# An Effective Off-Grid DOA Estimation Algorithm Using Difference Coarrays with Limited Snapshots
## 1 Introduction  
Direction-of-arrival (DOA) estimation has emerged as a critical signal processing technique with extensive applications across multiple domains, such as wireless communication, radar target detections, and satellite navigation, [1, 2, 3, 4, 5, 6]. Conventional subspace-based approaches, like multiple signal classification (MUSIC) and estimation of the signal parameters via rotational invariance techniques (ESPRIT), perform DOA estimation by applying eigendecomposition to the array covariance matrix, leveraging the orthogonality between the signal and noise subspaces [7, 8]. Recently, phase interferometry-based digital synchronous architectures and adaptive Direct Digital Synthesizer Phase-Locked Loop beam steering techniques have been proposed for DOA estimation [9, 10]. Although these methods can accuratly estimate the DOAs of sources with low computational complexity, they are limited to resolving fewer sources than the number of sensors.Alternately, various methods have been developed to estimate the DOAs of more sources than the number of sensors by utilizing non-uniform linear arrays (NULAs) [11]. The difference coarrayprovides expanded degrees-of-freedom (DOF), enabling an \(N\) element array to resolve more than \(N\) sources [12; 13]. One representative coarray-based approach is the SS-MUSIC algorithm, which employs an augmented sample covariance matrix and requires a large number of snapshots [14; 15]. The SS-MUSIC algorithm relies on an on-grid assumption, where true DOAs are presumed to lie precisely on predefined grid points. This basis mismatch inherently limits estimation accuracy in practical off-grid scenarios. An algorithm combining OptSpace and coarray ESPRIT was developed to perform DOA estimation effectively [16]. The OptSpace algorithm has been utilized to fill the midding elements in the coarray domain, effectively mapping the steering matrix of NLAs onto that of ULAs. This enables the application of ULA-based methods, like ESPRIT, within the coarray domain to achieve accurate DOA estimation. A source localization technique called coarray MUSIC-Group delay was developed in [17], utilizing the high-resolution properties of the group delay in the phase spectrum of the coarray MUSIC algorithm.A computationally efficient adaptive method was developed in [18], utilizing the least mean square (LMS) principle to perform coarray-based DOA estimation. Contemporary NULA off-grid estimators demonstrate superior source resolvability, with techniques like search-based methods and Enhanced Principal-Singular-Vector Utilization for Modal Analysis (EPUMA) successfully identifying more sources than the the number of available array elements [19; 20]. However, when multiple target within the field of view change rapidly, the number of data snapshot that can be collected becomes limited, intensifying the problem of limited snapshots. Under these conditions, the above-mentioned methods perform poorly due to the significant corruption of the signal and noise subspaces. Coarray-based methodology suffers from significant computational burdens, representing a major practical constraint [21; 22; 23; 24; 25; 26; 27; 28; 29]. Recently, numerous off-grid estimation techniques have been proposed, including sparse recovery, sparse Bayesian learning (SBL), and maximum-likelihood-estimation (MLE)-based methods [21; 22; 23]. Our prior research introduced an advanced coarray processing approach that incorporates Bayesian inference for enhanced off-grid DOA estimation [24]. These methods can overcome the off-grid problem. However, the use of dense sampling grids for DOA search in these approaches imposes a significant computational burden. A Nuclear-Norm-Minimization-based MUSIC (NNM-MUSIC) algorithm was developed in [25] for DOA estimation using a coprime electromagnetic vector sensor (EMVS) array, leveraging coarray interpolation techniques.