New sparse array for non‐circular sources with increased degrees of freedom

Recently, sparse arrays have received considerable attention as they provide larger array aperture and increased degrees-of-freedom (DOFs) compared to uniform linear arrays. These features are essential to enhance the direction-of-arrival estimation performance. However, most of the existing sparse arrays are mainly designed for circular sources and realize limited increment in DOFs for non-circular sources. In this letter, a new sparse array configuration for non-circular sources is presented, which significantly increases the achievable DOFs and improves the direction-of-arrival estimation performance. The proposed geometry comprises two effectively configured uniform linear arrays that exploit the characteristics of non-circular sources and extend the array aperture. For a given number of sensors, its virtual array is advantageously a hole-free uniform linear array. Moreover, the precise sensor locations, achievable DOFs, and optimal distribution of physical sensors are determined analytically by closed-form expressions. Owing to these benefits, the proposed array efficiently resolve multiple sources in under-determined conditions and achieves better direction-of-arrival estimation performance than its counterpart structures. Simulation results validate the superiority of the proposed configuration.

New sparse array for non-circular sources virtual aperture. Recently, a novel sparse array for non-circular sources (SANC) [11] is proposed that achieves more DOFs and an increased number of continuous virtual locations in the resultant co-array. Nevertheless, it involves an exhaustive search algorithm to determine the locations of physical sensors and lacks closed-form expressions for the antenna positions and array DOFs. Hence, it cannot be easily designed for an arbitrary number of elements. Besides, it has holes in the resultant co-array, which restricts the DOA estimation accuracy. To address these challenges, we proposed a new sparse array for NCS (NSANCS), which embeds all the useful properties. In particular, it attains more number of uniform DOFs (uDOFs) as well as benefited with closed-form expressions for the precise sensor locations and achievable DOFs. Advantageously, the extended virtual array of the proposed configuration is always kept as a hole-free ULA. Thanks to the closed-form expressions in nested array [7], MISC array [9], and improved nested array [10], the letter also formulates analytical expressions to determine uDOFs in these configurations for NCS based on extended array aperture.

NSANCS:
The proposed sparse array configuration consists of two ULAs effectively concatenated to achieve a larger hole-free resulting coarray for NCS, as depicted in Figure 1. The total number of elements M is M 1 + M 2 . The first ULA has M 1 sensors with an inter-element spacing of M 2 d, which is then followed by another ULA of M 2 elements holding an inter-element spacing of d, where d is the unit inter-sensor spacing and is typically fixed to the half-wavelength (λ/2). The inter-ULA separation is equal to M 2 d. Correspondingly, the set S NSANCS representing locations of physical sensors in the NSANCS array, can be expressed as follows: Table 1 illustrates the optimal distribution of sensors in NSANCS that achieves maximum uDOFs, under the constraint of a given number of elements, that is, M = M 1 + M 2 . Arithmetic mean-geometric mean inequalities can obtain the optimal solutions summarized in Table 1.
Signal model: Consider Z narrowband stationary sources incident on M-element array from different directions {θ 1 , θ 2 , . . . , θ Z } with the corresponding source powers, p = [σ 1 2 , σ 2 2 , . . . , σ Z 2 ] T , where (.) T is the transpose. The received signal x(t ) at time t can be expressed as where s(t ) = [s 1 (t ), s 2 (t ), . . . , s Z (t )] T . The components of noise vector n(t ) are the additive white gaussian noise, with zero value of mean and covariance of σ n 2 I, where I represents the identity matrix. A = [a(θ 1 ), a(θ 2 ), . . . , a(θ Z )] is the M × Z steering matrix with a(θ z ) is steering vector of z-th source, (z = 1, 2, . . . , Z), which is given by where l m (m = 0, 1, . . . , M − 1) is the distance between m-th and first sensor in the antenna array configuration, l m ∈ S, where S represents the corresponding set of physical locations. For the proposed array, l m ∈ S NSANCS . The covariance matrix corresponding to x(t ) is given by where E[.] is the statistical expectation, (.) H shows conjugate transpose.
In practice, the covariance matrix is averaged over T snapshotŝ Now E[x p (t )x q * (t )] with (.) * symbolizing the conjugation indicates the p-th row and q-th column of R x , which is given by where {l p , l q } ∈ S. The difference set can be defined as The difference set L forms the basis of a virtual array, and the elements in L signifies the relative location of the virtual array. It is wellknown that the EC of CS is zero, and a non-zero value of EC implies that the sources are non-circular, that is, ASK, PAM, BPSK, AM. In contrast to CS, NCS carries more valuable information that can be exploited to extend virtual array and improve the DOA estimation performance.
Since the EC of NCS is not zero By utilizing the non-circular characteristics of sources, the array aperture can be enlarged. Therefore The equivalent manifold matrix is represented by Hence, Correspondingly, the extended covariance matrix is given by Following this, BB H extends the array aperture; according to (8), the virtual array manifold of an extended array for NCS can be obtained by In general, to realize enhanced uDOFs, the number of continuous virtual locations in the resultant co-array reaches to the maximum. Therefore, the proposed configuration aims to achieve a higher number of continuous virtual sensors as well as produce a hole-free resulting ULA.

Degrees of freedom:
One of the key benefits of the proposed array is its convenient construction due to the availability of closed-form expressions to obtain not only the exact sensor locations but also uDOFs. We also formulate the closed-form expressions based on an extended virtual array defined in (17) for determining the corresponding uDOFs in nested, improved nested, and MISC arrays. The generalized expression of uDOFs in NSANCS, nested and improved nested arrays for NCS is given by: When M is even, uDOFs = where c assume the values 1, 2, and 2 + M 1 to provide the corresponding uDOFs as given below: ⎧ ⎨ ⎩ When c = 1, uDOFs for nested array. When c = 2, uDOFs for improved nested array. When c = 2 + M 1 , uDOFs for NSANCS array.
According to (17), the uDOFs of MISC array [9] for NCS with an extended array aperture can be obtained by where . denotes the integral part of the rational number in the square brackets and % symbolizes the remainder. Figure 2 shows the geometric distribution of physical and virtual sensors of typical array structures based on the extended array aperture, where M = 5. It is observed from Figure 2 that the proposed NSANCS configuration achieves a higher number of continuous virtual positions. It is also evident that only the NSANCS array realizes hole-free virtual ULA among all the arrays illustrated in Figure 2. Although SANC also achieves the same number of continuous virtual sensors, its resulting coarray is filled with holes. Besides, the antenna positions and uDOFs in SANC cannot be expressed in closed-form. Furthermore, Table 2 lists the uDOFs of typical array structures for NCS and their corresponding sensor locations (normalized by d) when the physical array contains five elements. It is found that the NSANCS configuration outperforms nested array [7], improved nested array [10], MISC array [9], and MRA [5] by achieving more number of uDOFs. Moreover, SANC obtains the same number of uDOFs as the proposed array, but at the expense of computational burden.
Numerical results: This section conducts experimental works to examine the superior performance of NSANCS.
In the first simulation, we demonstrate the ability of NSANCS to accurately resolve several sources in under-determined scenarios, that is, Z > M, using the spatial smoothing multiple signal classification (SS-MUSIC) algorithm [7]. Consider Z = 9 sources with an equal power incident on the array configurations given in Table 2. Since improved nested array and MISC array have the same antenna locations and achievable DOFs for M = 5, we considered MISC array for comparison. The nine sources are uniformly distributed in [−36 • , 36 • ], as indicated by the vertical grid-line. Figure 3 lists all the DOAs estimated by typical sparse arrays. It is observed from Figure 3 that the proposed NSANCS array can clearly and accurately identify all the incident sources compared to other array configurations. It can be noticed that the SANC array also resolves the incident sources correctly. However, its resulting virtual array has holes, and it is well-known that the holes cause ambiguity in parameter estimation when a large number of sources are taken into account.
In the second part, the DOA estimation is performed for the abovediscussed array configurations. The root mean square error (RMSE) is employed as performance metric, which can be defined as an average over 600 independent trials: whereθ n,z symbolizes the estimated DOA of z-th source, (z = 1, 2, . . . , Z), in n-th Monte-Carlo trial, n = 1, 2, . . . , 600. Firstly, the DOA estimation is examined in terms of signal-to-noise ratio (SNR), as depicted in Figure 4. The SNR is assumed over the range  Figure 4 that the proposed array exhibits better DOA estimation performance over an increasing range of SNR as compared to other array geometries.
Likewise, we performed the DOA estimation with reference to the number to snapshots, as shown in Figure 5. We keep the same parameters as in the previous case, except the range of snapshots is considered over a fixed SNR at 0 dB. It is observed from Figure 5 that the DOA estimation accuracy of the proposed array configuration is higher than the other arrays, and owns a smaller RMSE value when the number of snapshots is greater than or equal to 150.

Conclusion:
We proposed a new array configuration for NCS that achieves more uDOFs and results in a hole-free virtual ULA. Via utilizing the characteristics of NCS and exploiting extended array aperture, the NSANCS reveals significant improvement in the DOA estimation performance compared to its sparse array counterparts. Unlike MRA and SANC, the NSANCS array is benefited from closed-form expressions to determine the exact antenna positions and achievable DOFs. The letter also presents analytical expressions to obtain uDOFs for the nested array, improved nested array, and MISC array based on the extended virtual array. Furthermore, the optimal distribution of physical sensors is presented for NSANCS array to maximize its uDOFs capacity. Numerical results validate the effectiveness of the NSANCS configuration.