A new approach to the analytical implementation of the Rosenblatt transformation is described. It leverages the properties of the empirical probability density function, which is the standard estimate of an unknown density. As such its utility is to applications where training data is available for the unknown density. These applications include data-driven algorithms for detection/classification and other statistical signal processing problems where the underlying probabilistic description of the data is unknown. As an illustration, an application to anomaly detection is described in detail using Gaussian and radar datasets.

A classification rule based on the cumulant generating function of the training data, called the Cumulant Generating Function Classifier (CGFC), has been recently proposed, and has shown promising performance in terms of improved classification accuracy and robustness against noises. This paper first presents a new information-theoretical explanation of CGFC which indeed makes a classification by minimizing sample mutual information. The original CGFC is a type of global model, and a new variant, called Local-CGFC, is further introduced in this paper to achieve a local classification rule. Experimental studies on real-life datasets demonstrate the effectiveness of the proposed classifier and further illustrate its great potential for a number of real-world applications.

Detection of known signals of interest that are embedded in colored noise involves whitening the received samples and matched-filtering. In many applications, due to computational constraints, it is critical to select only a subset of the received samples for detection. This paper addresses the problem of selecting only a given number of temporal or spatial samples while maximizing detection performance for deterministic signals in first-order autoregressive Gaussian noise. The direct solution of this entails a combinatorial search, where the deflection coefficient is evaluated for each possible combination of sparse samples. This approach is infeasible when the number of samples is large since the number of possible combinations increases factorially with the number of samples. We present an efficient method to whiten Gaussian noise samples and express deflection coefficient in a form that is amenable to dynamic programming. Exploiting dynamic programming, we specify a feasible and efficient procedure to find optimal sparse samples where the number of computational steps increases linearly with the number of samples. Also, conditions under which uniform sampling is optimal is given.

Many signal processing algorithms utilize singular vectors of the data received by a sensor array or eigenvectors of the sample covariance matrix. The performance of such subspace-based algorithms can be substantially improved by using the optimal signal or noise subspace estimate instead of using the subspaces spanned by the eigenvectors or the singular vectors. The optimal signal and noise subspaces are estimated by exploiting the shift-invariant structure of the sensor array. We develop a methodology to find the optimal subspaces in sparse arrays that possess shift-invariant structure and interpolate the optimal subspaces to match the subspaces of the fully populated arrays with an equal aperture. These shift-invariant sparse arrays’ direction of arrival estimation performance is superior compared to the methods that use eigenvectors or singular vectors directly. Our method also encompasses any symmetric array that is not shift-invariant, thus broadening the class of sparse arrays where our method can be applied.

We address the problem of sparse sampling pattern design to maximize detection for a deterministic signal in colored noise. We model a colored noise as a continuous-time autoregressive process, which is obtained by passing a white noise through a causal linear-time invariant filter. This noise modeling is crucial to the development of the optimal sampling pattern design for a given number of sensors. We obtain a closed form expression for the whitening filter and consequently, for the Kullback-Leibler divergence at the whitening filter output, which is the detection index. The optimum sampling pattern is obtained by evaluating the detection index at Nyquist sampling rate, rank ordering the samples, and selecting the maximum values. We present some examples to illustrate the proposed procedure. We also extend our method to two-dimensional sampling. The advantage of our approach is its low computational complexity for both one-dimensional and two-dimensional cases and that optimality is guaranteed.

Coprime sensor arrays (CSAs) achieve the resolution of a fully populated uniform linear array (ULA) with the same aperture using fewer sensors. The conventional CSA product beamformer suffers from a smaller array gain due to the reduced number of sensors. This paper derives that the conditional PDFs for detecting Gaussian signals in spatially white Gaussian noise with the CSA product processor are products of Bessel functions. The resulting ROCs are compared with those of the ULA energy detector for a conventional beamformer. The Bessel function CSA detection PDFs asymptotically converge to exponential distributions like the ULA detection PDFs, revealing that the detection gain of the nonlinear CSA processor is still proportional to the number of sensors. Monte Carlo simulations confirm the validity of the analytic results and the asymptotic approximations to the PDFs.

Co-prime sensor arrays (CSAs) interleave two uniform linear subarrays that are undersampled by co-prime factors. The resulting nonuniform array requires far fewer sensors to match the spatial resolution of a fully populated ULA of the same aperture. Choosing the co-prime undersampling factors as close to equal as possible minimizes the number of sensors in the CSA. However, the peak side lobe of the CSA is higher than the peak side lobe of the equivalent full uniform linear array (ULA). Increasing the number of sensors in the CSA subarrays by half while maintaining the interelement spacing gurarantees that the CSA peak side lobe is less than that of the full aperture ULA when both arrays use rectangular windows.