One-bit data with non-zero quantization thresholds suffice to recover the covariance of the original Gaussian data. State-of-the-art recovery methods require computing the one-bit correlation from the estimated covariance. However, this step exhibits trade-offs between the computational complexity and approximation error. This paper proposes the harmonic approximation of the one-bit Hermite law for the one-bit correlation. The proposed approximation features closed forms with finite terms and low computational complexity. Furthermore, its approximation errors are analyzed through the approximation error bounds (AEBs). The AEBs also lead to theoretical insights into the signal statistics, quantization thresholds, and the parameters in the proposed approximation. Numerical examples demonstrate the applicability of the proposed approximation.

Sparse arrays with fourth-order cumulant processing can identify up to $\mathcal{O}\left( {{N^4}} \right)$ source directions with only N physical sensors. To achieve this property, the fourth-order difference co-array is preferable to own a contiguous segment with no holes. However, existing sparse array designs either have holes in the co-array of size $\mathcal{O}\left( {{N^4}} \right)$ or own fewer than $\mathcal{O}\left( {{N^4}} \right)$ elements in the hole-free co-array. This paper proposes the half inverted nested array (HINA), which consists of a nested array and an inverted, scaled, and shifted nested array. By maximizing the size of the co-array, it can be shown that HINA with the optimal parameters possesses a hole-free fourth-order difference co-array of size $\mathcal{O}\left( {{N^4}} \right)$. Numerical examples demonstrate the improved DOA estimation performance of HINA.

Linear sparse arrays, where the sensors are placed with non-uniform spacing, identify more source direction-of-arrivals (DOAs) than sensors. In contrast, uniform linear arrays (ULAs) can only find fewer source DOAs than sensors. To resolve these DOAs in practice, the sources have to be enumerated first. However, existing source enumerators are either designed for ULAs or computationally challenging for sparse arrays. This paper proposes sparse array source enumeration via coarray subspace optimization (SASE-CSO). The SASE-CSO algorithm first establishes multiple batches of array outputs. In each batch, we estimate the projection matrix onto the signal subspace on the difference coarray. Next, we propose the coarray subspace optimization (CSO) to combine these estimated projection matrices. The explicit relation between the optimal solution to the CSO and the estimated projection matrices reduces the complexity for implementation. Finally, the sources are enumerated from the most significant gap in the eigenvalues of the optimal solution to the CSO. The SASE-CSO can enumerate up to U sources, where U denotes the largest element in the central ULA segment in the difference coarray. Numerical examples demonstrate that the SASE-CSO increases the probability of detection for more sources than sensors and two closely-spaced sources with unequal powers.

Sparse arrays can resolve more source direction-of-arrivals (DOAs) than sensors in array processing. This property is achieved with DOA estimators based on the difference coarray. However, sensor failures on sparse arrays could make these DOA estimators inapplicable. In the literature, array processing under sensor failures can be coped with array diagnosis. The failed sensors are detected first in array diagnosis, and the data on these detected failed sensors are removed next. However, errors in detecting the failure patterns degrade the performance. This paper presents coarray combinations for sparse arrays under sensor failures. In converting array data from the physical array to the difference coarray, coarray combination matrices (CCMs) aim to preserve the structures of the difference coarray under sensor failures. This condition corresponds to a system of equations with the CCMs. With prior knowledge about the failure patterns, a convex optimization problem is cast for this system of equations. The optimal CCM is pre-computed with closed-form expressions, avoids array diagnosis, and applies to coarray-based DOA estimators. Furthermore, numerical examples demonstrate that the proposed method reduces the errors in DOA estimation in comparison with other methods where the data on the detected failed sensors are removed.

One-bit quantization has received attention due to its simplicity, low cost, and ability to recover the autocorrelation of the unquantized data. In the past, the autocorrelation estimation from one-bit data can be done in two separate stages. One is the power estimation by using one-bit quantizers of non-zero thresholds, while the other focuses on estimating the normalized autocorrelation with one-bit quantizers of a zero threshold. However, the overall hardware cost increases in this approach. This paper presents an autocorrelation estimator based on a one-bit quantizer with a non-zero threshold. The proposed method depends purely on a one-bit quantizer and its output data. Our method first infers the power information and then estimates the normalized autocorrelation by polynomial root-finding. The autocorrelation estimate is obtained by combining the power and the normalized autocorrelation. Numerical simulations show that the proposed method exhibits similar behavior to an estimator based on the unquantized data, with a 5dB loss in the estimation error.

In array processing, the fourth-order difference co-array of sparse arrays with N sensors admits to resolve$\mathcal{O}(N^4)$source directions with proper assumptions. This$\mathcal{O}(N^4)$property is relevant to the size of the central uniform linear array (ULA) segment of the fourth-order difference co-array. However, among the existing sparse arrays, the fundamental limits for the sizes of the fourth-order difference co-array remain a topic for further study. This paper characterizes the lower and upper bounds of the size of the fourth-order difference co-array. It is proved that the ULA and the exponential array achieve the lower and upper bounds, respectively. We also propose the fourth-order redundancy to quantify the efficiency of the ULA segment in the fourth-order difference co-array. The fourth-order redundancy owns a provable lower bound depending only on N, providing further insights into the size of the central ULA segment of the fourth-order difference co-array. These bounds are validated through numerical examples.

One-bit quantization has attracted attention in massive MIMO, radar, and array processing, due to its simplicity, low cost, and capability of parameter estimation. Specifically, the shape of the covariance of the unquantized data can be estimated from the arcsine law and onebit data, if the unquantized data is Gaussian. However, in practice, the Gaussian assumption is not satisfied due to outliers. It is known from the literature that outliers can be modeled by complex elliptically symmetric (CES) distributions with heavy tails. This paper shows that the arcsine law remains applicable to CES distributions. Therefore, the normalized scatter matrix of the unquantized data can be readily estimated from one-bit samples derived from CES distributions. The proposed estimator is not only computationally fast but also robust to CES distributions with heavy tails. These attributes will be demonstrated through numerical examples, in terms of computational time and the estimation error. An application in DOA estimation with MUSIC spectrum is also presented.

In array processing, minimum redundancy arrays (MRA) can identify up to O(N2) uncorrelated sources (the O(N2) property) with N physical sensors, but this property is susceptible to sensor failures. On the other hand, uniform linear arrays (ULA) are robust, but they resolve only O(N) sources. Recently, the robust MRA (RMRA) was shown to possess the O(N2) property and to be as robust as ULA. But finding RMRA is computationally difficult for large N. This paper proposes a novel array geometry called the composite Singer array, which is related to a classic paper by Singer in 1938, and to other results in number theory. For large N, composite Singer arrays could own the O(N2) property and are as robust as ULA. Furthermore, the sensor locations for the composite Singer array can be readily computed by the proposed recursive procedure. These properties will also be demonstrated by using numerical examples.

Sparse arrays can identify O(N2) uncorrelated sources using N physical sensors. This property is because the difference coarray, defined as the differences between sensor locations, has uniform linear array (ULA) segments of length O(N2). It is empirically known that, for sparse arrays like minimum redundancy arrays, nested arrays, and coprime arrays, this O(N2) segment is susceptible to sensor failure, which is an important issue in practical systems. This paper presents the (k-)essentialness property, which characterizes the combinations of the failing sensors that shrink the difference coarray. Based on this, the notion of fragility is proposed to quantify the reliability of sparse arrays with faulty sensors, along with comprehensive studies of their properties. It is demonstrated through examples that there do exist sparse arrays that are as robust as ULA and at the same time, they enjoy O(N2) consecutive elements in the difference coarray.

One-bit quantization has become an important topic in massive MIMO systems, as it offers low cost and low complexity in the implementation. Techniques to achieve high performance in spite of the coarse quantizers have recently been advanced. In the context of array processing and direction-of-arrival (DOA) estimation also, one bit quantizers have been studied in the past, although not as extensively. This paper shows that sparse arrays such as nested and coprime arrays are more robust to the deleterious effects of one-bit quantization, compared to uniform linear arrays (ULAs); in fact, sparse arrays with one-bit quantizers are often found to be as good as ULAs with unquantized data. Nested and coprime arrays without quanitzers are known to be able to resolve more DOAs than the number of sensors, when sources are uncorrelated. It will be demonstrated that this continues to be true even with one-bit quantization.

In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA). Sparse arrays, such as nested arrays, coprime arrays, and minimum redundancy arrays (MRAs), have reduced mutual coupling compared to uniform linear arrays (ULAs). With N denoting the number of sensors, these sparse arrays offer O(N2) freedoms for source estimation because their difference coarrays have O(N2)-long ULA segments. These arrays have different shortcomings: coprime arrays have holes in the coarray, MRAs have no closed-form expressions, and nested arrays have relatively large mutual coupling. This paper introduces a new array called the super nested array, which has all the good properties of the nested array, and at the same time reduces mutual coupling significantly. For fixed N, the super nested array has the same physical aperture, and the same hole-free coarray as does the nested array. But the number of sensor pairs with separation λ/2 is significantly reduced. Many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays.

Coprime arrays, consisting of two uniform linear arrays whose inter-element separations are coprime, can resolve O(MN) sources using only O(M + N) sensors. However, holes in the coarray prevent us from using the full coarray in the MUSIC algorithm for DOA estimation. Through interpolation, it may be possible to use the remaining elements of the coarray to increase the degrees of freedom beyond what is captured in the contiguous ULA section in the coarray. Techniques like positive definite Toeplitz completion, array interpolation, and sparse recovery, manage to include all the information in the coarray, but they demand extra fine-tuned parameters and have individual drawbacks. In this paper, a simple and tractable convex framework via nuclear norm minimization is presented. This approach has no extra tuning parameters and overcomes several undesired issues of other techniques. Numerical examples indicate that, in many instances, the proposed method not only increases the estimation accuracy but also distinguishes more sources than other methods1.

This paper extends the use of coprime arrays and samplers for the case of moving sources. Space-time adaptive processing (STAP) plays an important role in estimating direction-of-arrivals (DOAs) and radial velocities of emitting sources. However, the detection performance is fundamentally limited by the array geometry and the temporal samplers at each sensor. Coprime arrays and coprime samplers offer an enhanced degree of freedom of O(MN) using only O(M + N) physical sensors or samples. In this paper, we propose coprime joint angle-Doppler estimation (coprime JADE), which incorporates both coprime arrays and coprime samplers with the STAP framework. Nonuniform time samples at different sensors can be used to generate a sampled autocorrelation matrix, from which we compute a spatial smoothed matrix. It will be proved that spatial smoothed matrices can be used in the MUSIC algorithm for parameter estimation. With sufficient snapshots, coprime JADE distinguishes O(M1N1M22) independent sources if it corresponds to coprime arrays and coprime samplers with coprime integers (M1;1) and (M2;N2), respectively. It is verified through simulations that coprime JADE resolves the angle-Doppler information better compared to other conventional algorithms.

Coprime DFT filter banks (coprime DFTFB) achieve the effect of an MN-DFTFB by using two DFTFBs of size only M and N, where M and N are coprime integers. However, coprime DFTFBs need to be designed properly, to avoid unwanted bumps in stopbands or unsatisfactory total spectrum coverage, quantified by overall amplitude responses. In this paper, a detailed theoretical analysis will be made on the tradeoffs between bumps and overall amplitude responses. It will be shown that the bump level at the center frequency fbof a bump, is approximately one-fourth of the overall amplitude response at fb. Then, a novel design will be introduced based on an optimization problem pertaining to overall amplitude responses. The original problem is relaxed to a computationally tractable optimization program, which can be solved with alternating minimization algorithms. It is verified with simulations that the new designs cover the spectrum completely.

Sparse arrays such as nested and coprime arrays use a technique called spatial smoothing in order to successfully perform MUSIC in the difference-coarray domain. In this paper it is shown that the spatial smoothing step is not necessary in the sense that the effect achieved by that step can be obtained more directly. In particular, with R̃ssdenoting the spatial smoothed matrix with finite snapshots, it is shown here that the noise eigenspace of this matrix can be directly obtained from another matrix R̃ which is much easier to compute from data.

This paper presents an approach to 3D rotation estimation using discrete spherical harmonic oscillator transforms (discrete SHOTs). Discrete SHOTs not only have simple and fast implementation methods but also are compatible with the existing angle estimation algorithms related to spherical harmonics. Discrete SHOTs of the rotated signal follow the same formulation to the Wigner-D matrix as spherical harmonics transforms. Thus, the spherical harmonics related algorithms could be utilized to discrete SHOTs without modification. Furthermore, compared to some existing methods, our approach with discrete SHOTs exhibits higher accuracy, higher precision and improved robustness to noise if the input signal is sampled uniformly on Cartesian grids. The phenomenon results from no interpolations in discrete SHOTs.

In this paper, the general form of the two-dimensional Fourier transform (2D FT) eigenfunctions is discussed. It is obtained from the linear combination of the 2D separable Hermite Gaussian functions (SHGFs). For example, the rotated Hermite Gaussian functions (RHGFs) for the rotated coordinate and the Laguerre Gaussian functions (LGFs) for the polar coordinate are two special cases of the general form. With the aid of the general form, we can achieve these 2D functions with perfect orthogonality. Finding the combination coefficients is equivalent to the multinomial expansion problem. Therefore, we can apply the fast Fourier transform and some recurrence relations to the coefficients. The computation cost is much less than the close-form coefficients, which is associated with the Jacobi polynomials.

A new transform, called the generalized fractional Fourier transform (gFrFT), is proposed. Originally, the eigenfunctions of the fractional Fourier transform (FrFT) are known as the Hermite Gaussian functions (HGFs). Besides, in optics, the HGFs are generalized to be the generalized Hermite Gaussian functions (gHGFs) and their adjoint functions (AgHGFs). Therefore, we can define the gFrFT by the eigenvalues of the FrFT and the eigenfunctions (gHGFs/AgHGFs) in the analysis or synthesis step. Four types of the gFrFT are defined and discussed. The integral forms of the gFrFTs are derived and they are closely related to some popular transforms, such as the Fourier transform (FT), the FrFT, and the complex linear canonical transform (CLCT). We can also extend the FT and the FrFT to the standard and elegant versions. Finally, some properties of the gFrFT are discussed.