Joint Estimation of 2D-DOA and Frequency Based on Space-Time Matrix and Conformal Array

Abstract

Each element in the conformal array has a different pattern, which leads to the performance deterioration of the conventional high resolution direction-of-arrival (DOA) algorithms. In this paper, a joint frequency and two-dimension DOA (2D-DOA) estimation algorithm for conformal array are proposed. The delay correlation function is used to suppress noise. Both spatial and time sampling are utilized to construct the spatial-time matrix. The frequency and 2D-DOA estimation are accomplished based on parallel factor (PARAFAC) analysis without spectral peak searching and parameter pairing. The proposed algorithm needs only four guiding elements with precise positions to estimate frequency and 2D-DOA. Other instrumental elements can be arranged flexibly on the surface of the carrier. Simulation results demonstrate the effectiveness of the proposed algorithm.

1. Introduction

Conformal array is the array mounted on the surface of the conformal carrier [1]. The advantage of the conformal array includes the alleviation of array weight, reduction of aerodynamic drag, reduction of radar cross-section (RCS), and the cover of the whole range of azimuth [2]. Thus the conformal array is of great interest in a variety of applications such as radar, sonar, wireless communication, and seismology.

Conventional algorithms for direction-of-arrival (DOA) estimation, such as multiple signal classification (MUSIC) [3] and estimation of signal parameters via rotational invariance techniques (ESPRIT) [4], are not suitable for conformal array because of the varying curvature of the conformal carrier. Based on array interpolation [5, 6], the manifold separation technique was applied to arbitrary array structure [7]. The above algorithm possesses good performance when the pattern of element is isotropic directional. However, due to the effect of the curvature of conformal carrier, each element of the conformal array has different patterns [8]. Thus, the algorithm proposed in [7] could not be used for conformal array. Not all elements can receive signals because of the “shadow effect” caused by metallic carrier. The subarray divided technique and interpolation transformation were adopted for DOA estimation of conformal array. But the interpolation accuracy of virtual manifold transformation is not high [9, 10]. Recently, DOA estimation algorithms for conformal array have been proposed. Based on ESPRIT and specially designed array structure, a blind DOA estimation algorithm was presented [11]. The iterative ESPRIT algorithm was proposed for joint DOA and polarization parameter estimation [12]. Nevertheless, the two algorithms need parameter pairing and special array design.

By combining the techniques of temporal filtering and spatial beam forming, the joint DOA and delay were estimated [13]. Similarly, the proposed approach in [14] was a novel twist of parameter estimation and filtering processes, in which two 1D frequencies and one 1D spatial MUSIC were employed to estimate the DOAs and frequencies, respectively. Two approximate maximum likelihood (ML) algorithms were proposed in the spatially correlated noise scenario for joint frequency and DOA estimation. However, MUSIC and ML algorithms used in [13–15] suffer from tremendous computational complexity and are only suitable for linear array or planar array, which could not be used for conformal array. The research of joint frequency and DOA estimation is rare for the conformal array.

As a useful analysis tool of data arrays, parallel factor (PARAFAC) analysis model [16, 17] is a generalization of low-rank matrix decomposition to three-way arrays (TWAs) or multiway arrays (MWAs), which was introduced in array signal processing to estimate DOA [18]. PARAFAC has been first introduced as a data analysis tool in psychometrics, where it has been used in arithmetic complexity, exploratory data analysis, and other fields. The PARAFAC model was developed by Sidiropoulos et al. [19]. Most parameter estimation algorithms for conformal array focus on DOA estimation. The frequency estimation is not researched as far as we known. For 2D-DOA estimation of conformal array, the algorithms based on ESPRIT suffer from parameter pairing problem. The algorithms based on MUSIC suffer from tremendous computational complexity. In this paper, a novel joint frequency and 2D-DOA estimation algorithm for conformal array is proposed. The spatial domain sampling and time domain sampling are utilized to construct the spatial-time matrix. The delay correlation function is used to suppress noise. Based on PARAFAC, the joint frequency and 2D-DOA estimation are accomplished without parameter pairing. Unlike the special array design in [9–12], the proposed algorithm needs only four guiding elements, other instrumental elements of the array can be arranged flexibly.

The organization of this paper is structured as follows. Section 2 introduces the structure of the conformal array and describes the snapshot data model. Section 3 contains the core contributions of this paper, including the construction of spatial-time matrix and the PARAFAC model used for high accuracy frequency and 2D-DOA estimation. Section 4 presents the simulation result. Section 5 summarizes our conclusions.

2. The Structure of the Conformal Array and the Snapshot Data Model

A curve array antenna mounted on an arbitrary shape carrier is shown in Figure 1. Assuming that there are 𝑀 elements on the surface of the carrier, the coordinate of each element can be represented as (𝑥_𝑚,𝑦_𝑚,𝑧_𝑚),??𝑚=1,2,…,𝑀. Four guiding elements 𝐩₁, 𝐩₂, 𝐩₃, and 𝐩₄ with precise positions are chosen, and 𝐩₁ is assumed to be the reference element. 𝐩_𝑚 represents the position vector of the 𝑚th element on the surface of the carrier. 𝐞_𝑥, 𝐞_𝑦, and 𝐞_𝑧 represent the unit vector of 𝑋-axis, 𝑌-axis, and 𝑍-axis, respectively. 𝐩_𝑚 can be expressed as 𝐩_𝑚=𝑥_𝑚𝐞_𝑥+𝑦_𝑚𝐞_𝑦+𝑧_𝑚𝐞_𝑧. (1)

Figure 1 

The structure of the conformal array.

Consider 𝐾 narrow far field incident signals impinging on the conformal array, which is shown in Figure 1. The elevation and azimuth of 𝑘th incident signal are denoted as (𝜃_𝑘,𝜑_𝑘), 𝑘=1,2,…,𝐾. Thus, the general form of the array output is represented as 𝑥_𝑚(𝑡)=^𝐾_𝑘=1𝑠_𝑘(𝑡)𝑔_𝑚exp−𝑗2𝜋𝑓_𝑘𝑐𝐩_𝑚•𝐮_𝑘, (2) where 𝑠_𝑘(𝑡) stands for the 𝑘th incident signal and 𝑔_𝑚 is the element pattern which is defined in the 𝑚th local coordinate. 𝑓_𝑘 is the frequency of the 𝑘th incident signal, 𝑢_𝑘 is the direction vector of the incident signal 𝑠_𝑘(𝑡).

Consider 𝐮_𝑘=sin𝜃_𝑘cos𝜑_𝑘𝐞_𝑥+sin𝜃_𝑘sin𝜑_𝑘𝐞_𝑦+cos𝜃_𝑘𝐞_𝑧. (3)

The time domain sampling is done for the signal of the array output. The number of the tapped delay line is 𝐿. Then the 𝑙th tapped delay of the signal output of the 𝑚th element can be represented as 𝑥_𝑚(𝑡−𝑙𝜏)=^𝐾_𝑘=1𝑠_𝑘(𝑡)𝑔_𝑚exp−𝑗𝜔_𝑘𝜏_𝑚𝑘+𝑙𝜏+𝑛_𝑚(𝑡−𝑙𝜏),0≤𝑙≤𝐿−1, (4) where 𝜏_𝑚𝑘 represents the time delay between the 𝑚th element 𝐩_𝑚 and reference element 𝐩₁ when the mth element receives the 𝑘th incident signal, 𝜏 is the time interval of sampling, 𝑛_𝑚 is the additional white Gaussian noise (AWGN) with zero mean and variance 𝜎², 𝜔_𝑘=2𝜋𝑓_𝑘, and 𝜏_𝑚𝑘=(𝐩_𝑚•𝐮_𝑘)/𝑐.

The condition that the same incident signal impinges on the global coordinate and local coordinate is also shown in Figure 1. The coordinate which is constructed by the original point 𝐩₁ is the global coordinate. The coordinate which is constructed by the original point 𝐩_𝑚 is the local coordinate. As shown in Figure 1, the elevation and azimuth of the incident signal in global coordinate and local coordinate are distinct. The pattern function 𝑔_𝑚(𝜃_𝑚,𝜑_𝑚) is defined in the local coordinate of each element. Thus, the pattern in the local coordinate of each element is different. Here the parameter component transformation from the global coordinate to the 𝑚th local coordinate can be accomplished by using the general Euler rotate method [8]. More details can be found in [20].

Due to the “shadow effect” caused by the metal, not all elements in the conformal array can receive the signal. Thus, the subarray divided technique [9, 10] is adopted and the whole array can be divided into several parts. The same parameter estimation mechanism is used for each part. For the sake of simplicity, it is assumed that all the elements can receive the signal.

3. Joint Frequency and 2D-DOA Estimation

3.1. The Construction of Spatial-Time Matrix

The delay correlation function between the 𝑚th element and 𝑛th element of the conformal array is constructed as follows: 𝑅_{𝑥𝑚𝑥𝑛}=𝐸𝑥_𝑚(𝑡)𝑥^∗_𝑛(=𝑡−𝜏)_𝐾𝑘=1𝐸𝑠_𝑘(𝑡)𝑠^∗_𝑘(𝑡−𝜏)×exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_𝑛𝑘𝑛−𝜏+𝐸_𝑚(𝑡)𝑛^∗_𝑛(=𝑡−𝜏)_𝐾𝑘=1𝑅_{𝑠𝑘𝑠𝑘}(𝜏)𝑔_𝑚𝑔_𝑛exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_𝑛𝑘×exp𝑗𝜔_𝑘𝜏+𝑅_{𝑛𝑚𝑛𝑛}(𝜏),𝜏>0, (5) where 𝐑_{𝑠𝑘𝑠𝑘}(𝜏)=𝐸[𝑠_𝑚(𝑡)𝑠^∗_𝑛(𝑡−𝜏)] represents the delay correlation function of 𝑠_𝑘(𝑡), and 𝐑_{𝑛𝑚𝑛𝑛}(𝜏)=𝐸[𝑛_𝑚(𝑡)𝑛^∗_𝑛(𝑡−𝜏)] represents the noise cross-correlation function between the 𝑚th element and 𝑛th element.

On condition that the incident signal is coming from narrow far field and the bandwidth of the incident signal is 𝐁, the delay correlation processing of the data that the elements receive takes a long period of time. Hence, 𝐑_{𝑠𝑘𝑠𝑘}(𝜏)=𝐸[𝑠_𝑚(𝑡)𝑠^∗_𝑛(𝑡−𝜏)]≠0. In the more general case, when 𝜏≪1/𝐵, the Gaussian white noise is not correlated during the time interval 𝜏. At the same time, the variety of the signal envelope can be neglected. Then we can suppress the noise without destroying the signal.

Consider 𝐑_{𝑛𝑚𝑛𝑛}(𝜏)=𝐸𝑛_𝑚(𝑡)𝑛^∗_𝑛(𝑡−𝜏)=𝜎²𝛿(𝜏)𝛿(𝑚−𝑛)=0, (6) where the delay correlation function is constructed between 𝑥₁(𝑡−𝜏) and the data that each element receives. Based on (5), we can obtain 𝐑_{𝑥𝑚𝑥1}(𝜏)=𝐸𝑥_𝑚(𝑡)𝑥^∗₁(=𝑡−𝜏)_𝐾𝑘=1𝑅_{𝑠𝑘𝑠𝑘}(𝜏)𝑔_𝑚𝑔₁exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_1𝑘×exp𝑗𝜔_𝑘𝜏,1≤𝑚≤𝑀. (7) Similarly, the delay correlation functions are constructed between 𝑥₂(𝑡−𝜏), 𝑥₃(𝑡−𝜏), 𝑥₄(𝑡−𝜏), and 𝑥₁(𝑡) and the data that each element receives, respectively.

Consider 𝐑_{𝑥𝑚𝑥2}(𝜏)=𝐸𝑥_𝑚(𝑡)𝑥^∗₂(=𝑡−𝜏)_𝐾𝑘=1𝐑_{𝑠𝑘𝑠𝑘}(𝜏)𝑔_𝑚𝑔₁exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_2𝑘×𝑔₂𝑔₁exp𝑗𝜔_𝑘𝜏𝐑,1≤𝑚≤𝑀,_{𝑥𝑚𝑥3}(𝜏)=𝐸𝑥_𝑚(𝑡)𝑥^∗₃(=𝑡−𝜏)_𝐾𝑘=1𝐑_{𝑠𝑘𝑠𝑘}(𝜏)𝑔_𝑚𝑔₁exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_3𝑘×𝑔₃𝑔₁exp𝑗𝜔_𝑘𝜏𝐑,1≤𝑚≤𝑀,_{𝑥𝑚𝑥4}(𝜏)=𝐸𝑥_𝑚(𝑡)𝑥^∗₄(=𝑡−𝜏)_𝐾𝑘=1𝐑_{𝑠𝑘𝑠𝑘}(𝜏)𝑔_𝑚𝑔₁exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_4𝑘×𝑔₄𝑔₁exp𝑗𝜔_𝑘𝜏𝐑,1≤𝑚≤𝑀,_{𝑥𝑚𝑥1}(0)=𝐸𝑥_𝑚(𝑡)𝑥^∗₁(𝑡=_𝐾𝑘=1)𝐑_{𝑠𝑘𝑠𝑘}(0)𝑔_𝑚𝑔₁exp−𝑗𝜔_𝑘𝜏_𝑚𝑘−𝜏_1𝑘+𝜎²𝐈,1≤𝑚≤𝑀. (8)

The spatial-time matrix is constructed as follows: 𝐑_𝑆(𝜏)=𝐑_𝑠1𝑠1(𝜏),𝐑_𝑠2𝑠2(𝜏),…,𝐑_{𝑠𝐾𝑠𝐾}(𝜏)^𝑇,𝐑_𝑓(𝜏)=𝐑_𝑥1𝑥1(𝜏),𝐑_𝑥2𝑥1(𝜏),…,𝐑_{𝑥𝑚𝑥1}(𝜏)^𝑇,𝐑₁(𝜏)=𝐑_𝑥1𝑥2(𝜏),𝐑_𝑥2𝑥2(𝜏),…,𝐑_{𝑥𝑚𝑥2}(𝜏)^𝑇,𝐑₂(𝜏)=𝐑_𝑥1𝑥3(𝜏),𝐑_𝑥2𝑥3(𝜏),…,𝐑_{𝑥𝑚𝑥3}(𝜏)^𝑇,𝐑₃(𝜏)=𝐑_𝑥1𝑥4(𝜏),𝐑_𝑥2𝑥4(𝜏),…,𝐑_{𝑥𝑚𝑥4}(𝜏)^𝑇,𝐑(0)=𝐑_𝑥1𝑥1(0),𝐑_𝑥2𝑥1(0),…,𝐑_{𝑥𝑚𝑥1}(0)^𝑇,𝐀=𝐚₁𝜔₁,𝐚₂𝜔₂,…,𝐚_𝐾𝜔_𝐾,𝐚_𝑘𝜔_𝑘=𝑔₁e^{(−𝑗(𝜔𝑘/𝑐)𝐩1•𝐮𝑘)},𝑔₂e^{(−𝑗(𝜔𝑘/𝑐)𝐩2•𝐮𝑘)},…,𝑔_𝑀e^{(−𝑗(𝜔𝑘/𝑐)𝐩𝑀•𝐮𝑘)𝑇}. (9)(10)(11)(12)(13)(14)(15)(16) The matrix 𝐀 is the manifold matrix and 𝐚 is the steering matrix. Then (9)–(14) can be written as 𝐑_𝑓(𝜏)=𝐀Φ_𝑓𝐑_𝑆(𝜏),𝐑₁(𝜏)=𝐀Φ₁Φ_𝑓𝐑_𝑆(𝜏),𝐑₂(𝜏)=𝐀Φ₂Φ_𝑓𝐑_𝑆(𝜏),𝐑₃(𝜏)=𝐀Φ₃Φ_𝑓𝐑_𝑆(𝜏),𝐑(0)=𝐀𝐑_𝑆(0), (17)(18)(19)(20)(21) where Φ_𝑓=diage^{(𝑗𝜔1𝜏)},e^{(𝑗𝜔2𝜏)},…,e^{(𝑗𝜔𝐾𝜏)𝑇},Φ₁𝑔=diag²𝑔₁e^{(𝑗(𝜔1/𝑐)(𝐩2−𝐩1)•𝐮1)},𝑔₂𝑔₁e^{(𝑗(𝜔2/𝑐)(𝐩2−𝐩1)•𝐮2)}𝑔,…,²𝑔₁e^{(𝑗(𝜔𝐾/𝑐)(𝐩2−𝐩1)•𝐮𝐾)𝑇},Φ₂𝑔=diag³𝑔₁e^{(𝑗(𝜔1/𝑐)(𝐩3−𝐩1)•𝐮1)},𝑔₃𝑔₁e^{(𝑗(𝜔2/𝑐)(𝐩3−𝐩1)•𝐮2)}𝑔,…,³𝑔₁e^{(𝑗(𝜔𝐾/𝑐)(𝐩3−𝐩1)•𝐮𝐾)𝑇},Φ₃𝑔=diag⁴𝑔₁e^{(𝑗(𝜔1/𝑐)(𝐩4−𝐩1)•𝐮1)},𝑔₄𝑔₁e^{(𝑗(𝜔2/𝑐)(𝐩4−𝐩1)•𝐮2)}𝑔,…,⁴𝑔₁e^{(𝑗(𝜔𝐾/𝑐)(𝐩4−𝐩1)•𝐮𝐾)𝑇},𝜂_𝑓𝑘=𝜔_𝑘𝜏=2𝜋𝑓_𝑘𝜂𝜏,_1𝑘=𝜔_𝑘𝑐𝐩₂−𝐩₁•𝑢_𝑖,𝜂_2𝑘=𝜔_𝑘𝑐𝐩₃−𝐩₁•𝑢_𝑖,𝜂_3𝑘=𝜔_𝑘𝑐𝐩₄−𝐩₁•𝑢_𝑖. (22)(23)(24)(25)(26)(27)(28)(29) The incident signal is narrowband; thus, 𝐑_{𝑠𝑘𝑠𝑘}(𝑛𝜏)=𝐑_{𝑠𝑘𝑠𝑘}((𝑛−1)𝜏). Equation (21) can be written in another form as 𝐑(𝜏)=𝐀𝐑_𝑆(𝜏). (30)

Equations (17)–(21) possess the analogous form. 𝐑_𝑓(𝜏), 𝐑₁(𝜏), 𝐑₂(𝜏), 𝐑₃(𝜏), and 𝐑(0) are sampled at a time delay 𝜏_𝑠 for 𝑁 (𝑁≥𝐾) times, 𝜏_𝑠=𝑇_𝑠,2𝑇_𝑠,…,𝑁𝑇_𝑠, and 𝑇_𝑠 is the time interval of sampling which should be less than the reciprocal of the highest frequency. Thus, the aliasing effect would not happen. The pseudosnapshot data matrix can be constructed by using (17)–(21) as follows: 𝐑_𝑓=𝐑𝑇_𝑠,𝐑2𝑇_𝑠,…,𝐑𝑁𝑇_𝑠,𝐑₁=𝐑₁𝑇_𝑠,𝐑₁2𝑇_𝑠,…,𝐑₁𝑁𝑇_𝑠,𝐑₂=𝐑₂𝑇_𝑠,𝐑₂2𝑇_𝑠,…,𝐑₂𝑁𝑇_𝑠,𝐑₃=𝐑₃𝑇_𝑠,𝐑₃2𝑇_𝑠,…,𝐑₃𝑁𝑇_𝑠,𝐑=𝐑_𝑓𝑇_𝑠,𝐑_𝑓2𝑇_𝑠,…,𝐑_𝑓𝑁𝑇_𝑠. (31) Equation (31) can be written as 𝐑_𝑓=𝐀Φ_𝑓𝐑_𝑆,𝐑₁=𝐀Φ₁Φ_𝑓𝐑_𝑆,𝐑₂=𝐀Φ₂Φ_𝑓𝐑_𝑆,𝐑₃=𝐀Φ₃Φ_𝑓𝐑_𝑆,𝐑=𝐀𝐑_𝑆, (32) where 𝐑_𝑆=𝐑_𝑆𝑇_𝑠,𝐑_𝑆2𝑇_𝑠,…,𝐑_𝑆𝑁𝑇_𝑠. (33)

3.2. PARAFAC

The element of a tensor 𝐗∈𝒞^{𝐶×𝐷×𝐸} is 𝑥_{𝑖,𝑗,𝑘} and its 𝑃-component PARAFAC decomposition is given by 𝑥_{𝑐,𝑑,𝑒}=^𝑀_𝑝=1𝑠_𝑐,𝑝𝑡_𝑑,𝑝𝑢_𝑒,𝑝, (34) where 𝑐=1,…,𝐶, 𝑑=1,…,𝐷, and 𝑒=1,…,𝐸. 𝑠_𝑐,𝑝, 𝑡_𝑑,𝑝, and 𝑢_𝑒,𝑝 stand for the elements of 𝐒∈𝒞^𝐶×𝑃, 𝐓∈𝒞^𝐷×𝑃, and 𝐔∈𝒞^𝐸×𝑃, respectively. The typical elements of the 𝐗_𝑐∈𝒞^𝐷×𝐸, 𝐗_𝑑∈𝒞^𝐸×𝐶, and 𝐗_𝑒∈𝒞^𝐶×𝐷are 𝑋_𝑐(𝑑,𝑒)∶=𝑋_𝑑(𝑐,𝑒)∶=𝑋_𝑒(𝑐,𝑑)∶=𝑥_{𝑐,𝑑,𝑒}, respectively. The three way array (TWA) 𝐗 is “sliced” in three different directions: 𝐗_𝑐=𝐓Λ_𝑐𝐒𝐔^𝑇𝐗,𝑐=1,…,𝐶,_𝑑=𝐔Λ_𝑑𝐓𝐒^𝑇𝐗,𝑑=1,…,𝐷,_𝑒=𝐒Λ_𝑒𝐔𝐓^𝑇,𝑒=1,…,𝐸. (35) For a given matrix 𝐒, Λ_𝑐(𝐒) means a diagonal matrix with the diagonals given by the 𝑐th row of the matrix. The symbol “⊙” is used to denote the Khatri-Rao (KR) product. As an example, the KR product of two matrices 𝐒∈𝒞^𝐶×𝑃 and 𝐓∈𝒞^𝐷×𝑃 of an identical number of columns is given by 𝐒⊙𝐓=𝐬₁⊗𝐭₁,…,𝐬_𝑃⊗𝐭_𝑃∈𝒞^{𝐶𝐷×𝑃}, (36) where “⊗” represents the Kronecker product. Then the definition of Kronecker product of two vectors 𝐬∈𝒞^𝐶 and 𝐭∈𝒞^𝐷 is given by 𝐬⊗𝐭=𝑠₁𝐭𝑠₂𝐭⋮𝑠_𝐶𝐭. (37)

Based on the definition of KR product, 𝐗^{𝐶𝐷×𝐸} can be expressed as 𝐗^{𝐶𝐷×𝐸}=𝐗_𝑐=1𝐗_𝑐=2⋮𝐗_𝑐=𝐶=𝐓Λ₁𝐒𝐓Λ₂𝐒⋮𝐓Λ_𝐶𝐒𝐔^𝑇=𝐒⊙𝐓𝐔^𝑇. (38) Similarly, the matrix 𝐗^{𝐷𝐸×𝐶} and 𝐗^{𝐸𝐶×𝐷} can be expressed as 𝐗^{𝐷𝐸×𝐶}=𝐓⊙𝐔𝐒^𝑇,𝐗^{𝐸𝐶×𝐷}=𝐔⊙𝐒𝐓^𝑇. (39)

Definition 1 (Kruskal rank [18]). The Kruskal rank, or k-rank of a given matrix 𝐒∈𝒞^𝐶×𝑃 denoted by krank (𝐒) is said to be equal to 𝑟 when every collection of 𝑟 columns of (𝐒) is linearly independent but there exists a collection of 𝑟+1 linearly dependent columns. However, the rank of 𝐒 is said to be the maximal number of linearly independent columns. The definition of rank(𝐒) is more relaxed than that of krank(𝐒); that is, krank(𝐒)≤rank(𝐒).

Theorem 2 (Kruskal theorem [19]). Given a PARAFAC model 𝐗∈𝒞^{𝐶×𝐷×𝐸}; then the decomposition of this PARAFAC model is unique to permutation and scaling when 𝑘_𝐒+𝑘_𝐓+𝑘_𝐔≥2𝑃+2. (40) The matrix 𝐗 is constructed by 𝐒, 𝐓, and 𝐔 as 𝐒=𝐒𝚷Δ₁,𝐓=𝐓𝚷Δ₂,𝐔=𝐔𝚷Δ₃, (41) where 𝚷 stands for the permutation matrix. Δ₁, Δ₂, and Δ₃ are diagonal scaling matrices satisfying Δ₁Δ₂Δ₃=𝐈, (42) where 𝐈 is a unit matrix.

The 𝑀×𝑀×5 TWA of the conformal array is constructed based on PARAFAC model. 𝐑()𝐑()𝐑()𝐑()𝐑()=𝐑𝐑=∶,∶,1∶,∶,2∶,∶,3∶,∶,4∶,∶,5^𝑓𝐑₁𝐑₂𝐑₃𝐑=𝐀Φ_𝑓𝐑_𝑆𝐀Φ₁Φ_𝑓𝐑_𝑆𝐀Φ₂Φ_𝑓𝐑_𝑆𝐀Φ₃Φ_𝑓𝐑_𝑆𝐀𝐑_𝑆+𝐐, (43) where 𝐐 represents the observed noise in practice. In order to make the decomposition of the PARAFAC unique, the following assumption must be satisfied: 𝐚_𝑖𝜔_𝑖≠𝐚_𝑗𝜔_𝑗. (44) In other words, for two incident signals 𝑠_𝑖(𝑡) and 𝑠_𝑗(𝑡) with different DOAs, the following condition must be satisfied 𝑔_𝑜𝑒e^{−𝑗2𝜋𝑓𝑖((𝐩𝑜𝑒•𝐮𝑖)/𝑐)}≠𝑔_𝑜𝑒e^{−𝑗2𝜋𝑓𝑗((𝐩𝑜𝑒•𝐮𝑗)/𝑐)}. (45) The k-rank of matrix 𝐃, 𝐀, and 𝐑_𝑆 are 𝑘_𝐷=min(5,𝐾), 𝑘_𝐴=𝐾, and 𝑘_𝑅𝑆=𝐾, respectively. Thus, when 𝐾≥2, the k-rank decomposition of the PARAFAC is unique.

Based on the definition of KR product, (43) can be written in three forms as follows: 𝐑=(𝐃⊙𝐀)𝐑_𝑆+𝐑𝐐,_𝑋=𝐑^𝑇_𝑆⊙𝐃𝐀^𝑇+𝐐_𝑋,𝐑_𝑌=𝐀⊙𝐑^𝑇_𝑆𝐃^𝑇+𝐐_𝑌, (46)(47)(48) where Λ𝐃=⁻¹Φ_𝑓Λ⁻¹Φ₁Φ_𝑓Λ⁻¹Φ₂Φ_𝑓Λ⁻¹Φ₃Φ_𝑓Λ⁻¹(1), (49) where Λ⁻¹(Φ_𝑓) means the row vector constructed by the diagonal elements of the diagonal matrix Φ_𝑓.

The matrices 𝐃, 𝐀, and 𝐑_𝑆 can be solved by the trilinear alternating least squares (TALS) regression algorithm. Due to the existence of the observed noise, (46) could be transformed as a least square problem as follows: min_{𝐃,𝐀,𝐑𝑆}𝐑−(𝐃⊙𝐀)𝐑_𝑆²_𝐹. (50) Then the least square estimator of 𝐑_𝑆 can be expressed as 𝐑_𝑆=argmin_𝐑𝑆𝐑−(𝐃⊙𝐀)𝐑_𝑆²_𝐹. (51) Similarly, the solutions of 𝐃 and 𝐀 can be expressed as 𝐀^𝑇=𝐑^𝑇_𝑆⊙𝐃^†𝐑_𝑋𝐃^𝑇=𝐀⊙𝐑^𝑇_𝑆^†𝐑_𝑌. (52) We can update the matrices 𝐃, 𝐀, and 𝐑_𝑆 alternately until the algorithm converges; the estimators of the three matrices can be acquired. In the following experiments, the COMFAC algorithm (the TALS regression algorithm is realized in practice) is used to fit the 𝑚×𝑚×5 TWA. The initialization and fitting of the COMFAC is done in compressed space. The Tucker 3 three-way model is used in data compression [21].

Now the ALS algorithm steps can be summarized as follows.(1)Initialize 𝐀⁽⁰⁾∈𝒞^𝑀×𝐾 and 𝐃⁽⁰⁾∈𝒞^5×𝐾.(2)Initialize 𝜀>0 and 𝑘=0.(3)If ‖𝜌^(𝑘+1)−𝜌^(𝑘)‖/𝜌^(𝑘)>𝜀, calculate matrices 𝐀, 𝐑_𝑆, and 𝐃 by (51)-(52). Update just one matrix at each time; then 𝑘→𝑘+1.(4)Else ‖𝜌^(𝑘+1)−𝜌^(𝑘)‖/𝜌^(𝑘)<𝜀; the iteration is terminated.

In this paper, 𝜌 stands for 𝐀, 𝐑_𝑆, and 𝐃, respectively.

3.3. The Joint Parameter Estimation

The matrix 𝐃 can be estimated by TALS regression algorithm; then the estimator of 𝜔_𝑓𝑘 is given by 𝜔_𝑘𝑖𝐃=angle^1𝑘𝐃_5𝑘. (53) Because 𝑟₁ and 𝑟₂ are real numbers, 𝐃_2𝑘/𝐃_1𝑘 is squared to avoid the ambiguity caused by the positive and negative values of 𝑟₁ and 𝑟₂. Then 𝜔_1𝑘1=−2𝑔angle²𝜃_𝑘,𝜑_𝑘𝑔₁𝜃_𝑘,𝜑_𝑘exp−𝑗𝜔_1𝑘²1=−2angleexp−𝑗2𝜔_1𝑘1=−2𝐃angle^2𝑘𝐃_1𝑘². (54) Similarly 𝜔_2𝑘1=−2𝐃angle^3𝑘𝐃_1𝑘²𝜔_3𝑘1=−2𝐃angle^4𝑘𝐃_1𝑘². (55)(56) Taking (26) into (53), the frequency 𝑓_𝑘 of the 𝑘th incident signal is expressed as 𝑓_𝑘=1𝐃2𝜋𝜏angle^1𝑘𝐃_5𝑘. (57) Assuming that 𝛾_1𝑘=sin(𝜃_𝑘)cos(𝜑_𝑘), 𝛾_2𝑘=sin(𝜃_𝑘)sin(𝜑_𝑘), and 𝛾_3𝑘=cos(𝜃_𝑘), and solving (27), (28), (29), (55), (56), and (57) simultaneously, we can obtain the equation as follows: 𝑐2𝜋𝑓_𝑘𝜔_1𝑘𝑑₁𝜔_2𝑘𝑑₂𝜔_3𝑘𝑑₃=𝑥₂−𝑥₁𝑦₂−𝑦₁𝑧₂−𝑧₁𝑥₃−𝑥₁𝑦₃−𝑦₁𝑧₃−𝑧₁𝑥₄−𝑥₁𝑦₄−𝑦₁𝑦₄−𝑦₁𝛾_1𝑘𝛾_2𝑘𝛾_3𝑘. (58) The solution of the equation is represented as 𝛾_1𝑘𝛾_2𝑘𝛾_3𝑘=𝑐2𝜋𝑓_𝑖𝑥₂−𝑥₁𝑦₂−𝑦₁𝑧₂−𝑧₁𝑥₃−𝑥₁𝑦₃−𝑦₁𝑧₃−𝑧₁𝑥₄−𝑥₁𝑦₄−𝑦₁𝑦₄−𝑦₁⁻¹𝜔_1𝑘𝑑₁𝜔_2𝑘𝑑₂𝜔_3𝑘𝑑₃. (59) There are two approaches to solve 𝜃_𝑘 and 𝜑_𝑘. The first approach is 𝜃_𝑘=arccos𝛾_3𝑘,𝜑_𝑘𝛾=arcsin^1𝑘cos𝜃_𝑘or𝜑_𝑘𝛾=arcsin^2𝑘sin𝜃_𝑘. (60) The second approach is 𝜑_𝑘𝛾=arctan^2𝑘𝛾_1𝑘,𝜃_𝑘𝛾=arccos^1𝑘cos𝜑_𝑘or𝜃_𝑘𝛾=arcsin^2𝑘sin𝜑_𝑘. (61) In this paper the first approach is employed.

In order to ensure the uniqueness of the estimated parameters, the following condition must be satisfied: 𝐩₂−𝐩₁≤𝑐2max𝑓_𝑖=min𝜆_𝑘2,𝐩₃−𝐩₁≤𝑐2max𝑓_𝑖=min𝜆_𝑘2,𝐩₄−𝐩₁≤𝑐2max𝑓_𝑖=min𝜆_𝑘2. (62) According to Theorem 2, the matrices 𝐃, 𝐀, and 𝐑_𝑆 have the same permutation matrix, that is, the 𝑘th column of matrix 𝐃 corresponds to the 𝑘th column of the manifold matrix 𝐀.

Based on the PARAFAC model and TALS regression algorithm, the joint frequency and 2D-DOA estimation can be accomplished. The algorithm steps are summarized as follows:(1)choose four guiding elements and obtain the accurate positions of the elements;(2)calculate the delay correlation function between the four elements mentioned above and the whole elements of the conformal array;(3)calculate the matrices 𝐑_𝑓(𝜏), 𝐑₁(𝜏), 𝐑₂(𝜏), 𝐑₃(𝜏), and 𝐑(0), respectively according to (5);(4)calculate the pseudosnapshot data matrix based on (17)–(21). 𝐑_𝑓(𝜏), 𝐑₁(𝜏), 𝐑₂(𝜏), 𝐑₃(𝜏), and 𝐑(0) are sampled at time delay 𝜏_𝑠, respectively;(5)construct the PARAFAC model based on (43);(6)estimate the matrix 𝐃 using TALS regression algorithm;(7)obtain 𝜔_𝑓𝑘, 𝜔_1𝑘, 𝜔_2𝑘, and 𝜔_3𝑘 by using the estimator of the matrix 𝐃 and calculating (53)–(56);(8)obtain the joint frequency and 2D-DOA estimation by calculate (57)–(59) or (57), (60), and (61).

4. Simulation Results

This section focuses on several sets of simulation results to demonstrate the performance of the proposed algorithm. In all simulation examples, 200 Monte Carlo trials are considered, and the root mean square error (RMSE) is used as our performance measure, which is defined as RMSE=1200²⁰⁰_𝜂=1𝜃_𝑘,𝜂−𝜃²+𝜑_𝑘,𝜂−𝜑², (63) where 𝜃_𝑖,𝑡 and 𝜑_𝑖,𝑡 are the elevation and azimuth estimators of the 𝑘th incident signal in the 𝜂th trial.

The successful rate is defined as the proportion of the number of the successful experiments to the total number of the experiments. For frequency estimation, a successful experiment is defined as the experiment with estimation error of less than 2?MHz; for DOA estimation, a successful experiment is defined as the experiment with estimation error of less than 2 degree.

In order to simplify the transformation from the global coordinate to the local coordinate, only the cylinder with single curvature is considered in the simulations. The array structure is shown in Figure 2 is only a subarray, which only covers 120^∘. Then three subarrays can estimate the whole space. The highest frequency of the incident signal is 𝑓_max=2?GHz, 𝜆_min=𝑐/𝑓_max. The radius of the cylinder is 5𝜆_min, the adjacent element spacing in the same intersecting surface of the cylinder is 𝜆_min/4, and the adjacent intersecting surface spacing is 𝜆_min/4, 𝜆_min stands for the shortest wavelength of the incident signal. The position of the reference element is 𝑝₁=(0,5𝜆,0).

Figure 2 

The structure of the cylindrical conformal array.

The element pattern is defined in the local coordinate of each element; thus, the transformation from the global coordinate to the local coordinate of elevation 𝜃_𝑘 and azimuth 𝜑_𝑘 of the incident signal must be accomplished. For the cylindrical conformal array, the corresponding Euler rotation angles are 𝐷=−Θ,𝐸=−𝜋/2,𝐹=0, (64) respectively. 𝐷 represents the rotation angle based on the right-hand rule in the first rotation and the 𝑍-axis is the rotation axis; 𝐸 represents the rotation angle in the second rotation and the 𝑌-axis is the rotation axis; and 𝐹 represents the rotation angle in the third rotation and the 𝑋-axis is the rotation axis. Θ represents the angle between the position vector of the element and the positive direction of 𝑋-axis.

The unit vector of the 𝑘th incident signal (𝜃_𝑘,𝜑_𝑘) in the global coordinate can be expressed as 𝑥=sin𝜃_𝑘cos𝜑_𝑘,𝑦=sin𝜃_𝑘sin𝜑_𝑘,𝑧=cos𝜃_𝑘. (65)

Based on the Euler rotation matrix, the unit vector in global coordinate can be transformed into the local coordinate of the 𝑚th element.

Consider 𝑥_𝑚𝑦_𝑚𝑧_𝑚^𝑇=𝑅𝐷_𝑚,𝐸_𝑚,𝐹_𝑚𝑥𝑦𝑧^𝑇, (66) where 𝑅𝐷_𝑚,𝐸_𝑚,𝐹_𝑚=××.cos𝐹sin𝐹0−sin𝐹cos𝐹0001cos𝐸0−sin𝐸010sin𝐸0cos𝐸cos𝐷sin𝐷0−sin𝐷cos𝐷0001 (67) Then the elevation 𝜃 and azimuth 𝜑 of the 𝑘th incident signal in the local coordinate of the 𝑚th element can be represented as 𝜑_𝑚𝑘=arctan𝑦_𝑚/𝑥_𝑚,𝜃_𝑚𝑘=arccos𝑧_𝑚. (68) Thus, the element pattern transformation from the global coordinate to the local coordinate is completed.

First, the curves of RMSE of frequency and angle and the successful probability of angle in different SNR are plotted in Figure 3 using the proposed algorithm. The number of snapshots is 200, the number of pseudosnapshots is 100. SNR∈[5,30], and the noise is AWGN which is independent of incident signals. The sampling frequency is 𝑓_𝑠=1/𝑇_𝑠=5?GHz. The elevation 𝜃, azimuth 𝜑, and frequency of these two narrow incident signals are (100^∘, 60^∘, 1?GHz) and (95^∘, 50^∘, 2?GHz), respectively, which are independent of each other. The number of elements are 10, including 4 guiding elements and other 6 instrumental elements. The element pattern used in simulation is the lowest order circular patch model [8, 22] 𝑔_𝜃𝜃,𝜑=𝐽₂𝜋𝑑𝜆sin𝜃−𝐽₀𝜋𝑑𝜆×𝜋sin𝜃cos𝜑−𝑗sin𝜑,0≤𝜃≤2,𝑔_𝜑𝜃,𝜑=𝐽₂𝜋𝑑𝜆sin𝜃+𝐽₀𝜋𝑑𝜆𝜋sin𝜃×cos𝜃sin𝜑−𝑗cos𝜑,0≤𝜃≤2,𝑔_𝜃𝜃,𝜑=𝑔_𝜑𝜃,𝜑=0,otherwise, (69) where 𝐽₀ and 𝐽₂ are the zeroth- and second-order Bessel functions of the first kind. The element pattern transformation is achieved with the method proposed above.

(a)

(b)

(c)

(d)

Figure 3 

The RMSE and successful probability with different SNR. (a) The successful probability of frequencies, (b) the RMSE of frequency, (c) the successful probability of angles, and (d) the RMSE of angles.

Figure 3 shows that the RMSE of frequency and angle decreases while the SNR increases; meanwhile, the success probability of frequency and angle increase as the SNR increases. The estimated performance of the two frequencies are very close which can be seen in Figures 3(a) and 3(b). As shown in Figure 3(a), the successful probability is larger than 80%, when the SNR is larger than 15?dB. The RMSE of frequency is less than 2?MHz, when SNR is larger than 10?dB as shown in Figure 3(b). In Figure 3(c) the successful probability of azimuths is slightly lower than that of the elevations. The success probability of the angle is almost 100% when the SNR reaches 15?dB. Figure 3(d) shows that the RMSE is smaller than 0.16^∘ when SNR is larger than 5?dB. The RMSE of azimuth is larger than that of elevation. The estimation results of frequency and elevation are applied to estimate azimuth. The phenomenon in Figure 3(d) can be interpreted as the existence of the estimated error of frequency and elevation. Next, the RMSE of frequency and angle as well as the success probability of angle in different snapshot number are depicted in Figure 4. SNR = 10?dB; other simulation conditions are identical with the previous experiment. In Figure 4, the RMSE of frequency and angle decreases as the snapshot number increases, and the success probability of frequency and angle rises as the snapshot number increases. As shown in Figure 4(a), the successful probability is larger than 80%, when the number of snapshot is larger than 500. Figure 4(b) shows that the RMSE of frequency 1 is slightly larger than that of frequency 2. When the snapshot number is larger than 500, the RMSE drops below 1?MHz, meaning that the proposed algorithm achieved three orders of magnitude reduction. Figures 4(c) and 4(d) show that the success probability of azimuth is lower than that of the elevation, and the RMSE of azimuth are larger than that of elevation. The reason why this happens is because the estimation error of frequency and elevation affects the estimation performance of azimuth.

(a)

(b)

(c)

(d)

Figure 4 

The RMSE and successful probability with different snapshot number. (a) The successful probability of frequencies, (b) the RMSE of frequency, (c) the successful probability of angles, and (d) the RMSE of angles.

5. Conclusion

Due to the varying curvature of the surface of the conformal carrier, the pattern of each element of the conformal array is different. Thus, the conventional algorithms could not be used for conformal array. In this paper, a novel joint frequency and 2D-DOA estimation algorithm with high accuracy are proposed. Both spatial and time sampling are utilized to construct the spatial-time matrix. The delay correlation function is used to suppress noise. The PARAFAC model is used for parameter estimation without parameter pairing. Only four elements are needed, and the positions of these elements should be known accurately. Other instrumental elements can be flexibly arranged on the surface of the conformal carrier. The algorithm proposed in this paper can be extended to estimate 2D-DOA straightly with little modification. The simulation results verify the effectiveness of the proposed algorithm. It can be expected that the proposed algorithm would have an application prospect in the parameter estimation of conformal array. In the future work, we will focus on the application of the proposed algorithm [23–26].

Acknowledgments

This work was supported in part by the National Science Foundation of China under Grant 61201410 and in part by Fundamental Research Focused on Special Fund Project of the Central Universities (Program no. HEUCF130804).