Chun-Lin Liu

Scatter matrix or covariance matrix estimation under data with outliers is crucial to signal processing applications. These data can be modeled as Cauchy or elliptically symmetric (ES) distributions, under which many estimators have been developed. However, these estimators primarily assume continuous-valued data. For implementation purposes, these data are quantized into discrete levels, which can be represented by finite bits. The complexity of the estimators is also restricted. These hardware constraints degrade the performance of existing scatter matrix estimators. This paper proposes the Cauchy onebit scatter matrix estimator (COSME). The COSME is founded on the proposed integral expressions of the marginal and joint probability for one-bit centered bivariate ES (CB-ES) data. These probabilities are simplified in closed forms for one-bit centered bivariate Cauchy (CB-Cauchy) data. For equal normalized quantization thresholds, these forms admit the COSME with lowcomplexity steps. In particular, these steps involve counting the occurrences of the events in the one-bit CB-Cauchy data and evaluating trigonometric functions. The minimum number of snapshots and the multivariate extension of the COSME are also investigated. Numerical examples demonstrate that the COSME exhibits shorter CPU time under the same number of snapshots and improved estimation performance under the same number of bits, compared to several existing estimators. Empirical results show the robustness of the COSME to Student’s t distributions and parameter variations.

Linear sparse arrays with fourth-order cumulant processing can resolve $\mathcal{O}(N^4)$ directions-of-arrival (DOAs) using $N$ physical sensors, provided that the fourth-order difference co-array $\Delta_4$ contains a contiguous segment of size $\mathcal{O}(N^4)$ . Furthermore, if $\Delta_4$ has no holes, then the received data can be fully exploited in subspace-based DOA estimators. However, few existing arrays attain large hole-free $\Delta_4$ . Many existing arrays designed for $\Delta_4$ are constructed from two smaller arrays, called the basis arrays. Nevertheless, such arrays either restrict the basis arrays to certain types or have no guarantee of hole-free $\Delta_4$ . This paper proposes the half-inverted (HI) arrays, parameterized by two basis arrays $\mathbb{S}^{(1)}$ and $\mathbb{S}^{(2)}$ , the shifting parameter $M$ , and the scaling parameter $\sigma$ . An HI array consists of $\mathbb{S}^{(1)}$ and an inverted, scaled, and shifted version of $\mathbb{S}^{(2)}$ . HI arrays are guaranteed with hole-free $\Delta_4$ over a range of $(M,\sigma)$ pairs. This property unifies several existing arrays with hole-free $\Delta_4$ and admits an optimization problem over $(M,\sigma)$ . The half-inverted general hole-free (HIGH) scheme is defined as the HI array with a closed-form and optimized $(M,\sigma)$ pair determined by the second-order co-arrays of the basis arrays. The HIGH scheme enjoys a large hole-free $\Delta_4$ . The shift-scale representation (SSR) is presented to study $\Delta_4$ of HI arrays visually. From these results, the half-inverted array based on second-order optimization and extended shift (HI-SOES) is proposed. For a fixed $N$ , HI-SOES synthesizes a hole-free $\Delta_4$ larger than an existing array. Numerical examples demonstrate the DOA estimation performance of HI arrays and existing arrays.

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 letter 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.

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.

Linear time-invariant (LTI) systems find ubiquitous applications in digital signal processing [1], image processing [2], communication [3], and array processing [4], to name a few. An LTI system can be characterized by its impulse response, which is defined as the output response to an impulse input. For an arbitrary input signal, the output signal of an LTI system is obtained by the convolution of the input signal with the impulse response. This property is closely related to transform-based methods in signal processing, such as discrete-time Fourier transforms, z-transforms, and discrete Fourier transforms. Interested readers are referred to [1] and the references therein.

Sparse arrays, where the sensors are properly placed with nonuniform spacing, are able to resolve more uncorrelated sources than sensors. This ability arises from the property that the difference coarray, defined as the differences between sensor locations, has many more consecutive integers (hole-free) than the number of sensors. In some implementations, it might be preferable that a) the arrays be symmetric, b) that the arrays be maximally economic, that is, each sensor be essential, and c) that the coarray be hole-free. The essentialness property of a sensor means that if it is deleted, then the difference coarray changes. Existing sparse arrays, such as minimum redundancy arrays (MRA), nested arrays, and coprime arrays do not satisfy these three criteria simultaneously. It will be shown in this paper that Cantor arrays meet all the desired properties mentioned above, based on a comprehensive study on the structure of the difference coarray. Even though Cantor arrays were previously proposed in fractal array design, their coarray properties have not been studied earlier. It will also be shown that the Cantor array has a hole-free difference coarray of size N ^{log2 3} ≈ N ¹⁵⁸⁵ where N is the number of sensors. This is unlike the sizes of difference coarrays of the MRA, nested array, coprime array (all O(N ² )), and uniform linear arrays (O(N)) ¹ .

The concept of correlation subspaces was recently introduced in array processing literature by Rahmani and Atia. Given a sensor array, its geometry determines the correlation subspace completely, and the covariance matrix of the array output is constrained in a certain way by the correlation subspace. It has been shown by Rahmani and Atia that this knowledge about the covariance constraint can be exploited to improve the performance of DOA estimators. In this paper, it is shown that there is a simple closed form expression for the basis vectors of the correlation subspace. Thus, computation of this subspace is greatly simplified. Another fundamental observation is that, this expression is closely related to the difference coarray. Thirdly, the paper also shows an interesting logical connection between correlation subspaces, redundancy averaging, and rectification, which are popularly used in DOA estimation1.