12}, and

**A**

_{13}are expressed as

**x**

_{k}as $[{A}_{k1},{A}_{k2},{A}_{k3}]$,

**x**

_{k}subarray as

**Φ**

_{z}is rotation invariance operator, which can be written as

**z**

_{m}and

**z**

_{m−}

_{1}at (θ

_{i}, φ

_{i}), we have

**z**

_{1}as $[{B}_{11},{B}_{12},{B}_{13}]$;

**B**

_{11},

**B**

_{12}, and

**B**

_{13}are written as

**z**

_{1}as $[{B}_{m1},{B}_{m2},{B}_{m3}]$,

**z**

_{m}$[{B}_{m1},{B}_{m2},{B}_{m3}]$ can be expressed as

**Φ**

_{x}is rotation invariance operator, which can be written as

#### 3.2. Generalized Signal Vector

^{H}; ${\overline{s}}_{1},{\overline{s}}_{2}$ and ${\overline{s}}_{3}$ are written as

**x**

_{k}can be expressed as a combination of the generalized signal vector and generalized steering matrix

**x**-axis can be express as

**n**

_{X}(t) is the noise vector of the URA along the

**x**-axis, which can be expressed as

**z**

_{m}can be expressed as

_{θi}, M

_{φi}can be express as

**Λ**,

**M**

_{θ}, and

**M**

_{φ}can be written as

#### 3.3. Nominal Angles Estimation

**X**

_{1}and

**X**

_{2}along the direction of the

**x**-axis.

**X**

_{1}is constituted by arrays from

**x**

_{1}to

**x**

_{K−}

_{1}; while

**X**

_{2}contains arrays from

**x**

_{2}to

**x**

_{K}. Thus,

**X**

_{1}(t) is M(K − 1) × 1 dimensional received vector of

**X**

_{1}also equals a vector containing elements from 1 to MK − M row of $\overline{X}$(t); whereas

**X**

_{2}(t) is M(K − 1) × 1 dimensional received vector of

**X**

_{2}containing elements from M + 1 to MK of $\overline{X}$(t).

**X**

_{1}(t) and

**X**

_{2}(t) can be written as

**n**

_{X}

_{1}(t) is noise vector of

**X**

_{1}containing elements from 1 to MK − M row of

**n**

_{X}(t),

**n**

_{X}

_{2}(t) is noise vector of

**X**

_{2}containing elements from M to MK row of

**n**

_{X}(t). $\left[{A}_{x1}^{1},{A}_{x2}^{1},{A}_{x3}^{1}\right]$ is the generalized steering matrix of subarray

**X**

_{1}(t), which can be expressed as

**X**

_{2}(t). From Equation (22) we can obtain

**X**

_{1}and

**X**

_{2}. According to the ESPRIT principle, combing the vector

**X**

_{1}(t) and

**X**

_{2}(t), we obtain a vector with rotational invariance property as

**X**

_{12}(t) can be expressed as

**X**

_{12}(t) which can be expressed as

**Λ,**

**M**

_{θ}

**Λ**and

**M**

_{φ}

**Λ**. The three parts all has q elements. Each part corresponds to respective eigenvectors. Under the assumption of small angular spread, we can obtain M

_{θi}< 1 and M

_{φi}< 1. Therefore, subspace spanned by eigenvectors corresponding to the largest q eigenvalues is equal to subspace spanned by

**A**

_{x}

_{1}. Suppose

**E**

_{x}is 2M(K − 1) × q dimensional matrix with columns as the eigenvectors of the covariance matrix ${R}_{x}^{12}$ corresponding to the q largest eigenvalues. Accordingly, there exists a q × q nonsingular matrix

**T**satisfying the following relation

**E**

_{x}

_{1}and

**E**

_{x}

_{2}denote matrices selecting upper and lower MK − M rows of

**E**

_{x}

**Φ**

_{x}is elements of

**Ф**

_{z}.

**Ω**

_{x}can be obtained as

^{+}denotes pseudo-inverse operator, then the nominal elevation of the sources can be get from

_{i}is the ith eigenvalues of

**Ω**

_{x}, angle(•) denotes argument of complex variable.

**Z**

_{1}and

**Z**

_{2}along the direction of

**z**-axis.

**Z**

_{1}is constituted by arrays from

**z**

_{1}to

**z**

_{M−}

_{1}; while

**Z**

_{2}contains arrays from

**z**

_{2}to

**z**

_{M}.$Z$

_{1}(t) is K(M − 1) × 1 dimensional receive vector of

**Z**

_{1}containing elements from 1 to K(M − 1) row of $\overline{Z}$(t); whereas

**Z**

_{2}(t) is K(M − 1) × 1 dimensional received vector of

**Z**

_{2}containing elements from K + 1 to MK row of $\overline{Z}$(t).

**n**

_{Z}

_{1}(t) and

**n**

_{Z}

_{2}(t) have the similar definition as

**n**

_{X}

_{1}(t) and

**n**

_{X}

_{2}(t). The received vectors

**Z**

_{1}(t) and

**Z**

_{2}(t) can be expressed as

**Z**

_{1}(t), which can be expressed as

**Z**

_{2}(t) $\left[{B}_{z1}^{2},{B}_{z2}^{2},{B}_{z3}^{2}\right]$ can be obtained as

**Z**

_{1}and

**Z**

_{2}. Combing the vector

**Z**

_{1}(t) and

**Z**

_{2}(t), we obtain a vector with rotational invariance property as

**Z**

_{12}(t) can be expressed as

**Z**

_{1}(t) and

**Z**

_{2}(t) can be expressed as

**Z**

_{12}(t) can be expressed as

**E**

_{z}is 2K(M − 1) × q matrix with columns as the eigenvectors of the covariance matrix ${R}_{z}^{12}$ corresponding to the q largest eigenvalues. As the same as

**E**

_{x},

**E**

_{z}is the same subspace spanned by ${B}_{z1}$. Accordingly, there exists a q × q nonsingular matrix

**Q**satisfying the following relation

**E**

_{z}

_{1}and

**E**

_{z}

_{2}denote matrices selecting upper and lower MK − K rows of

**E**

_{z}

**Φ**

_{z}is elements of

**Ф**

_{x}.

**Ω**

_{z}can be obtained as

_{i}is the ith eigenvalues of

**Ω**

_{z}.

#### 3.4. Angle Matching Method

**Ω**

_{x}and

**Ω**

_{z}are calculated, we need to match the right θ

_{i}to right φ

_{i}. We can obtain cost function by applying Capon principle to subarray

**x**

_{1}firstly. Then angle matching can be obtained by substituting all the possible pairs into the cost function.

**x**

_{1}is $\left[{A}_{11},{A}_{12},{A}_{13}\right]$, the Capon principle with regard to the subarray

**x**

_{1}can be expressed as

**x**

_{1}. Equation (75) can be solved through minimization of Lagrange function as follows

**w**and set the result equal to 0, we obtain

**w**

_{opt}can be obtained as follows

**Ω**

_{x}and

**Ω**

_{z}are calculated. Now we summarize the angle matching procedure as follows:

_{i}(i = 1, 2,…, q) from Equation (60). Select one elevation angle ${\widehat{\phi}}_{i}={\phi}_{i}$ form set $\{{\phi}_{1},{\phi}_{2},\cdots ,{\phi}_{q}\}$ at random. Superscript

_{^}denotes the angle already determined. Substitute ${\widehat{\phi}}_{i}$ to the eigenvalues set of

**Ω**

_{z}:$\{angle({\mu}_{1}),angle({\mu}_{2}),\cdots ,angle({\mu}_{q})\}$ and calculate q matching azimuth angles ${\theta}_{j}$ $(j=1,2,\cdots ,q)$ from Equation (74). So we get q possible pairs $({\theta}_{j},{\widehat{\phi}}_{i})$ $(j=1,2,\cdots ,q)$.

#### 3.5. Computational Procedure and Complexity Analysis

**E**

_{x}and

**E**

_{z}corresponding to the q largest eigenvalues through eigendecomposition of ${\widehat{R}}_{x}^{12}$ and ${\widehat{R}}_{z}^{12}$. Divide

**E**

_{x}into

**E**

_{x}

_{1},

**E**

_{x}

_{2}and divide

**E**

_{z}into

**E**

_{z}

_{1},

**E**

_{z}

_{2}.

**Ω**

_{x}and

**Ω**

_{z}from Equations (59) and (73), calculate eigenvalues ${\mu}_{i}$ and ${\eta}_{i}$ (i = 1, 2, …, q) through eigendecomposition of

**Ω**

_{x}and

**Ω**

_{z}.

_{i}and nominal elevation φ

_{i}from Equations (60) and (74).

^{2}K

^{2}) and the alternating projection technique with respect to cost functions which is O(M

^{4}K

^{4}+ 2M

^{2}K

^{2}). Zhou’s algorithm uses TLS-ESPRIT to calculate nominal elevation and 1D searching to find nominal azimuth. Computational cost of Zhou’s algorithm [24] mainly contains calculation of the sample covariance matrix O(4NM

^{2}), eigendecomposition and inversion of the sample covariance matrix O(16M

^{3}) and 1D searching O(8M

^{3}). Computational cost of the proposed algorithm mainly contains calculation of the sample covariance matrix O[N(MK − K)

^{2}+ N(MK − M)

^{2}], eigendecomposition of ${\widehat{R}}_{x}^{12}$ and ${\widehat{R}}_{z}^{12}$ O[8 (MK − K)

^{3}+ 8(MK − M)

^{3}], eigendecomposition of

**Ω**

_{x}and

**Ω**

_{z}O(q

^{3}) and angle matching O(q

^{2}). The main computational complexity of the three algorithms are shown in Table 1.

## 4. Results and Discussion

_{θ}and RMSE

_{φ}denote the RMSE of nominal azimuth and nominal elevation respectively, which can be expressed as

_{θ}and RMSE

_{φ}curves with SNR varying from −5 dB to 30 dB. The figures also show the estimation of the COMET [23] using the URA, Zhou’s algorithm [24] using arrays

**x**

_{1}and

**x**

_{2}and the Cramer–Rao lower bound (CRLB). As shown from Figure 2a,b, when SNR changes from −5 dB to 0 dB, COMET [23] presents better performance than the proposed algorithm. As SNR increases, RMSE

_{θ}and RMSE

_{φ}of all algorithms decrease. The proposed algorithm has better performance than other algorithms with SNR ranging from 10 dB to 30 dB. Thus, it can be concluded that our method has a good performance when SNR is at high levels. Estimation of our method is based on acquiring signal subspace through eigendecomposition of covariance matrix of received vector. The accuracy of eigendecomposition would deteriorate at low SNR, while COMET [23] separates the noise and signal power through covariance matching fitting matrix by alternating projection technique firstly, as a result, perform better at low SNR.

_{θ}curves as nominal azimuth changing from 0° to 180° with nominal elevation fixed at 20°, while Figure 3b shows RMSE

_{φ}curves as nominal elevation changing from 0° to 180° with nominal azimuth fixed at 20°. As can be seen, both RMSE

_{θ}and RMSE

_{φ}increase markedly near the boundary region. Generally, the error of the boundary region estimated by our method is acceptable.

_{θ}and RMSE

_{φ}of both Gaussian and uniform sources increase with the distance d from λ/20 to λ/2. When d is λ/2, RMSE

_{θ}and RMSE

_{φ}of Gaussian sources reach 0.62 and 0.44, those of uniform sources reach 0.79 and 0.67, which are still satisfactory results. It can be concluded that the proposed algorithm shows satisfactory performance with small distance between adjacent sensors.

^{2}. Utilizing double parallel arrays which contains only 2M sensors can estimate M sources simultaneously. We consider seven URA with M varying from 4 to 10. The total number of sources is set at M

^{2}which is a theoretical upper limit of the trail. The (a, b)th source is set at [20° + (a − 1)10°, 20° + (b − 1)10°]. All sources are Gaussian and have same power. SNR is 15 dB, number of snapshots is 200, MC = 100, d = λ/10, ${\sigma}_{\theta}={\sigma}_{\varphi}=$2°. Covariance coefficients of Gaussian sources are set at 0.5. When all sources are estimated successfully, the difference between estimators is accuracy which can be described by RMSE. When not all sources are estimated successfully, RMSE cannot differentiate performance of estimators. Thus, an indicator reflecting the number of sources detected is needed to measure the performance of different estimators. Estimation is regarded as effective when the estimated angles satisfying $\sqrt{{({\widehat{\theta}}_{i}-{\theta}_{i})}^{2}+{({\widehat{\phi}}_{i}-{\phi}_{i})}^{2}}\le $ 5°. Define detection probability as N

_{d}/M

^{2}where N

_{d}is number of source estimated effectively. So in theory the detection probability of the proposed algorithm is (M − 1)/M, COMET is 1, Zhou’s algorithm [24] is 1/M.

**Ω**

_{x}and

**Ω**

_{z}is 64 × 64 dimensional, which all need eigendecomposition. The deterioration of COMET [23] is also closely related to the number of sources. COMET [23] separates unknown variables of each source based on alternating projection technique, and then formulate equation sets of unknown variables. In the separating process, 64 inversion operations of a 64 × 64 dimensional matrix are executed on the condition M = 8 and total source is 64. Consequently, the errors of the separating process deliver to equations set and affect the validity of estimation.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 2.**(

**a**) RMSE

_{θ}estimated by three algorithms for 2D ID sources vesus SNR; (

**b**) RMSE

_{φ}estimated by three algorithms for 2D ID sources vesus SNR.

**Figure 3.**(

**a**) RMSE

_{θ}estimated with θ changing from 0° to 180° while φ is fixed at 20°; (

**b**) RMSE

_{φ}estimated with φ changing from 0° to 180° while θ is fixed at 20°.

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**MDPI and ACS Style**

Wu, T.; Deng, Z.; Li, Y.; Huang, Y. Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays. *Sensors* **2018**, *18*, 3600.
https://doi.org/10.3390/s18113600

**AMA Style**

Wu T, Deng Z, Li Y, Huang Y. Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays. *Sensors*. 2018; 18(11):3600.
https://doi.org/10.3390/s18113600

**Chicago/Turabian Style**

Wu, Tao, Zhenghong Deng, Yiwen Li, and Yijie Huang. 2018. "Two-Dimensional DOA Estimation for Incoherently Distributed Sources with Uniform Rectangular Arrays" *Sensors* 18, no. 11: 3600.
https://doi.org/10.3390/s18113600