Open AccessArticle

Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation

He Ma

^1,2

Libao Liu

³,

Zhihong Gan

^1,2,

Yang Gao

^1,2

and

Xingpeng Mao

^1,2,*

School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China

Key Laboratory of Marine Environmental Monitoring and Information Processing, Ministry of Industry and Information Technology, Harbin 150001, China

School of Science, Harbin Institute of Technology, Weihai 264209, China

Author to whom correspondence should be addressed.

Remote Sens. 2024, 16(24), 4711; https://doi.org/10.3390/rs16244711

Submission received: 15 November 2024 / Revised: 13 December 2024 / Accepted: 15 December 2024 / Published: 17 December 2024

(This article belongs to the Topic Advanced Array Antenna Design and Signal Processing Techniques)

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Abstract

Direction of arrival (DOA) estimation plays a critical role in remote sensing, where it aids in identifying and tracking multiple targets across complex environments, from atmospheric monitoring to resource mapping. Leveraging difference covariance array (DCA) for DOA estimation has become prevalent, particularly with sparse arrays capable of resolving more targets than the number of sensors. This paper proposes a new hole-free sparse array configuration for remote sensing applications to achieve improved DOA estimation performance using DCA. By symmetrically placing a minimum redundancy array (MRA) and its complementary MRA on both sides of a sparse uniform linear array (ULA), this configuration maximizes degrees of freedom (DOFs) and minimizes mutual coupling effects. Expressions for calculating sensor positions and optimal element allocation methods to maximize DOFs are derived. Simulation experiments in various scenarios have shown the advantages of the proposed array in DOA estimation, including a strong ability to estimate multi-targets, high angular resolution, low estimation error, and strong robustness to mutual coupling.

Keywords:

DOA estimation; degrees of freedom; difference covariance array; sparse linear array

1. Introduction

In the past decade, direction of arrival (DOA) estimation has gained increasing importance in remote sensing applications such as navigation [1,2], military surveillance [3,4], and meteorology [5,6], where precise detection and tracking of multiple targets are essential. Antenna array configuration is a core component of radar systems in remote sensing, responsible for transmitting and receiving signals to estimate target directions. The choice of antenna array directly impacts DOA estimation performance, with various configurations, including one-dimensional (1D) linear arrays and two-dimensional (2D) planar arrays, being explored for their suitability in different remote sensing scenarios. Among these, the traditional 1D uniform linear array (ULA) [7] has been favored for its simplicity, ease of implementation, and cost-effectiveness.

However, standard ULA configurations face challenges in high-density remote sensing environments. Their performance can be limited by mutual coupling effects between array sensors [8], constraints on the number of detectable targets [9], and fixed degrees of freedom (DOFs) [10], which restrict the array’s resolution and target-tracking abilities. In response, sparse arrays have been developed to increase DOFs with fewer sensors, making them particularly valuable in remote sensing scenarios requiring wide-area monitoring and multi-target detection [11,12,13,14]. Sparse linear arrays, which can reduce the number of sensors while maintaining high DOA performance, serve as the foundation for extending sparse arrays into 2D applications, offering versatility in applications such as angle of arrival (AOA) and angle of departure (AOD) estimation [15].

When applied in DOA estimation for remote sensing, the effective use of sparse array difference covariance array (DCA) can enhance estimation accuracy and substantially increase the number of detectable sources [10,16]. A DCA organizes distinct cross-correlation values in the covariance matrix of signals received by the sparse array, classifying sparse linear arrays into hole-free and hole arrays based on continuity. The performance of a hole array [17,18,19,20,21] in remote sensing is hindered by non-consecutive lags that prevent the direct application of ULA-based DOA estimation methods, such as spatial smoothing MUSIC (SS-MUSIC) [22]. This limitation necessitates more complex algorithms for arrays with holes [23]. On the other hand, the continuous DCA of a hole-free sparse array, when combined with the SS-MUSIC algorithm, can enhance the estimation accuracy and substantially increase the number of detectable sources. The maximum number of detectable targets is equal to half of the array’s DOFs minus one, calculated as

(D O F - 1) / 2

In practice, we prefer to use simple and effective algorithms to achieve accurate DOA results, reduce computation time and complexity, and estimate a greater number of targets. Hence, we are more concerned with hole-free arrays. Such arrays have been developed over the last few decades, including coprime [24,25], super nested [26,27], modified nested [28], multi-level [29,30], multi-order [31,32], and other arrays. Here, some arrays studied in recent years are introduced in detail. The enhanced nested array (ENA) [33] improves the DOFs when the number of sensors is even. The one-side extended nested array (OSENA) [34] greatly improves the DOFs in the case of larger-scale arrays. The augmented nested array (ANA) [35] and the improved nested array (INA) [36] increase the degrees of freedom when there is no limit to the number of sensors. Compared with ANA, INA has more DOFs but is more affected by mutual coupling. In addition, improved array configurations should be designed to achieve better performance.

In this paper, a new type of sparse linear array without holes is proposed. The proposed array improves on the minimum redundancy array (MRA) [37] and nested array (NA) [10] and avoids their major shortcomings. The MRA does not have an exact expression and cannot be applied in a mass sensors scenario. The NA contains a dense ULA, and thus it is highly influenced by mutual coupling. To overcome the shortcomings of MRA, the proposed array adopts a two-level structure, and each layer contains a ULA that has no upper limit on the number of elements. This means the array can be applied to large-scale array scenarios by increasing the number of elements in ULA. Furthermore, the proposed array uses symmetric complementary MRA to replace a part of the dense ULA in NA. This elimination of the dense ULA structure reduces the array’s exposure to mutual coupling, thus avoiding the disadvantage of NA.

The proposed array configuration can increase the number of DOFs and reduce the effect of the mutual coupling, causing an expected improvement in DOA estimation accuracy. The main contributions of this paper are as follows:

(a): We define the complementary MRA (CMRA) and find that the new array is a hole-free array configuration when the MRA and its corresponding CMRA are placed symmetrically along the central axis. Then, a novel hole-free array, named a symmetric complementary minimum redundancy array (SC_MRA), is introduced and discussed in detail.
(b): Based on SC_MRA and NA, the nested symmetric complementary minimum redundancy array (NSC_MRA) is proposed, inheriting all the advantages of NA and reducing the mutual coupling by redistributing the sensors of the dense ULA part of a classical NA.
(c): An optimal element allocation method is derived to maximize DOFs for the NSC_MRA configuration, offering flexibility and enhanced target resolution in remote sensing applications requiring high spatial coverage.

The remaining part of this paper is organized as follows: In Section 2, some basic conceptions are introduced to help understand the proposed array. In Section 3, the derivation of the geometric characteristics of the proposed array is presented, and the optimal distribution of sensors that maximize the total DOFs is discussed. In Section 4, we analyze the performance of the proposed array and obtain the theoretical advantages of the proposed array compared with other arrays in DOA estimation. In this section, a numerical simulation of DOA estimation is also provided, consistent with the analysis results. Finally, conclusions are drawn in Section 5.

2. Basic Conceptions

To facilitate the illustration of the proposed array configuration, some basic conceptions are introduced briefly in this section.

2.1. Difference Covariance Arrays

Assume the set of real array sensor positions is

P

, in which all elements are the true positions of the sensors in the 1D coordinate system. In reality, the difference between any two elements of

P

is usually a multiple of the unit spacing. Therefore, we define an array configuration set

P

to describe the position relationship between sensors simply and intuitively.

P = \frac{P}{d} = \{p_{1}, p_{2}, \dots, p_{N}\}

(1)

where

d

is the unit spacing, which usually equals

λ / 2

, and

λ

is the carrier wavelength. The

n

th element of

P

p_{n}

, equals the ratio of the

n

th element of

P

d

n \leq N

and

n \in N^{+}

, where

N

is the sensor number and

N^{+}

denotes the set of positive integers. The elements in

P

are usually integers.

For a 1D linear array specified by

P

, we define its virtual array with virtual elements located at the following set:

D = \{p_{i} - p_{j}, p_{i}, p_{j} \in P\}

(2)

According to [10], the virtual array

D

is the DCA of the received signal of

P

. Constructing

D

involves vectorizing the covariance matrix of the received signal. It should be noted that

D

contains

N^{2}

elements; however, some of them are repeated. In other words, multiple virtual elements are associated with the same virtual sensor location. Note that DCA is symmetric, and every

p_{i} - p_{j}

in the

D

position set has a corresponding

{- (p}_{i} - p_{j})

in the same set. When the arrays receive a spatially wide sense stationary signal, the correlation of the received signal can be calculated from all the differences in the set

D

If the original array is properly designed, the number of elements in the DCA equals the number of DOFs, which can be larger than the number of physical sensors. The DCA provides a powerful tool for DOA estimation, whose benefits can be fully realized by properly designing the original array.

In [10], it has been shown that for an array of

N

sensors, the maximum number of elements in the DCA can be expressed as:

F_{\max, N} = N (N - 1) + 1

(3)

2.2. Sparse Array Processing

The signal processing model of sparse array DOA estimation has been proven many times [38]. A brief description of the signal processing model for sparse array DOA estimation is provided in this section. Assume that

K

far-field stationary targets are incident at an angle

θ_{k}

k = 1, \dots, K

on a

N

-sensor sparse linear array. Then, the received signal at time

t

can be expressed as:

X (t) = A S (t) + n (t)

(4)

where

A = [a (θ_{1}), a (θ_{1}), \dots, a (θ_{K})]

is the

N \times K

array manifold matrix with the column

a (θ_{K})

a (θ_{K}) = {[a_{1} (θ_{K}), a_{2} (θ_{K}), \dots, a_{N} (θ_{K})]}^{T}

is the

N \times 1

steering vector,

θ_{k}

is the incoming angle of the

k

th target, and

a_{n} (θ_{K}) = e^{j 2 π \sin θ_{k} p_{n} d / λ}

S (t)

denotes the

K \times 1

target vector at time

t

t = 1, \dots, T

, where

T

is the number of available samples.

n (t)

denotes the

N \times 1

white noise vector. It is assumed that all sensors exhibit identical noise characteristics and receive noise power with the same value, denoted as

σ^{2}

. The auto-correlation matrix is generated by collecting

T

snapshots and can be expressed as:

R_{X X} = E \{X (t) X^{H} (t)\} = A R_{S S} A^{H} + {σ^{2} I}_{N}

(5)

where

{(•)}^{H}

denotes the Hermitian transpose of a matrix.

R_{S S} = E \{S (t) S^{H} (t)\} = diag {h}

diag {•}

stands for the diagonal matrix,

h

denotes the equivalent target vector with a single snapshot,

h = {[ς_{1}^{2}, ς_{2}^{2}, \dots, ς_{K}^{2}]}^{T}

ς_{k}^{2}

denotes the

k

th target power, and

{(•)}^{T}

denotes the transpose of a matrix.

I_{N}

is an

N \times N

identity matrix.

The covariance matrix

{\tilde{R}}_{X X}

can be expressed as:

{\tilde{R}}_{X X} = \frac{1}{T} \sum_{t = 1}^{T} X (t) X^{H} (t)

(6)

Then,

{\tilde{R}}_{X X}

is vectorized as follows:

vec ({\tilde{R}}_{X X}) = vec (E \{X X^{H}\}) = (A^{*} ⨀ A) h + {σ^{2} vec (I}_{N})

(7)

where

vec (•)

denotes the vectorization operator, which means converting a matrix into a column vector,

⨀

denotes the Khatri–Rao product. Compared with Equation (5), the term

(A^{*} ⨀ A)

in Equation (7) can be regarded as the array manifold matrix of a longer virtual ULA whose sensors are located at the DCA. This approach leads to an increase in DOFs beyond the number of physical sensors.

2.3. Signal Model with Mutual Coupling

In real-world applications, the mutual coupling effect occurs when the antenna sensors interact with each other. The matrix

C

is the mutual coupling matrix (MCM) for an array. Technically, the MCM should be specific to each sensor set, taking into account physical dimensions and feeding points [39]. It is empirically observed that MCM behavior is approximately a function of sensor separations only. This coupling is mostly inversely proportional to the distance between the array sensors.

The perturbed signal at the output of the sensors can be described as:

X (t) = C A S (t) + n (t),

(8)

where

C

can be approximated by the

B

-band symmetric Toeplitz matrix, whose

(i, j)

th element is a function of the distance between the sensor positions

p_{i}

and

p_{j}

B \in N

, where

N

denotes the set of natural numbers, and

p_{i}, p_{j} \in P

. Therefore,

C

can be written as:

C = [\begin{matrix} \begin{matrix} c_{|p_{1} - p_{1}|} & c_{|p_{1} - p_{2}|} \end{matrix} & \dots & \begin{matrix} c_{|p_{1} - p_{N - 1}|} & c_{|p_{1} - p_{N}|} \end{matrix} \\ \begin{matrix} c_{|p_{2} - p_{1}|} & c_{|p_{2} - p_{2}|} \\ ⋮ & ⋮ \\ c_{|p_{N - 1} - p_{1}|} & c_{|p_{N - 1} - p_{2}|} \end{matrix} & \begin{matrix} \dots \\ \dots \\ \dots \end{matrix} & \begin{matrix} c_{|p_{2} - p_{N - 1}|} & c_{|p_{2} - p_{N}|} \\ ⋮ & ⋮ \\ c_{|p_{N - 1} - p_{N - 1}|} & c_{|p_{N - 1} - p_{N}|} \end{matrix} \\ \begin{matrix} c_{|p_{N} - p_{1}|} & c_{|p_{N} - p_{2}|} \end{matrix} & \dots & \begin{matrix} c_{|p_{N} - p_{N - 1}|} & c_{|p_{N} - p_{N}|} \end{matrix} \end{matrix}]

(9)

where

|p_{i} - p_{j}| = l

l \in N

. If

l = 0

c_{l} = 1

; if

0 < l \leq B

|c_{l}| > |c_{l + 1}|

; if

l > B

c_{l} = 0

. And, in general, the magnitudes of the coupling coefficients are inversely proportional to their sensor separations [26,27], i.e.,

|c_{l} {/ c}_{k}| = k / l

In addition, the coupling leakage can be evaluated by:

γ = \frac{{| | C - d i a g (C) | |}_{F}}{{| | C | |}_{F}}

(10)

where

{| | • | |}_{F}

stands for the Frobenius norm of a matrix,

d i a g (•)

means that only the diagonal elements of the matrix are kept, and the remaining elements are zero.

{| | C - d i a g (C) | |}_{F}

is the energy of all off-diagonal components, which characterizes the level of mutual coupling. A higher

γ

means a heavier mutual coupling. In Section 4.2, γ is used as a parameter to compare the mutual coupling characteristics of different arrays in simulations.

3. Proposed Array Design with MRA, Complementary MRA, and NA

3.1. Symmetric Complementary Minimum Redundant Array

The MRA array discusses how to achieve a larger array aperture with a limited number of sensors. For an N-sensor MRA, the distance between any two sensor pairs is an integer multiple of the minimum unit distance, and these multiples contain all the integers in [−

M_{N}

M_{N}

], where

M_{N}

is the largest number in the configuration of the N-sensor MRA. Therefore, the MRA is a sparse hole-free linear array that maximizes the aperture for a given number of physical sensors, and its DCA is a filled ULA [37]. The maximum achievable aperture and DOFs of the MRA are equivalent to those of the ULA composed of

M_{N} + 1

sensors.

F_{MRA, N} = 2 M_{N} + 1 = F_{ULA, (M_{N} + 1)}

(11)

where

F_{ULA, (M_{N} + 1)}

is the number of DOFs of the ULA configuration composed of

M_{N} + 1

sensors. Specifically, take a seven-sensor MRA as an example. In reference [40], its configuration can be found as {0, 1, 2, 6, 10, 14, 17}. The array element (

N

) of the seven-sensor MRA is 7, and the largest number in the configuration (

M_{7}

) is 17, so that it has 2

\times

17 + 1 DOFs. The same DOFs as the 7-sensor MRA is the ULA with array element

M_{7} + 1

, i.e., an 18-sensor ULA. The DOFs of the 18-sensor ULA is 2

\times

18 − 1, and it is equal to 2

\times

17 + 1.

As the number of sensors increases, the time it takes for an MRA to perform an accurate position calculation increases exponentially. Therefore, only the situation when

3 \leq N \leq 30

is calculated in [40]. Based on these MRAs, a new hole-free array configuration is proposed and named as complementary MRA (CMRA). The number of CMRA sensors corresponding to an MRA with

N

sensors is

N'

(

N^{'} = M_{N} + 1 - N

). According to [40],

M_{3} = 3

and

M_{30} = 287

. Thus,

1 \leq N^{'} \leq 258

. The configuration set of the

N^{'}

-sensor CMRA is given by:

P_{CMRA, N^{'}} = ∁_{Z} P_{MRA, N}

(12)

where

Z

stands for the set of integers

{0,1, 2, \dots, M_{N}}

P_{MRA, N}

is the MRA configuration set, and

∁_{Z} P_{MRA, N}

is the absolute complement of

P_{MRA, N}

Z

. In addition,

M_{N}

is defined as the maximum element in the

N

-sensor MRA array configuration set (

P_{MRA, N}

), while

N_{N^{'}}

is the minimum element in the array configuration set of the

N^{'}

-sensor CMRA (

P_{CMRA, N^{'}}

). Table 1 shows examples of MRA configurations and the corresponding CMRA configurations of the number of sensors

N = 3

to 6.

By combining the MRA and CMRA, it can be found that the new generating array is a symmetric complementary array when the MRA and its corresponding CMRA are placed symmetrically along the central axis. For notational convenience, the symmetric complementary MRA will be abbreviated as SC_MRA. After the complementary symmetrical placement of all the MRAs listed in reference [40], it can be found that all the newly composed SC_MRAs are also hole-free arrays.

A schematic diagram of the SC_MRAs is shown in Figure 1. The configuration is divided into two parts: part A (MRA) and part B (CMRA), which are placed symmetrically with the intermediate axis. In Figure 1, ● means a sensor is put in a specific position, the number in ● corresponds to the elements of

P_{MRA, N}

and

P_{CMRA, N^{'}}

, while

⨂

means there is no sensor in that position. To simplify the expressions in the formula subscripts, “SCM” is used as the abbreviation for the SC_MRA. The configuration set of the

({N + N}^{'})

-sensor SC_MRA is given by:

P_{SCM, {N + N}^{'}} = P_{MRA, N} \cup \{P_{CMRA, N^{'}} + M_{N}\}

(13)

From Figure 1, it is clearly shown that the array length of the

N

-sensor part A is

M_{N}

, while the array length of the corresponding

N'

-sensor part B is

M_{N} - N_{N^{'}} - 1

. There is a gap of two-unit element spacing between these two parts. Therefore, the total array length of the SC_MRA is

M_{N} + M_{N} - N_{N^{'}} - 1 + 2 = 2 M_{N} - N_{N^{'}} + 1

. For a hole-free array, the DOFs are equal to twice the array length plus one. Hence, the number of DOFs of the

({N + N}^{'})

-sensor SC_MRA is given by:

F_{SCM, {N + N}^{'}} = 2 (2 M_{N} - N_{N^{'}} + 1) + 1 = {4 M}_{N} - 2 N_{N^{'}} + 3

(14)

3.2. Nested Symmetric Complementary Minimum Redundant Array

Although the SC_MRA proposed in Section 3.1 is a sparse array without holes, its predominance over the most common arrays is not obvious. To further enhance the DOA estimation performance, we combined the SC_MRA with NA to obtain a new array, named the nested symmetric complementary minimum redundancy array (NSC_MRA). In this section, the array configuration of NSC_MRA is introduced.

Unlike an MRA, an NA is easy to construct and can obtain exact expressions of sensor positions and available DOFs in the condition of a known number of sensors. An

N

-sensor two-level NA is a concatenation of two ULAs, an

N_{1}

-sensor dense ULA and an

N_{2}

-sensor sparse ULA, where

N_{1} + N_{2} = N

. The sensor spacing of the dense ULA is

(N_{1} + 1)

times that of the sparse ULA. Thus, the configuration set of the

N

-sensor NA is given by:

\begin{matrix} P_{NA, N} = P_{ULA, N_{1}} \cup P_{ULA, N_{2}} \\ P_{ULA, N_{1}} = \{p_{i} | p_{i} = 1, 2, \dots, N_{1}\} \\ P_{ULA, N_{2}} = \{p_{j} (N_{1} + 1) | p_{j} = 1, 2, \dots, N_{2}\} \end{matrix}

(15)

Figure 2 shows the 14-sensor NA configuration with

N_{1} = N_{2} = 7

as an example, where ● and

⨂

have the same meanings as in Figure 1. Level 1 of the NA is the dense ULA that is defined as locations from 1 to

N_{1} = 7

, while level 2 is the sparse ULA from

N_{1} + 1 = 8

N_{2} (N_{1} + 1) = 56

. However, the number of redundancies of the DCA is still high in the NA. Hence, it is possible to modify the NA to obtain more DOFs.

To increase the DOFs and array aperture, the NSC_MRA is proposed by combining an MRA, CMRA, and sparse ULA. The first step is to define the CMRA and SC_MRA, which were introduced in Section 3.1. Then, we used part A in the SC_MRA to replace the dense ULA in the NA. Finally, the center of the sparse ULA is taken as the symmetry axis, and the CMRA of part B is placed symmetrically with the axis of the extension symmetry of part A. The proposed array adopts an NA nested structure. The combination of part A (MRA) and part B (CMRA) is level 1, and the sparse ULA is level 2. Assume that

N_{A}

is the number of sensors in part A,

N_{A}^{'}

is the number of sensors in part B,

N_{1}

is the number of sensors in level 1 (

{N_{1} = N}_{A} + N_{A}^{'} = M_{N_{A}} + 1

), and

N_{2}

is the number of sensors in level 2 (sparse ULA). Similarly, “NSCM” is used as the abbreviation for the NSC_MRA in the formula subscripts. Then, the array configuration set of the

N

-sensor NSC_MRA can be given by:

\begin{matrix} P_{NSCM, N} = P_{PartA, N_{A}} \cup P_{ULA, N_{2}} \cup P_{PartB, N_{A}^{'}} \\ P_{PartA, N_{A}} = \{p_{i} + 1\} \\ P_{ULA, N_{2}} = {p_{j} (M_{N_{A}} + 2)} = {p_{j} (N_{1} + 1)} \\ P_{PartB, N_{A}^{'}} = \{(N_{2} + 1) (M_{N_{A}} + 2) - (p_{k} + 1)\} = \{(N_{2} + 1) (N_{1} + 1) - (p_{k} + 1)\} \end{matrix}

(16)

where

p_{i} {\in P}_{MRA, N_{A}}

p_{j} = 1, 2, \dots, N_{2}

p_{k} {\in P}_{CMRA, N_{A}^{'}}

. Since part A is the MRA,

3 \leq N_{A} \leq 30

. Part B is the CMRA; thus,

1 \leq N_{A}^{'} \leq 258

. Level 2 is the sparse ULA; thus,

N_{2} \geq 2

N_{A}, N_{A}^{'}, {N_{1}, N}_{2} \in N^{+}

Figure 3 provides a preview of the 14-sensor NSC_MRA model. Compared to the NA in Figure 2, the proposed NSC_MRA is sparser, and this is because the number of sensor pairs with small separations (

λ / 2

2 λ / 2

3 λ / 2

) of the NSC_MRA is less under the same sensor number conditions. The NSC_MRA exhibits a distinct pattern where the number of sensor pairs with a separation of

λ / 2

is three, contrary to seven for the NA. Similarly, the number of sensor pairs with a separation of

2 λ / 2

is three, contrary to six for the NA, and the number of sensor pairs with a separation of

3 λ / 2

is three, contrary to five for the NA. Therefore, this rearrangement significantly reduces the adverse effects of mutual coupling on the DOA estimation. Meanwhile, the NSC_MRA has a larger physical aperture than the NA with the same number of sensors, thus increasing the number of DOFs and the length of the physical aperture.

The disadvantage of the above-mentioned array design is that

M_{N_{A}}

in Equation (16) cannot cover all positive integer sets, which restricts the allocation of array elements in each part. To overcome this challenge, the proposed arrays need to be extended to a generic form. The first step is to add part D between the sparse ULA and CMRA, where part D is a ULA with

N_{D}

sensors. Simultaneously, the separation between part A and the sparse ULA increases. Then, an empty part C is formed, whose length is

N_{D} + 1

, that is, the spacing between part A and the sparse ULA is increased from 1 to

N_{D} + 1

. In Figure 4, a 20-sensor NSC_MRA with parameters

{N_{1} = N}_{2} = 10

is shown as an example.

The sensor configurations of proposed array NSC_MRAs are given by:

\begin{matrix} P_{NSCM, N} = P_{PartA, N_{A}} \cup P_{ULA, N_{2}} \cup P_{PartD, N_{D}} \cup P_{PartB, N_{A}^{'}} \\ P_{Part, N_{A}} = \{p_{i} + 1\} \\ P_{ULA, N_{2}} = {p_{j} (M_{N_{A}} + N_{D} + 2)} = {p_{j} (N_{1} + 1)} \\ P_{PartD, N_{D}} = \{N_{2} (M_{N_{A}} + N_{D} + 2) + p_{k}\} = \{N_{2} (N_{1} + 1) + p_{k}\} \\ P_{PartB, N_{A}^{'}} = \{{(N}_{2} + 1) (M_{N_{A}} + N_{D} + 2) - (p_{r} + 1)\} = {{(N}_{2} + 1) (N_{1} + 1) - (p_{r} + 1)} \end{matrix}

(17)

where

p_{i} {\in P}_{MRA, N_{A}}

p_{j} = 1, 2, \dots, N_{2}

p_{k} = 1, 2, \dots, N_{D}

p_{r} {\in P}_{CMRA, N_{A}^{'}}

N_{A}

N_{A}^{'}

N_{2}

have the same meaning as those in Equation (16), and

N_{D}

is the number of sensors in part D. In addition, it should be noted that the concept of

N_{1}

is changed to

{N_{1} = N}_{A} + N_{A}^{'} {+ N}_{D}

(meanwhile, it is equal to

M_{N_{A}} + N_{D} + 1

). It is assumed that

3 \leq N_{A} \leq 30

N_{2} \geq 2

, and

N_{A}, N_{A}^{'}, N_{1}, N_{D}, N_{2} \in N^{+}

3.3. Optimal Element Allocation Method

From Equation (17), it can be noticed that the first sensor of the NSC_MRA is designed to be placed at 1, and the last sensor is at

{(N}_{2} + 1) (N_{1} + 1) - (N_{N_{A}^{'}} + 1)

. As the NSC_MRA proposed in Section 3.2 is a hole-free array, the number of DOFs in the corresponding DCA can be expressed as:

\begin{matrix} F_{NSCM, N} = 2 {{(N}_{2} + 1) (N_{1} + 1) - (N_{N_{A}^{'}} + 1) - 1} + 1 \\ = {2 (N}_{2} N_{1} + N_{1} + N_{2} - N_{N_{A}^{'}} - 1) + 1 \\ = {2 (N}_{2} N_{1} + N - N_{N_{A}^{'}}) - 1 \end{matrix}

(18)

The proposed array is a hole-free array with a continuous DCA, which improves the maximum number of detectable targets. For the proposed array with

N

sensors, its maximum number of detectable targets is

{(F}_{NSCM, N} - 1) / 2

The proposed NSC_MRA has a nested structure; therefore, the array’s DOF number varies with the number of sensors assigned to each level. Specifically, the configuration sets (

P_{NSCM, 15}

) and DOFs (

F_{NSCM, 15}

) of the 15-sensor NSC_MRA for all sensor assignments are listed in Table 2.

Under the constraint of a fixed total number of sensors

N = N_{1} + N_{2}

N_{1}

and

N_{2}

have a certain numerical relationship that maximizes

F_{NSCM, N}

. The total number of DOFs is determined by the array element allocation, where the product of

N_{1} \times N_{2}

plays a crucial role. By optimizing this product, we can maximize

F_{NSCM, N}

When

N

is even, the optimal element allocation method that maximizes the DOFs is

N_{1} = N_{2} = N / 2

. This ensures that the product

N_{1} \times N_{2}

is maximized, leading to the maximum possible DOFs. In this case, Equation (18) can be rewritten as:

F_{NSCM, N_{even}} = 2 ({(\frac{N}{2})}^{2} + N - N_{N_{A}^{'}}) - 1 = \frac{N^{2} - 2}{2} + 2 N - 2 N_{N_{A}^{'}}

(19)

When

N

is odd, both

N_{1} = \frac{N - 1}{2}

N_{2} = \frac{N + 1}{2}

and

N_{1} = \frac{N + 1}{2}

N_{2} = \frac{N - 1}{2}

can achieve the maximum DOFs. In array design, mutual coupling should be taken into account in addition to maximizing the degrees of freedom. The sensor spacing of part A is much smaller than that of the sparse ULA. The number of NAs should be minimized while ensuring the maximum DOFs. Thus, when N is odd, the optimal element allocation method that maximizes the DOFs is

N_{1} = \frac{N - 1}{2}

and

N_{2} = \frac{N + 1}{2}

. In this case, Equation (18) can be rewritten as:

F_{NSCM, N_{odd}} = 2 (\frac{N - 1}{2} \times \frac{N + 1}{2} + N - N_{N_{A}^{'}}) - 1 = \frac{N^{2} - 3}{2} + 2 N - 2 N_{N_{A}^{'}}

(20)

Under the constraint of the total number of array sensors, the DOFs are not only related to the numerical relationship between

N_{1}

and

N_{2}

but also related to the selection of

N_{N_{A}^{'}}

, which can be up to two at least. Thus Equations (19) and (20) can be modified as follows:

F_{NSCM, N_{even}} = (N^{2} - 2) / 2 + 2 N - 4 = (N^{2} - 10) / 2 + 2 N

(21)

F_{NSCM, N_{odd}} = (N^{2} - 3) / 2 + 2 N - 4 = (N^{2} - 11) / 2 + 2 N

(22)

The maximum number of detectable targets for hole-free array is

{(F}_{NSCM, N} - 1) / 2

. Thus, when

N

is even, the maximum number of detectable targets for the proposed array is

(N^{2} - 12) / 4 + N

; when

N

is odd, it is

(N^{2} - 13) / 4 + N

4. Performance Analysis

In this section, the performances of the proposed array (NSC_MRA), such as the array aperture, DOFs, weight function, and coupling leakage, are analyzed. In addition, the proposed NSC_MRA is compared with several widely studied alternatives, NA [10], ENA [33], OSENA [34], ANA [35], and INA [36]. For a fair comparison, all array configurations considered in this section consist of 15 sensors. The scenarios considered in this section focus exclusively on stationary targets, where all targets have a speed of 0.

4.1. Array Aperture and Degrees of Freedom

Figure 5 shows the array configurations and weight functions for NA, ENA, OSENA, ANA, INA, and the proposed NSC_MRA. Each subfigure is divided into two parts. The upper part is the array configuration, whose elements are placed in a way that corresponds to its array configuration set

P

, which is defined in Section 2.1. The numbers in the upper parts are

p_{n}

, which means the ratio of the actual sensor positions to the unit spacing,

p_{n} \in P

. The bottom half describes the weight function

ω (l)

, whose horizontal coordinate is the natural number

l

and the ordinate is

ω (l)

. The number of sensor pairs whose spatial separation equals

l

[26,27] can be defined as:

ω (l) = |\{(p_{1}, p_{2}) \in P^{2}, p_{1} - p_{2} = l\}|

(23)

where

| • |

stands for counting the number of element pairs in the set.

It is observed from Figure 5 that the

ω (l)

of the NSC_MRA does not miss any integer elements from 0 to the end, which means that the proposed array is a hole-free array with contiguous DCAs. It can solve the problem well when the number of targets is larger than the number of sensors.

In addition, the value of

F_{u}

in the figure is marked for ease of understanding, where

F_{u}

denotes the number of one-side uniform DOFs of the DCA. For hole-free arrays, the length of the array apertures is equal to its

F_{u}

. Looking at the array configurations and

F_{u}

in Figure 5, it can easily be concluded that the NSC_MRA provides a large array aperture, whose length of an array is only smaller than the INA but larger than the others, illustrating that the proposed NSC_MRA is good at resolving closely spaced targets.

In DOA estimation, the performance is also often closely related to the number of DOFs, which is equal to the length of the virtual continuous DCA. Thus,

F

can be calculated by

F_{u}

(

F {= 2 F}_{u} - 1

). From the results, it can be noted that the NSC_MRA has lower DOFs than the INA but higher DOFs than the others. This means that, theoretically, the DOA estimation performance of the NSC_MRA is only second to that of INA in the absence of mutual coupling.

4.2. Mutual Coupling Effect

In practice, the mutual coupling effect between sensor pairs increases significantly with the reduction in spatial separation. The specific values of

ω (1)

ω (2)

, and

ω (3)

are labelled in Figure 5. In addition, the non-negative covariance array weight functions corresponding to

l > 3

are also provided in Figure 5, distinguished by the height of the blue line. As can be seen from Figure 5, the NSC_MRA has only three sensor pairs with a separation of

λ / 2

, and the number of sensor pairs with a separation of

2 λ / 2

and

3 λ / 2

are both three. The NSC_MRA has a similar performance as the ANA and is much better than the other arrays.

To confirm the advantage of the NSC_MRA against mutual coupling, the case of significant mutual coupling is taken into account. Accordingly, the specific parameters in the coupling model given by Equation (9) are set as

{c_{1} = 0.5 e}^{j π / 3}

and

c_{l} = c_{1} / l

Figure 6 shows the mutual coupling matrices of six different 15-sensor arrays. The coupling leakages for each array configuration calculated according to Equation (10) are also shown in the upper right corner of the respective subfigure. In Figure 6, the darker the color, the more energy it has in the corresponding entry. The results show that the NA has the highest value of

γ

, indicating that it is the most affected by mutual coupling. On the other hand, the NSC_MRA exhibits the lowest value of

γ

, suggesting that it is the least impacted by mutual coupling compared to all the other arrays tested. It can be concluded that the NSC_MRA shows enhanced mutual coupling adaptation than the other arrays, especially for large mutual coupling.

4.3. Numerical Simulations

In Section 4.1, some important properties of the proposed array (NSC_MRA) were analyzed, and theoretical analysis results were obtained. In this section, the simulation results are presented to evaluate the DOA estimation performance by using the SS-MUSIC algorithm. SS-MUSIC is a particularly popular estimation algorithm in DOA estimation [22]. Here, the array types are the same as those in Section 4.1.

4.3.1. Spatial Spectrum Comparison of Different Arrays

Figure 7 shows the SS-MUSIC spectra of six different 15-sensor arrays. The arrays are evaluated for 51 targets at

[- 60 ° : 2.4 ° : 60 °]

with an SNR of 0 dB, and the number of snapshots

L

is 1000. The peak search process is executed with the same step size of 0.01°. The minimum unit distance between sensors is set to

d = λ / 2

It can be observed from Figure 7 that the NA and ENA show false peaks at

[- 90 °, - 60 °]

and

[60 °, 90 °]

due to the lower number of DOFs. The OSENA has no spurious peaks at

[- 90 °, - 60 °]

and

[60 °, 90 °]

. However, at around ±60°, there is a phenomenon that the peaks of the two real target directions fuse, resulting in no peak at the correct target direction and a spurious peak between the two real target directions. The ANA, INA, and NSC_MRA can successfully identify 51 targets. There are no false peaks in the ANA, INA, and NSC_MRA. Each spectrum peak corresponds to the true target direction, but some peaks of the ANA and NSC_MRA near ±60° are slightly lower. The simulation results show that the NSC_MRA outperforms all the other arrays except the INA in terms of DOA resolution in the absence of mutual coupling, confirming that the proposed array NSC_MRA can solve the DOA estimation problem well when the number of targets is larger than the number of sensors.

4.3.2. Performance of Distinguishing the Targets with Similar Angles of Incidence

Two equal-power uncorrelated targets impinge onto the array from directions of

θ_{1} = - 0.3 °

and

θ_{2} = 0.3 °

. The SNR is set to 0 dB, and the estimation process employs

1000

snapshots to estimate the correlations by using sample averages. For DOA estimation, the same method (SS-MUSIC) is used for six different arrays, and the resulting MUSIC spectrum is plotted in Figure 8.

As can be seen in the simulation results in Figure 8, all arrays exhibit two peaks around the true DOAs, and the specific ability to separate the two peaks is different. Specifically, the strength of the ability to distinguish two close targets can be seen by the lowest point of the trough between the peaks of the two real targets in the figure. In the case of estimating two close targets, the NA and ENA have the highest trough due to the smallest DOFs, that is, their performances in terms of DOA estimation for close targets are the worst. Moreover, it is found that the more DOFs the array has, the bigger the waveform height will be. A bigger waveform height means that the peaks are more clearly identified. From the figure, the INA has the lowest trough, which corresponds to its maximum DOFs. The NSC_MRA and ANA have the same DOFs, their troughs are similar, and both are lower than the other arrays except the INA.

To ensure the generalization of the simulation results, we conducted

Q = 500

Monte Carlo trials to measure the probability of correct and false detections. Two equal-power uncorrelated targets from

θ_{1} = - 0.3 °

and

θ_{2} = 0.3 °

are used. The search range is from

- 90 °

90 °

and the search step size is

0.01 °

. The acceptable angle error is set to

0.05 °

. If the estimated directions

({\tilde{θ}}_{q, 1}, {\tilde{θ}}_{q, 2})

from the

q

th trial satisfy

|{\tilde{θ}}_{q, 1} - θ_{1}| \leq 0.05 °

and

|{\tilde{θ}}_{q, 1} - θ_{2}| \leq 0.05 °

, we record the resolution probability

ξ_{q}

for that trial as 1. Otherwise,

ψ_{q}

is set to

0

. The overall resolution probability

Ψ

is then calculated using the following:

Ψ = \frac{1}{Q} \sum_{q = 1}^{Q} ψ_{q}

(24)

Under consistent conditions, the resolution improves as the array aperture increases. From the results in Figure 5, it is evident that the proposed array design covers a wider spatial aperture, leading to superior resolution performance. In order to verify this advantage, we conducted multiple Monte Carlo experiments. The results are shown in Figure 9.

Figure 9 illustrates the probability of correct detections (

Ψ

) as a function of SNR for various array configurations under conditions without mutual coupling. The results demonstrate that the performance of the proposed array is comparable to that of the ANA configuration and slightly inferior to the INA configuration across the entire SNR range. This is consistent with the DOFs provided by each array design. Notably, as the SNR increases, all arrays exhibit an improvement in the probability of correct detections. At lower SNR levels, the proposed array still maintains competitive performance, demonstrating its robustness and reliability under challenging noise conditions.

The simulation results show that the proposed array, NSC_MRA, has a high angular resolution because of its large physical aperture.

4.3.3. RMSE of NSC_MRA When the Number of Sensors in Different Levels Changes

In Section 3.3, for the proposed array, the influence of different array element allocations at each level on the DOFs is theoretically discussed for the case when the total sensor number is fixed. The optimal element allocation method of NSC_MRA is given. In this subsection, the DOA estimation simulation experiments are implemented using the array configurations in Table 2 to verify the conclusions in Section 3.3. Simulation results were obtained through

500

Monte Carlo experiments.

To further demonstrate the DOA estimation performance, the DOA accuracy was evaluated in terms of root mean square error (RMSE) from

500

Monte Carlo trials. The RMSE is defined as

RMSE = \sqrt{\frac{1}{500 K} \sum_{q = 1}^{500} \sum_{k = 1}^{K} {| {\tilde{θ}}_{q, k} - θ_{k} |}^{2}}

(25)

where

{\tilde{θ}}_{q, k}

denotes the DOA estimate of the

k

th target in the

q

th Monte Carlo trial, and

θ_{k}

is the

k

th target’s true incoming direction angle.

The DOA estimate’s RMSE versus SNR is shown in Figure 10, and the DOA estimate’s RMSE versus the number of snapshots is shown in Figure 11. The search range is from

- 90 °

90 °

, and the search step size is

0.01 °

. For ease of observation, we drew the RMSE of arrays with the same DOFs using the same color but with different line shapes. From the results, the array with the lowest RMSE is consistent with the variety with the highest DOFs in Table 2, and the array with the highest RMSE is consistent with the variety with the lowest DOFs. The array element allocation that minimizes the RMSE is

N_{1} = \frac{N - 1}{2}

N_{2} = \frac{N + 1}{2}

and

N_{1} = \frac{N + 1}{2}

N_{2} = \frac{N - 1}{2}

. The simulation results verify the conclusions in Section 3.3.

4.3.4. DOA Estimation Performance of Different Arrays Without Mutual Coupling

In this section, the DOA estimation performance of the proposed array is assessed through comprehensive simulation experiments and by conducting comparisons with other arrays without mutual coupling. The search range is from

- 90 °

90 °

, and the search step size is

0.01 °

The DOA estimate’s RMSE versus SNR is shown in Figure 12. In Figure 12, the array configuration has

K = 21

targets with a spatial location range of

[- 60 ° : 6 ° : 60 °]

. The range of SNR is set from

- 10

dB to

10

dB. From the results, it can be seen that the NSC_MRA achieves lower RMSEs than the others for all cases except the INA. It can be concluded that the proposed array significantly improves the estimation accuracy under different SNR environments.

The probability of correct detections versus SNR is shown in Figure 13. The results indicate that the correct detection probability of all the arrays improves as the SNR increases. At each SNR, the resolution probability of the proposed array is comparable to that of the ANA, slightly lower than the INA, and higher than the other arrays, demonstrating the reliability of the proposed array.

A comparison between Figure 12 and Figure 13 reveals a strong relationship between the RMSE performance and resolution capability. Specifically, the arrays with a smaller RMSE values in Figure 12 consistently achieve higher probabilities of correct detections in Figure 13 under identical simulation conditions, demonstrating superior target resolution ability. This finding underscores the interplay between the RMSE and resolution capability as critical indicators of estimation performance. These improvements are attributed to the increased degrees of freedom provided by the array configuration, which effectively reduces the RMSE and enhances the resolution capability. The results highlight the advantages of arrays with higher degrees of freedom, enabling superior estimation performance characterized by both a reduced RMSE and an improved resolution.

The DOA estimation performances of the different array configurations with different snapshot numbers are shown in Figure 14. The SNR is fixed at 0 dB and

N = 15

. In this simulation, the number of targets is greater than the number of sensors (

K = 21

). It can be observed that the RMSE reduces as the number of snapshots increases. The NSC_MRA configuration achieves a lower RMSE than the other arrays except the INA for all cases.

For cases when the number of sensors of the different arrays is different, their DOFs are shown in Table 3. In general, the DOFs play a decisive role in the performance of DOA estimation.

The DOA estimation performances are shown in Figure 15. The SNR is fixed at

0

dB, the snapshot number is fixed at

1000

, and the number of targets is greater than the number of sensors (

K = 21

). The simulation results show that the RMSE of the NA and ENA is relatively higher than that of the other arrays because they have fewer DOFs for the same

N

. This is because the ENA has two more DOFs of DCA than those of the NA when

N

is even and the ENA has the same DOFs as the NA when

N

is odd. Therefore, the DOA estimation performance of the ENA is slightly better than that of the NA when

N

is even. The design of the OSENA imposes stricter constraints on the number of sensors. The OSENA requires

N \geq 13

, so there is no array configuration, DOFs, or RMSE for

N = 12

in Table 3 and Figure 15. When

N

is small, the OSENA has relatively low DOFs, so the DOA estimation performance is poor. When

N

is increased, the DOA estimation performance greatly improves. The DOFs of the INA are always the highest, so its RMSE is also the lowest. The ANA and NSC_MRA have the same DOFs, so their RMSEs are also similar, second only to the INA.

It can be concluded that the proposed array NSC_MRA can significantly improve the estimation accuracy because it has a large number of DOFs, as analyzed in Section 4.1.

4.3.5. DOA Estimation Performance of Different Arrays in the Presence of Mutual Coupling

In the following simulations, the impact of mutual coupling on the RMSE performance, snapshot number, and coupling coefficient modulus

a

are simulated and analyzed. The mutual coupling model given in Equation (9) is described by coefficients

{c_{1} = 0.2 e}^{j π / 3}

and

c_{l} = c_{1} / l

. The parameters are set to SNR

= 0

dB,

L = 1000

, and

K = 21

. Targets are set with a spatial location range of

[- 60 ° : 6 ° : 60 °]

. The search range is from −90° to 90°, and the search step size is 0.01°.

Figure 16 shows the RMSE of the DOA estimates concerning SNR, where the performance of the NSC_MRA is compared with the other arrays. Different from Section 4.2, the results in this experiment demonstrate that the NSC_MRA achieves the lowest RMSE among all the tested arrays. This behavior can be attributed to the fact that the NSC_MRA has a few small, separated sensor pairs. Figure 17 shows the RMSE curves vs. the number of snapshots for the different array configurations. The results demonstrate that the proposed NSC_MRA achieves the lowest RMSE, which means that the proposed array is robust to mutual coupling effects.

Figure 18 shows the RMSE of the DOA estimates versus the coupling coefficient modulus

a

. In this simulation, the specific parameters in the coupling model given by Equation (9) are set to

{c_{1} = a \times 0.2 e}^{j π / 3}

and

c_{l} = c_{1} / l

. The increase in

a

leads to a corresponding increase in the RMSE due to a more severe mutual coupling effect introduced by higher values of

a

. The INA produces the highest performance at

a = 0

, while the NA and ENA produce the lowest performance. This is because the number of DOFs has a significant impact on the estimation performance when the mutual coupling is weak. When

a = 1

, the proposed array has the lowest RMSE and the NA has the highest RMSE, which is due to the fact that one of their coupling leakages is the smallest and the other is the largest. This is because when the influence of mutual coupling is large, the DOA estimation performance is no longer only determined by DOFs, and the performance is more determined by the size of the coupling leakage. Compared to the other evaluated arrays, the NSC_MRA provides the minimum RMSE, especially for strong mutual coupling.

Based on the results, it can therefore be concluded that the NSC_MRA performs much better than the other arrays with mutual coupling effects.

5. Conclusions and Discussion

This paper proposes a new hole-free array, named nested symmetric complementary minimum redundancy array (NSC_MRA), designed to achieve high DOFs and enhanced DOA estimation performance. The main research conclusions are summarized as follows:

(1): Improvement in Degrees of Freedom (DOF): Compared with the nested array (NA), the proposed NSC_MRA achieves $N - 5$ more DOFs when the array sensor number $N$ is odd and $N - 4$ more DOFs when $N$ is even. This significant improvement highlights the advantage of the proposed array configuration in scenarios requiring larger DOFs.
(2): Simulation Results of DOA Estimation Performance: To validate the effectiveness of the proposed array, the SS-MUSIC algorithm was used for a variety of simulation experiments. Taking a 15-sensor array as an example, the performance of the proposed array was compared with five sparse linear arrays from other studies. The results show that the proposed array achieves a larger target number and higher angular resolution, demonstrating its superior DOA estimation capabilities.
(3): Robustness to Mutual Coupling: While the RMSE of the proposed array is slightly worse than that of the improved nested array (INA) in the absence of mutual coupling, it exhibits the best RMSE performance when mutual coupling effects are considered. This indicates that the proposed array offers significantly stronger robustness to mutual coupling compared to other sparse arrays.

These findings collectively highlight the advantages of the NSC_MRA in applications requiring high-precision DOA estimation with strong robustness to mutual coupling.

Author Contributions

Conceptualization, H.M. and X.M.; methodology, H.M. and X.M.; software, H.M.; validation, H.M. and X.M.; formal analysis, H.M. and X.M.; investigation, H.M. and X.M.; resources, H.M. and X.M.; data curation, H.M. and X.M.; writing—original draft preparation, H.M. and X.M.; writing—review and editing, H.M., L.L., Z.G., Y.G. and X.M.; visualization, H.M. and X.M.; supervision, L.L. and X.M.; project administration, X.M.; funding acquisition, X.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the Key Project of National Natural Science Foundation of China (No. 61831009), National Natural Science Foundation of China (No. 62371479), and the Aeronautical Science Foundation of China (No. 2024Z073077002).

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Schematic diagram of the SC_MRA.

Figure 2. An illustration of a 14-sensor NA configuration containing a 7-sensor dense ULA and a 7-sensor sparse ULA.

Figure 3. An illustration of a 14-sensor NSC_MRA configuration containing a 4-sensor MRA, a 3-sensor CMRA, and a sparse 7-sensor ULA.

Figure 4. An illustration of a 20-sensor NSC_MRA configuration with optimal DOFs under the constraint of the total number of array sensors.

Figure 5. The array configurations and non-negative covariance array weight functions of 15-sensor arrays. (a) NA; (b) ENA; (c) OSENA; (d) ANA; (e) INA; and (f) the proposed method (NSC_MRA).

Figure 6. The magnitudes of the mutual coupling matrices of 15-sensor arrays and their respective coupling leakage. (a) NA; (b) ENA; (c) OSENA; (d) ANA; (e) INA; and (f) the proposed method (NSC_MRA).

Figure 7. Spectrum of SS-MUSIC for 15-sensor arrays without mutual coupling when

K = 51

sources are located at

[- 60 ° : 2.4 ° : 60 °]

, SNR

= 0

dB, and

L = 1000

. (a) NA; (b) ENA; (c) OSENA; (d) ANA; (e) INA; and (f) the proposed method (NSC_MRA).

Figure 7. Spectrum of SS-MUSIC for 15-sensor arrays without mutual coupling when

K = 51

sources are located at

[- 60 ° : 2.4 ° : 60 °]

, SNR

= 0

dB, and

L = 1000

. (a) NA; (b) ENA; (c) OSENA; (d) ANA; (e) INA; and (f) the proposed method (NSC_MRA).

Figure 8. Spectrum of SS-MUSIC for 15-sensor arrays without mutual coupling when

K = 2

targets are located at

[- 0.3 °, 0.3 °]

, SNR = 0 dB, and

L = 1000

Figure 8. Spectrum of SS-MUSIC for 15-sensor arrays without mutual coupling when

K = 2

targets are located at

[- 0.3 °, 0.3 °]

, SNR = 0 dB, and

L = 1000

Figure 9. Probability of correct detections vs. SNR for different array configurations without mutual coupling when

K = 2

targets are located at

[- 0.3 °, 0.3 °]

N = 15

, and

L = 1000

. The acceptable angle error is set to 0.05°.

Figure 9. Probability of correct detections vs. SNR for different array configurations without mutual coupling when

K = 2

targets are located at

[- 0.3 °, 0.3 °]

N = 15

, and

L = 1000

. The acceptable angle error is set to 0.05°.

Figure 10. RMSE (in degrees) curves vs. SNR for different arrays in Table 2 without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

L = 1000

Figure 10. RMSE (in degrees) curves vs. SNR for different arrays in Table 2 without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

L = 1000

Figure 11. RMSE (in degrees) curves vs. the number of snapshots for different arrays in Table 2 without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

S N R = 0

dB.

Figure 11. RMSE (in degrees) curves vs. the number of snapshots for different arrays in Table 2 without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

S N R = 0

dB.

Figure 12. RMSE (in degrees) curves vs. SNR for different array configurations without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

L = 1000

Figure 12. RMSE (in degrees) curves vs. SNR for different array configurations without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

L = 1000

Figure 13. Probability of correct detections vs. SNR for different array configurations without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

L = 1000

. The acceptable angle error is set to 0.15°.

Figure 13. Probability of correct detections vs. SNR for different array configurations without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

, and

L = 1000

. The acceptable angle error is set to 0.15°.

Figure 14. RMSE (in degrees) curves vs. the number of snapshots for different array configurations without mutual coupling when K = 21 targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

S N R = 0

dB, and

c_{l} = 0

Figure 14. RMSE (in degrees) curves vs. the number of snapshots for different array configurations without mutual coupling when K = 21 targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

S N R = 0

dB, and

c_{l} = 0

Figure 15. RMSE (in degrees) curves vs.

N

for different array configurations without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

, SNR

= 0

dB, and

L = 1000

Figure 15. RMSE (in degrees) curves vs.

N

for different array configurations without mutual coupling when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

, SNR

= 0

dB, and

L = 1000

Figure 16. RMSE (in degrees) curves vs. SNR for different array configurations in the presence of mutual coupling.

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

L = 1000

{c_{1} = 0.2 e}^{j π / 3}

, and

c_{l} = c_{1} / l

Figure 16. RMSE (in degrees) curves vs. SNR for different array configurations in the presence of mutual coupling.

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

L = 1000

{c_{1} = 0.2 e}^{j π / 3}

, and

c_{l} = c_{1} / l

Figure 17. RMSE (in degrees) curves vs. the number of snapshots for different array configurations in the presence of mutual coupling.

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

S N R = 0

dB,

{c_{1} = 0.2 e}^{j π / 3}

, and

c_{l} = c_{1} / l

Figure 17. RMSE (in degrees) curves vs. the number of snapshots for different array configurations in the presence of mutual coupling.

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

S N R = 0

dB,

{c_{1} = 0.2 e}^{j π / 3}

, and

c_{l} = c_{1} / l

Figure 18. RMSE (in degrees) curves vs.

a

for different array configurations when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

L = 1000

{c_{1} = a \times 0.2 e}^{j π / 3}

, and

c_{l} = c_{1} / l

Figure 18. RMSE (in degrees) curves vs.

a

for different array configurations when

K = 21

targets are located at

[- 60 ° : 6 ° : 60 °]

N = 15

L = 1000

{c_{1} = a \times 0.2 e}^{j π / 3}

, and

c_{l} = c_{1} / l

Table 1. The array configurations of MRAs and corresponding CMRAs.

$N$	$P_{MRA, N}$	$P_{CMRA, N^{'}}$	$M_{N}$	$N_{N^{'}}$
3	0 1 3	2	3	2
4	0 1 4 6	2 3 5	6	2
5	0 1 4 7 9	2 3 5 6 8	9	2
6	0 1 2 6 10 13	3 4 5 7 8 9 11 12	13	3

Table 2. The array configurations and DOFs of 15-sensor NSC_MRA when the number of sensors in different levels changes.

$N_{1} (N_{A}, N_{A}^{'})$	$N_{2}$	$P_{NSCM, 15}$	$F_{NSCM, 15}$
4(3,1)	11	1 2 4 5 10 15 20 25 30 35 40 45 50 55 57	113
5(3,2)	10	1 2 4 6 12 18 24 30 36 42 48 54 60 61 63	125
6(3,3)	9	1 2 4 7 14 21 28 35 42 49 56 63 64 65 67	133
7(4,3)	8	1 2 5 7 8 16 24 32 40 48 56 64 66 68 69	137
8(4,4)	7	1 2 5 7 9 18 27 36 45 54 63 64 66 68 69	137
9(4,5)	6	1 2 5 7 10 20 30 40 50 60 61 62 64 66 67	133
10(5,5)	5	1 2 5 8 10 11 22 33 44 55 57 59 60 62 63	125
11(5,6)	4	1 2 5 8 10 12 24 36 48 49 51 53 54 56 57	113
12(5,7)	3	1 2 5 8 10 13 26 39 40 41 43 45 46 48 49	97
13(5,8)	2	1 2 5 8 10 14 28 29 30 31 33 35 36 38 39	77

Table 3. The DOFs of different arrays when the number of sensors changes.

$N$	NA	ENA	OSENA	ANA	INA	NSC_MRA
12	83	85	--	91	93	91
13	97	97	87	105	107	105
14	111	113	111	121	123	121
15	127	127	135	137	139	137

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Ma, H.; Liu, L.; Gan, Z.; Gao, Y.; Mao, X. Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation. Remote Sens. 2024, 16, 4711. https://doi.org/10.3390/rs16244711

AMA Style

Ma H, Liu L, Gan Z, Gao Y, Mao X. Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation. Remote Sensing. 2024; 16(24):4711. https://doi.org/10.3390/rs16244711

Chicago/Turabian Style

Ma, He, Libao Liu, Zhihong Gan, Yang Gao, and Xingpeng Mao. 2024. "Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation" Remote Sensing 16, no. 24: 4711. https://doi.org/10.3390/rs16244711

APA Style

Ma, H., Liu, L., Gan, Z., Gao, Y., & Mao, X. (2024). Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation. Remote Sensing, 16(24), 4711. https://doi.org/10.3390/rs16244711

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Hole-Free Symmetric Complementary Sparse Array Design for High-Precision DOA Estimation

Abstract

1. Introduction

2. Basic Conceptions

2.1. Difference Covariance Arrays

2.2. Sparse Array Processing

2.3. Signal Model with Mutual Coupling

3. Proposed Array Design with MRA, Complementary MRA, and NA

3.1. Symmetric Complementary Minimum Redundant Array

3.2. Nested Symmetric Complementary Minimum Redundant Array

3.3. Optimal Element Allocation Method

4. Performance Analysis

4.1. Array Aperture and Degrees of Freedom

4.2. Mutual Coupling Effect

4.3. Numerical Simulations

4.3.1. Spatial Spectrum Comparison of Different Arrays

4.3.2. Performance of Distinguishing the Targets with Similar Angles of Incidence

4.3.3. RMSE of NSC_MRA When the Number of Sensors in Different Levels Changes

4.3.4. DOA Estimation Performance of Different Arrays Without Mutual Coupling

4.3.5. DOA Estimation Performance of Different Arrays in the Presence of Mutual Coupling

5. Conclusions and Discussion

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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$N$	NA	ENA	OSENA	ANA	INA	NSC_MRA
12	83	85	--	91	93	91
13	97	97	87	105	107	105
14	111	113	111	121	123	121
15	127	127	135	137	139	137

$N$	NA	ENA	OSENA	ANA	INA	NSC_MRA
12	83	85	--	91	93	91
13	97	97	87	105	107	105
14	111	113	111	121	123	121
15	127	127	135	137	139	137

$N$	NA	ENA	OSENA	ANA	INA	NSC_MRA
12	83	85	--	91	93	91
13	97	97	87	105	107	105
14	111	113	111	121	123	121
15	127	127	135	137	139	137