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In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to recover $X$ exactly with noise-free measurements, or to construct a good approximation of $X$ with noisy measurements? Existing solutions to these problems involve sampling the elements uniformly and at random, and can guarantee exact recovery of the unknown matrix only with high probability. In this paper, we present a \textit{deterministic} sampling method for matrix completion. We achieve this by choosing the sampling set as the edge set of an asymmetric Ramanujan bigraph, and constrained nuclear norm minimization is the recovery method. Specifically, we derive sufficient conditions under which the unknown matrix is completed exactly with noise-free measurements, and is approximately completed with noisy measurements, which we call "stable" completion. The conditions derived here are only sufficient and more restrictive than random sampling. To study how close they are to being necessary, we conducted numerical simulations on randomly generated low rank matrices, using the LPS families of Ramanujan graphs. These simulations demonstrate two facts: (i) In order to achieve exact completion, it appears sufficient to choose the degree $d$ of the Ramanujan graph to be $\geq 3r$. (ii) There is a "phase transition," whereby the likelihood of success suddenly drops from 100\% to 0\% if the rank is increased by just one or two beyond a critical value. The phase transition phenomenon is well-known and well-studied in vector recovery using $\ell_1$-norm minimization. However, it is less studied in matrix completion and nuclear norm minimization, and not much understood., The original submission 1908.00963 has been split into two parts. The replacement submission is Part-1 of the revised version. Part-2 can also be found on arXiv

A coprime sensor array (CSA) interleaves two undersampled uniform linear arrays with coprime undersampling factors and has recently found broad applications in signal detection and estimation. CSAs commonly use the product processor by multiplying the scanned responses of two colinear subarrays to estimate the spatial power spectral density (PSD) of the received signal. This paper proposes a new CSA processor, the CSAmin processor, which chooses the minimum between the two CSA subarray periodograms at each bearing to estimate the spatial PSD. The proposed CSAmin processor resolves the CSA subarray spatial aliasing equally well as the product processor. For an extended aperture CSA, the CSAmin reduces the peak sidelobe height and total sidelobe area over the product processor for the same CSA geometry. Moreover, unlike the PSD estimate from the product processor, the PSD estimate from the CSAmin is guaranteed to be positive semidefinite. This paper derives the probability density function, the complementary cumulative distribution function (CCDF, or tail distribution), and the first two moments of the CSAmin PSD estimator in closed form for Gaussian sources in white Gaussian noise. Numerical simulations verify the derived CSAmin statistics and demonstrate that the CSAmin improves the performance over the product processor in detecting narrowband Gaussian sources in the presence of loud interferers and noise. The CSAmin spatial PSD estimate achieves lower variance than the product processor estimate, and keeps the PSD estimate unbiased for white Gaussian processes and asymptotically unbiased for nonwhite Gaussian processes.

This paper presents an extension of the integrated time-chirp rate analysis technique, known as the parameterized centroid frequency-chirp rate distribution (PCFCRD), for noisy multicomponent linear frequency modulated signals analysis. The PCFCRD is based on a newly defined correlation function and its auto term accumulation is two-dimensionally coherent. The principle, cross term characteristic, implementation, properties, parameter selection criterion, antinoise performance, and estimation accuracy are analyzed for the PCFCRD. In this paper, comparisons with the coherently integrated cubic phase function, maximum likelihood method, and Lv's distribution are performed. With mathematic analyses and numerical simulations, we demonstrate that the PCFCRD outperforms these three representative counterparts. Further, by combining with analyses of the PCFCRD, several unclear mechanisms of the constant delay introduction are mathematically interpreted, especially quantitative influences on the chirp rate resolution and theoretical antinoise performance.

In this paper, we propose a widely linear reproducing kernel Hilbert space (WL-RKHS) for nonlinear regression with complex-valued signals. Our approach is a nonlinear extension of WL signal processing that has been proven to be more versatile than linear systems for dealing with complex-value signals. To be able to use the WL concept in kernel methods, we need to introduce a pseudo-kernel to complement the standard kernel in RKHS, which is not defined in previous RKHS approaches in the existing literature. In this paper, we present WL-RKHS, its properties, and the kernel and pseudo-kernel designs. We illustrate the need of the pseudo-kernel with simply verifiable examples that allow understanding the intuitions behind this kernel. We conclude this paper, showing that in the all-relevant nonlinear equalization problem the pseudo-kernel plays a significant role and previous approaches that do not rely on this kernel clearly underperform.

The product of the scanned responses of two colinear interleaved arrays estimates the spatial power spectral density of the received signal. This estimate has mathematical structure similar to the classical periodogram and hence is called the generalized periodogram. The generalized periodogram includes sparse arrays such as coprime arrays and two level nested arrays. This paper derives the analytical expected value of the generalized periodogram output for any signal. The generalized periodogram expected value is the convolution of the true spectral density with a windowing function. This result reveals that the expected product spectrum is not guaranteed to be positive definite. The paper also derives the covariance function of the generalized periodogram for a white Gaussian process. Finally, the paper illustrates the implicit equivalence between the product spectrum and estimation of the correlation function from the products of appropriately spaced pairs of sensors.

In this paper, a novel model-based compressive sampling (CS) technique for natural images is proposed. Our algorithm integrates a universal hidden Markov tree (uHMT) model, which captures the relation among the sparse wavelet coefficients of images, into both sampling and recovery steps of CS. At the sampling step, we employ the uHMT model to devise a nonuniformly sparse measurement matrix $\boldsymbol{\Phi }_{\text{uHMT}}$ . In contrast to the conventional CS sampling matrices, such as dense Gaussian, Bernoulli or uniformly sparse matrices that are oblivious to the signal model and the correlation among the signal coefficients, the proposed $\boldsymbol{\Phi }_{\text{uHMT}}$ is designed based on the signal model and samples the coarser wavelet coefficients with higher probabilities and more sparse wavelet coefficients with lower probabilities. At the recovery step, we integrate the uHMT model into two state-of-the-art Bayesian CS recovery schemes. Our simulation results confirm the superiority of our proposed HMT model-based nonuniform compressive sampling and recovery , referred to as uHMT-NCS , over other model-based CS techniques that solely consider the signal model at the recovery step. This paper is distinguished from other model-based CS schemes in that we take a novel approach to simultaneously integrating the signal model into both CS sampling and recovery steps. We show that such integration greatly increases the performance of the CS recovery, which is equivalent to reducing the required number of samples for a given reconstruction quality.

This paper describes the adaptive detection of point-like targets in a Gaussian environment assuming a group symmetric structure for the interference covariance matrix. This special configuration enforces block-sparsity and permits splitting of the original observation space into lower-dimensional subspaces, each characterized by its own nuisance parameters. Hence, the Principle of Invariance is invoked to select decision rules enjoying some symmetry defined by a suitable group of transformations which leave the decision problem unaltered. All the invariant tests can be expressed as functions of a maximal invariant statistic whose derivation is among the key results of this paper. Finally, some common design criteria are applied to come up with adaptive architectures that share invariance, and their behavior is assessed to highlight the interplay between sample size and detection performance in comparison with conventional decision schemes.

Measuring the Time delay of Arrival (TDOA) between a set of sensors is the basic setup for many applications, such as localization or signal beamforming. This paper presents the set of TDOA matrices, which are built from noise-free TDOA measurements, not requiring knowledge of the sensor array geometry. We prove that TDOA matrices are rank-two and have a special SVD decomposition that leads to a compact linear parametric representation. Properties of TDOA matrices are applied in this paper to perform denoising, by finding the TDOA matrix closest to the matrix composed with noisy measurements. The paper shows that this problem admits a closed-form solution for TDOA measurements contaminated with Gaussian noise which extends to the case of having missing data. The paper also proposes a novel robust denoising method resistant to outliers, missing data and inspired in recent advances in robust low-rank estimation. Experiments in synthetic and real datasets show significant improvements of the proposed denoising algorithms in TDOA-based localization, both in terms of TDOA accuracy estimation and localization error.

The stability problems of the least mean fourth (LMF) algorithm put a limitation on its tracking capability. The paper investigates the possibility of solving this problem via stabilization of the algorithm. The analysis is done for a Markov plant. It is found that the available stable normalized LMF (NLMF) algorithm has a tracking limitation for high signal-to-noise ratio. The paper presents a new stable NLMF algorithm that is free of this limitation. Mean-square stability of the algorithm is proved. Expressions are derived for the minimum steady-state mean square deviation (MSD) and the corresponding convergence time. The new algorithm outperforms the available stable NLMF algorithm in both the transient and steady states. The new algorithm is also compared with the NLMS algorithm when the adaptation parameter of each algorithm is set to the value that minimizes its steady-state MSD. For large initial MSD, the algorithm outperforms the NLMS algorithm, even for Gaussian noise. For small initial MSD, the algorithm outperforms the NLMS algorithm for sub-Gaussian noise, while the situation is opposite for Gaussian noise. Analytical results are supported by simulations.

Reduced-rank adaptive processing (RRAP) has received considerable attention in recent years. One of the key problems with this technique is how to determine an appropriate dimension for the reduced-rank subspace. In this paper we study in detail four types of rank selection methods for the widely used RRAP approach-multistage Wiener filter (MWF). All of these methods have a computational complexity of order O(1), compared with other existing methods with an order of O(i) or even O(i 2 ) at the ith stage of the MWF. The main idea underlying these methods is to find a recursive algorithm to calculate the stopping criterion. The first two algorithms, based on the generalized discrepancy principle (GDP) and error estimation (EE) respectively, are fast versions of existing approaches. The last two algorithms, based on fast Ritz values estimation (RVE) and new information (NI) respectively, are new and proposed in this paper. In addition to requiring less computation, simulation results show that the proposed algorithms have a higher accuracy in adaptive beamforming applications for both narrowband and broadband signals.

In compressive sensing (CS), the restricted isometry property (RIP) is an important condition on measurement matrices which guarantees the recovery of sparse signals with undersampled measurements. It has been proved in the prior works that both random (e.g., i.i.d. Gaussian, Bernoulli, $\ldots$ ) and Toeplitz matrices satisfy the RIP with high probability. However, structured matrices, such as banded Toeplitz matrices have drawn more attention since their structures have the advantage of fast matrix multiplication which may decrease the computational complexity of recovery algorithms. In this paper, we show that banded block Toeplitz matrices satisfy the RIP condition with high probability. Banded block Toeplitz matrices can be used in the sparse multi-channel source separation. The banded block Toeplitz matrices decrease the computational complexity while they have fewer number of non-zero entries in comparison to the same dimensional banded Toeplitz matrices. Furthermore, our simulation results show that banded block Toeplitz matrices outperform banded Toeplitz matrices in signal estimation. The analytical RIP bound for banded block Toeplitz matrices is provided in this paper and the RIP bound of sparse Gaussian matrices is also obtained as an upper bound for banded block Toeplitz matrices. Our simulation and analytical results show that sparse Gaussian random matrices do satisfy the RIP condition with high probability. The probability of satisfying the RIP depends on the probability of zero entries.

This paper is concerned with Gaussian approximations to the posterior probability density function (PDF) in the update step of Bayesian filtering with nonlinear measurements. In this setting, sigma-point approximations to the Kalman filter (KF) recursion are widely used due to their ease of implementation and relatively good performance. In the update step, these sigma-point KFs are equivalent to linearizing the nonlinear measurement function by statistical linear regression (SLR) with respect to the prior PDF. In this paper, we argue that the measurement function should be linearized using SLR with respect to the posterior rather than the prior to take into account the information provided by the measurement. The resulting filter is referred to as the posterior linearization filter (PLF). In practice, the exact PLF update is intractable but can be approximated by the iterated PLF (IPLF), which carries out iterated SLRs with respect to the best available approximation to the posterior. The IPLF can be seen as an approximate recursive Kullback-Leibler divergence minimization procedure. We demonstrate the high performance of the IPLF in relation to other Gaussian filters in two numerical examples.

We consider the problem of approximating optimal in the Minimum Mean Squared Error (MMSE) sense nonlinear filters in a discrete time setting, exploiting properties of stochastically convergent state process approximations. More specifically, we consider a class of nonlinear, partially observable stochastic systems, comprised by a (possibly nonstationary) hidden stochastic process (the state), observed through another conditionally Gaussian stochastic process (the observations). Under general assumptions, we show that, given an approximating process which, for each time step, is stochastically convergent to the state process, an approximate filtering operator can be defined, which converges to the true optimal nonlinear filter of the state in a strong and well defined sense. In particular, the convergence is compact in time and uniform in a completely characterized measurable set of probability measure almost unity, also providing a purely quantitative justification of Egoroff's Theorem for the problem at hand. The results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating optimal nonlinear filters, such as approximate grid based techniques, known to perform well in a variety of applications., EXTENDED version of an original paper published in the IEEE Transactions on Signal Processing; 37 pages

Joint source-channel coding schemes based on analog mappings for point-to-point channels have recently gained attention for their simplicity and low delay. In this paper, these schemes are extended either to scenarios with or without side information at the decoders to transmit multiple descriptions of a Gaussian source over independent parallel channels. They are based on a lattice scaling approach together with bandwidth reduction analog mappings adapted for this multiple description scenario. The rationale behind lattice scaling is to improve performance through bandwidth expansion. Another important contribution of this paper is the proof of the separation theorem for the communication scenario of multiple description with side information at the decoders. This proof allows us to compare the performance of the proposed schemes with the theoretical limits when considering AWGN channels. Finally, when the channels are parallel slow-fading Rayleigh, the proposed schemes are optimal in terms of distortion exponent.

International audience; This paper addresses the problem of the Clutter Subspace Projector (CSP) estimation in the context of a disturbance composed of a Low Rank (LR) heterogeneous clutter , modeled here by a Spherically Invariant Random Vector (SIRV), plus a white Gaussian noise (WGN). In such context, the corresponding LR adaptive filters and detectors require less training vectors than classical methods to reach equivalent performance. Unlike classical adaptive processes, which are based on an estimate of the noise Covariance Matrix (CM), the LR processes are based on a CSP estimate. This CSP estimate is usually derived from a Singular Value Decomposition (SVD) of the CM estimate. However, no Maximum Likelihood Estimator (MLE) of the CM has been derived for the considered disturbance model. In this paper, we introduce the fixed point equation that MLE of the CSP satisfies for a disturbance composed of a LR-SIRV clutter plus a zero mean WGN. A recursive algorithm is proposed to compute this solution. Numerical simulations validate the introduced estimator and illustrate its interest compared to the current state of art.

Most conventional target tracking algorithms assume that one target can generate at most one detection per scan. However, in many practical target tracking applications, one target may generate multiple detections in one scan, because of multipath propagation, or high sensor resolution or some other reason. If the multiple detections from the same target can be effectively utilized, the performance of the multitarget tracking system can be improved. However, the challenge is that the uncertainty in the number of targets and the measurement set-to-target association will increase the complexity of tracking algorithms. To solve this problem, the random finite set (RFS) modeling and the random finite set statistics (FISST) are used in this paper. Without any extra approximation beyond those made in the standard probability hypothesis density (PHD) filter, a general multi-detection PHD (MD-PHD) update formulation is derived. It is also established in this paper that, with certain reasonable assumptions, the proposed MD-PHD recursion can function as a generalized extended target PHD or multisensor PHD filter. Furthermore, a Gaussian Mixture (GM) implementation of the proposed MD-PHD formulation, called the MD-GM-PHD filter, is presented. The proposed MD-GM-PHD filter is demonstrated on a simulated over-the-horizon radar (OTHR) scenario.

The famous mathematician S. Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). For any fixed integer q, this is a sequence in n with periodicity q. Ramanujan showed that many standard arithmetic functions in the theory of numbers, such as Euler's totient function φ(n) and the Mobius function μ(n), can be expressed as linear combinations of c_q(n), 1 ≤ q ≤ ∞. In the last ten years, Ramanujan sums have aroused some interest in signal processing. There is evidence that these sums can be used to extract periodic components in discrete-time signals. The purpose of this paper and the companion paper (Part II) is to develop this theory in detail. After a brief review of the properties of Ramanujan sums, the paper introduces a subspace called the Ramanujan subspace S_q and studies its properties in detail. For fixed q, the subspace S_q includes an entire family of signals with properties similar to c_q(n). These subspaces have a simple integer basis defined in terms of the Ramanujan sum c_q(n) and its circular shifts. The projection of arbitrary signals onto these subspaces can be calculated using only integer operations. Linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns, which makes it easy to identify periods. In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.

The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞

In this two-part paper, we consider the problem of adaptive multidimensional/multichannel signal detection in Gaussian noise with unknown covariance matrix. The test data (primary data) is assumed as a collection of sample vectors, arranged as the columns of a rectangular data array. The rows and columns of the signal matrix are both assumed to lie in known subspaces, but with unknown coordinates. Due to this feature of the signal structure, we name this kind of signal as the double subspace signal. Part I of this paper focuses on the adaptive detection in homogeneous environments, while Part II deals with the adaptive detection in partially homogeneous environments. Precisely, in this part, we derive the generalized likelihood ratio test (GLRT), Rao test, Wald test, as well as their two-step variations, in homogeneous environments. Three types of spectral norm tests (SNTs) are also introduced. All these detectors are shown to possess the constant false alarm rate (CFAR) property. Moreover, we discuss the differences between them and show how they work. Another contribution is that we investigate various special cases of these detectors. Remarkably, some of them are well-known existing detectors, while some others are still new. At the stage of performance evaluation, conducted by Monte Carlo simulations, both matched and mismatched signals are dealt with. For each case, more than one scenario is considered.

The paper investigates a new stability problem of the least mean fourth (LMF) algorithm, which is the dependence of the algorithm stability on the time-variation of the target weights of the adaptive filter. The analysis is done in the context of tracking a Markov plant with a stationary white Gaussian input. It is found that the algorithm diverges if the mean square increment of the plant parameter vector exceeds a threshold value that depends on the step-size, input variance, and noise moments. The paper also derives a closed form of the steady-state mean square deviation without the usual assumption of a strong noise. Comparison of the tracking capabilities of the LMF and LMS algorithms is provided. The comparison is done in terms of the minimum mean square deviation attained by each algorithm over the stability range of its step-size. Gaussian, uniform, and binary distributions of the noise are considered. Conditions that make one algorithm outperform the other are determined. Analytical results are supported by simulations.

Given $N$ noisy measurements denoted by ${\mathbf y}$ and an overcomplete Gaussian dictionary, ${\mathbf A}$ , the authors of [1] establish the existence and the asymptotic statistical efficiency of an unbiased estimator unaware of the locations of the nonzero entries, collected in set $\mathcal {I}$ , in the deterministic $L$ -sparse signal ${\mathbf x}$ . More precisely, there exists an estimator ${\hat{\mathbf {x}}}({\mathbf y}, {\mathbf A})$ unaware of set $\mathcal {I}$ with a variance reaching the oracle-Cramer–Rao Bound in the asymptotic scenario, i.e., for $N,L\rightarrow \infty$ and $L/N \rightarrow \alpha \in (0,1)$ . As was noted in the paper “Fundamental limits and constructive methods for estimation and sensing of sparse signals” by B. Babadi, the existence proof remains true even though Lemma 3.5 and (20) the paper “Asymptotic achievability of the Cramer–Rao bound for noisy compressive sampling” are inexact. In this note, the exact closed-form expression of the variance of the estimator ${\hat{\mathbf {x}}}({\mathbf y}, {\mathbf A})$ is provided, and its practical usefulness is numerically illustrated with the orthogonal matching pursuit estimator.

The Probability Hypothesis Density Filter (PHD) is a multitarget tracker for recursively estimating the number of targets and their state vectors from a set of observations. The PHD filter is capable of working well in scenarios with false alarms and missed detections. Two distinct PHD filter implementations are available in the literature: the Sequential Monte Carlo Probability Hypothesis Density (SMC-PHD) and the Gaussian Mixture Probability Hypothesis Density (GM-PHD) filters. The SMC-PHD filter uses particles to provide target state estimates, which can lead to a high computational load, whereas the GM-PHD filter does not use particles, but restricts to linear Gaussian mixture models. The SMC-PHD filter technique provides only weighted samples at discrete points in the state space instead of a continuous estimate of the probability density function of the system state and thus suffers from the well-known degeneracy problem. This paper proposes a B-Spline based Probability Hypothesis Density (S-PHD) filter, which has the capability to model any arbitrary probability density function. The resulting algorithm can handle linear, non-linear, Gaussian, and non-Gaussian models and the S-PHD filter can also provide continuous estimates of the probability density function of the system state. In addition, by moving the knots dynamically, the S-PHD filter ensures that the splines cover only the region where the probability of the system state is significant, hence the high efficiency of the S-PHD filter is maintained at all times. Also, unlike the SMC-PHD filter, the S-PHD filter is immune to the degeneracy problem due to its continuous nature. The S-PHD filter derivations and simulations are provided in this paper.

In this paper, we consider the delay-constrained beamforming control for downlink multi-user MIMO (MU- MIMO) systems with imperfect channel state information at the transmitter (CSIT). The delay-constrained control problem is formulated as an infinite horizon average cost partially observed Markov decision process. To deal with the curse of dimensionality, we introduce a virtual continuous time system and derive a closed-form approximate value function using perturbation analysis w.r.t. the CSIT errors. To deal with the challenge of the conditional packet error rate (PER), we build a tractable closed- form approximation using a Bernstein-type inequality. Based on the closed-form approximations of the relative value function and the conditional PER, we propose a conservative formulation of the original beamforming control problem. The conservative problem is non-convex and we transform it into a convex problem using the semidefinite relaxation (SDR) technique. We then propose an alternating iterative algorithm to solve the SDR problem. Finally, the proposed scheme is compared with various baselines through simulations and it is shown that significant performance gain can be achieved., Comment: 14 pages, 7 figures, 1 table. This paper has been accepted by the IEEE Transactions on Signal Processing

Randomized (dithered) quantization is a method capable of achieving white reconstruction error independent of the source. Dithered quantizers have traditionally been considered within their natural setting of uniform quantization. In this paper we extend conventional dithered quantization to nonuniform quantization, via a subterfage: dithering is performed in the companded domain. Closed form necessary conditions for optimality of the compressor and expander mappings are derived for both fixed and variable rate randomized quantization. Numerically, mappings are optimized by iteratively imposing these necessary conditions. The framework is extended to include an explicit constraint that deterministic or randomized quantizers yield reconstruction error that is uncorrelated with the source. Surprising theoretical results show direct and simple connection between the optimal constrained quantizers and their unconstrained counterparts. Numerical results for the Gaussian source provide strong evidence that the proposed constrained randomized quantizer outperforms the conventional dithered quantizer, as well as the constrained deterministic quantizer. Moreover, the proposed constrained quantizer renders the reconstruction error nearly white. In the second part of the paper, we investigate whether uncorrelated reconstruction error requires random coding to achieve asymptotic optimality. We show that for a Gaussian source, the optimal vector quantizer of asymptotically high dimension whose quantization error is uncorrelated with the source, is indeed random. Thus, random encoding in this setting of rate-distortion theory, is not merely a tool to characterize performance bounds, but a required property of quantizers that approach such bounds., Comment: submitted to IEEE Transactions on Signal Processing, 2012

In extended/group target tracking, where the extensions of the targets are estimated, target spawning and combination events might have significant implications on the extensions. This paper investigates target spawning and combination events for the case that the target extensions are modeled in a random matrix framework. The paper proposes functions that should be provided by the tracking filter in such a scenario. The results, which are obtained by a gamma Gaussian inverse Wishart implementation of an extended target probability hypothesis density filter, confirms that the proposed functions improve the performance of the tracking filter for spawning and combination events. Funding Agencies|Swedish Research Council|621-2010-4301|Swedish Foundation for Strategic Research (SSF)|| CADICS ETT CUAS

In this paper, we study the performance of mean square error (MSE) and Gaussian entropy criteria for linear and widely linear complex filtering. The MSE criterion has been extensively studied, and with a widely linear filter form, it can take into account the full second-order statistics of the input signal. However, it cannot exploit the full second-order statistics of the error, and doubles the dimension of the parameter vector to be estimated. In this paper, we introduce the use of Gaussian entropy criterion such that full second-order statistics of the error can be taken into account, and compare the performance of the Gaussian entropy and MSE criteria for a linear and widely linear filter implementation in batch and adaptive implementations. Detailed performance analysis with numerical examples is presented to investigate the relationship and performance differences of the two criteria in diverse scenarios.

The recursive procedure to compute the posterior Cramer-Rao lower bound (PCRLB) for sequential Bayesian estimators, derived by Tichavsky , provides an off-line performance bound for a general nonlinear filtering problem. Since the corresponding Fisher information matrix (FIM) is obtained by taking the expectation with respect to all the random variables, this PCRLB is not well suited for online adaptive resource management for dynamic systems. For online estimation performance evaluation in a nonlinear system, the concept of conditional PCRLB was proposed by Zuo in 2011. In this paper, two other online conditional PCRLBs are proposed which are alternatives to the one proposed by Zuo Numerical examples are provided to show that the three online bounds, namely the conditional PCRLB proposed by Zuo and the two conditional PCRLBs proposed in this paper, are very close to one another.

In this paper we describe an alternative to standard nonnegative matrix factorization (NMF) for nonnegative dictionary learning, i.e., the task of learning a dictionary with nonnegative values from nonnegative data, under the assumption of nonnegative expansion coefficients. A popular cost function used for NMF is the Kullback-Leibler divergence, which underlies a Poisson observation model. NMF can thus be considered as maximization of the joint likelihood of the dictionary and the expansion coefficients. This approach lacks optimality because the number of parameters (which include the expansion coefficients) grows with the number of observations. In this paper we describe variational Bayes and Monte-Carlo EM algorithms for optimization of the marginal likelihood, i.e., the likelihood of the dictionary where the expansion coefficients have been integrated out (given a Gamma prior). We compare the output of both maximum joint likelihood estimation (i.e., standard NMF) and maximum marginal likelihood estimation (MMLE) on real and synthetical datasets. In particular we present face reconstruction results on CBCL dataset and text retrieval results over the musiXmatch dataset, a collection of word counts in song lyrics. The MMLE approach is shown to prevent overfitting by automatically pruning out irrelevant dictionary columns, i.e., embedding automatic model order selection.

International audience; Independent component analysis (ICA) and blind source separation (BSS) deal with extracting a number of mutually independent elements from a set of observed linear mixtures. Motivated by various applications, this paper considers a more general and more flexible model: the sources can be partitioned into groups exhibiting dependence within a given group but independence between two different groups. We argue that this is tantamount to considering multidimensional components as opposed to the standard ICA case which is restricted to one-dimensional components. The core of the paper is devoted to the statistical analysis of the blind separation of multidimensional components based on second-order statistics, in a piecewise-stationary model. We develop the likelihood and the associated estimating equations for the Gaussian case. We obtain closed-form expressions for the Fisher information matrix and the Cramér-Rao bound of the de-mixing parameters, as well as the mean-square error (MSE) of the component estimates. The derived MSE is valid also for non-Gaussian data. Our analysis is verified through numerical experiments, and its performance is compared to classical ICA in various dependence scenarios, quantifying the gain in the accuracy of component recovery in presence of multidimensional components.

Given a U-invariant sampling scheme on a Hilbert space H. This paper characterizes atomic subspaces A of H such that every signal x ∈ A can be reconstructed from its generalized samples acquired with the U-invariant sampling scheme. If signal recovery is possible a linear filter is derived which reconstructs the signal from the samples. Moreover, the paper gives necessary and sufficient conditions on a sequence of sampling functions such that it forms a pseudoframe or pseudo-Riesz basis for a given atomic subspace A.

In this paper, we consider the joint blind source separation (JBSS) problem and introduce a number of algorithms to solve the JBSS problem using the independent vector analysis (IVA) framework. Source separation of multiple datasets simultaneously is possible when the sources within each and every dataset are independent of one another and each source is dependent on at most one source within each of the other datasets. In addition to source separation, the IVA framework solves an essential problem of JBSS, namely the identification of the dependent sources across the datasets. We propose to use the multivariate Gaussian source prior to achieve JBSS of sources that are linearly dependent across datasets. Analysis within the paper yields the local stability conditions, nonidentifiability conditions, and induced Cramer-Rao lower bound on the achievable interference to source ratio for IVA with multivariate Gaussian source priors. Additionally, by exploiting a novel nonorthogonal decoupling of the IVA cost function we introduce both Newton and quasi-Newton optimization algorithms for the general IVA framework.

In this paper we expand our recently introduced concept of UW-OFDM (unique word orthogonal frequency division multiplexing). In UW-OFDM the cyclic prefixes (CPs) are replaced by deterministic sequences, the so-called unique words (UWs). The UWs are generated by appropriately loading a set of redundant subcarriers. By that a systematic complex number Reed Solomon (RS) code construction is introduced in a quite natural way, because an RS code may be defined as the set of vectors, for which a block of successive zeros occurs in the other domain w.r.t. a discrete Fourier transform. (For a fixed block different to zero, i.e., a UW, a coset code of an RS code is generated.) A remaining problem in the original systematic coded UW-OFDM concept is the fact that the redundant subcarrier symbols disproportionately contribute to the mean OFDM symbol energy. In this paper we introduce the concept of non-systematic coded UW-OFDM, where the redundancy is no longer allocated to dedicated subcarriers, but distributed over all subcarriers. We derive optimum complex valued code generator matrices matched to the BLUE (best linear unbiased estimator) and to the LMMSE (linear minimum mean square error) data estimator, respectively. With the help of simulations we highlight the advantageous spectral properties and the superior BER (bit error ratio) performance of non-systematic coded UW-OFDM compared to systematic coded UW-OFDM as well as to CP-OFDM in AWGN (additive white Gaussian noise) and in frequency selective environments., IEEE Tranactions on Signal Processing, Vol. 60, No. 1, 2012. arXiv admin note: text overlap with arXiv:1202.1398

In this paper, we consider the problem of detecting a signal in Gaussian noise with unknown covariance matrix, in partially homogeneous environments where the test and training data samples share the same noise covariance matrix up to an unknown scaling factor. One solution to this problem is the adaptive subspace detector (ASD) with a single or multiple observations. However, the probabilities of false alarm and detection of this ASD have not been obtained yet. In this paper, these expressions are derived on the basis of a single observation, which are confirmed with Monte Carlo simulations. It is shown that the ASD has a constant false alarm rate property with respect to both the shared noise covariance matrix structure and the independent scaling of the noise in the test data. In addition, we prove that for the First-Order model where the signal of interest is assumed to be a deterministic but unknown vector, the ASD derived with the generalized likelihood ratio test is consistent with that derived with an ad hoc two-step design procedure.

This paper addresses the robust H∞ filtering problem for a class of discrete time-varying Markovian jump systems with randomly occurring nonlinearities and sensor saturation. Two kinds of transition probability matrices for the Markovian process are considered, namely, the one with polytopic uncertainties and the one with partially unknown entries. The nonlinear disturbances are assumed to occur randomly according to stochastic variables satisfying the Bernoulli distributions. The main purpose of this paper is to design a robust filter, over a given finite-horizon, such that the H∞ disturbance attenuation level is guaranteed for the time-varying Markovian jump systems in the presence of both the randomly occurring nonlinearities and the sensor saturation. Sufficient conditions are established for the existence of the desired filter satisfying the H∞ performance constraint in terms of a set of recursive linear matrix inequalities. Simulation results demonstrate the effectiveness of the developed filter design scheme.

In a recent paper, the second-order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing, which are associated with the two main types of quaternion properness. In this paper, we consider the problem of determining, from a finite number of independent vector observations, whether a quaternion Gaussian vector is proper or not. Specifically, we derive three generalized likelihood ratio tests (GLRTs) for testing the two main kinds of quaternion properness and show that the GLRTs reduce to the estimation of three previously proposed quaternion improperness measures. Interestingly, the three GLRT statistics (improperness measures) can be interpreted as an estimate of the entropy loss due to the quaternion improperness. Additionally, we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown, which results in the problem of estimating the principal \BBC-properness direction, i.e., the pure unit quaternion minimizing the \BBC -improperness measure. Although this estimation problem is not convex, we propose a technique based on successive convex approximations, which can be solved in closed form. Finally, some simulation examples illustrate the performance and practical application of the proposed tests.

Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. The primary mathematical tool employed in these methods is the notion of the Reproducing Kernel Hilbert Space. However, so far, the emphasis has been on batch techniques. It is only recently, that online techniques have been considered in the context of adaptive signal processing tasks. Moreover, these efforts have only been focussed on real valued data sequences. To the best of our knowledge, no adaptive kernel-based strategy has been developed, so far, for complex valued signals. Furthermore, although the real reproducing kernels are used in an increasing number of machine learning problems, complex kernels have not, yet, been used, in spite of their potential interest in applications that deal with complex signals, with Communications being a typical example. In this paper, we present a general framework to attack the problem of adaptive filtering of complex signals, using either real reproducing kernels, taking advantage of a technique called \textit{complexification} of real RKHSs, or complex reproducing kernels, highlighting the use of the complex gaussian kernel. In order to derive gradients of operators that need to be defined on the associated complex RKHSs, we employ the powerful tool of Wirtinger's Calculus, which has recently attracted attention in the signal processing community. To this end, in this paper, the notion of Wirtinger's calculus is extended, for the first time, to include complex RKHSs and use it to derive several realizations of the Complex Kernel Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS offers significant performance improvements over several linear and nonlinear algorithms, when dealing with nonlinearities., 15 pages (double column), preprint of article accepted in IEEE Trans. Sig. Proc

In this paper, we consider the problem of the accuracy of estimating the location and other attributes of a moving single molecule whose trajectory is imaged by fluorescence microscopy. As accuracy in parameter estimation is closely related to the Fisher information matrix, we first give a general expression of the Fisher information matrix for the estimated parameters for a single object moving in three-dimensional (3D) space. Explicit Cramér-Rao lower bound (CRLB) expressions are then obtained from the Fisher information matrix for a single object moving in the two-dimensional (2D) focus plane with the object trajectory being either linear or circular. We also investigate how extraneous noise sources, pixelation, parameters of the detection system and parameters of the trajectory affect the limit of the accuracy. The results obtained in this paper provide insights that enable the experimentalists to optimize their experimental setups for tracking single molecules in order to achieve the best possible accuracy. They are also applicable to the general problem of tracking an object using quantum limited detectors.

This paper proposes the use of a synchronized linear transform, the synchronized short-time-Fourier-transform (sSTFT), for time-frequency analysis of anechoic mixtures. We address the short comings of the commonly used time-frequency linear transform in multichannel settings, namely the classical short-time-Fourier-transform (cSTFT). We propose a series of desirable properties for the linear transform used in a multichannel source separation scenario: stationary invertibility, relative delay, relative attenuation, and finally delay invariant relative windowed-disjoint orthogonality (DIRWDO). Multisensor source separation techniques which operate in the time-frequency domain, have an inherent error unless consideration is given to the multichannel properties proposed in this paper. The sSTFT preserves these relationships for multichannel data. The crucial innovation of the sSTFT is to locally synchronize the analysis to the observations as opposed to a global clock. Improvement in separation performance can be achieved because assumed properties of the time-frequency transform are satisfied when it is appropriately synchronized. Numerical experiments show the sSTFT improves instantaneous subsample relative parameter estimation in low noise conditions and achieves good synthesis.

In this paper, we address the problem of data-aided (DA) and nondata-aided (NDA) per-antenna signal-to-noise ratio (SNR) estimation over wireless single-input multiple-output (SIMO) channels from linearly modulated signals. Under constant channels and additive white Gaussian noise (AWGN), we first derive the DA maximum-likelihood (ML) SNR estimator in closed-form expression. The performance of the DA ML estimator is analytically carried out by deriving the closed-form expression of its bias and variance. Besides, in order to compare its performance with the fundamental limit, we derive the DA Cramer-Rao lower bound (CRLB) in closed-form expression. In the NDA case, the expectation-maximization (EM) algorithm is derived to iteratively maximize the log-likelihood function. The performance of the NDA ML estimator is empirically assessed using Monte Carlo simulations. Moreover, we introduce an efficient algorithm, which applies to any one/two-dimensional M-ary constellation, to numerically compute the NDA CRLBs. In this paper, the noise components are assumed to be spatially uncorrelated over all the antenna elements and temporally white. In both cases, we show that our new inphase and quadrature I/Q-based estimators offer substantial performance improvements over the single-input single-output (SISO) ML SNR estimator due to the optimal usage of the statistical dependence between the antenna branches, and that it reaches the corresponding CRLB over a wide SNR range. We also show that the use of the I/Q-based ML estimators can lead to remarkable performance improvements over the moment-based estimators for the same antenna-array size. Moreover, it is shown that SIMO configurations can contribute to decreasing the required number of iterations of the EM algorithm.

In this paper, we derive for the first time the analytical expressions of the exact Cramer-Rao lower bounds (CRLBs) of the carrier frequency and the carrier phase from square quadrature amplitude modulated (QAM) signals, assuming the noise power and the signal amplitude to be completely unknown. The signal is assumed to be corrupted by additive white Gaussian noise (AWGN). The main contribution of this paper consists in deriving the analytical expressions for the non-data-aided (NDA) Fisher information matrix (FIM) for higher-order square QAM-modulated signals. We prove that the problem of estimating the synchronization parameters is separable from the one of estimating the signal and the noise powers by showing that the FIM is block diagonal. Besides, we show analytically that the phase CRLB is higher than the frequency CRLB, implying that it is much easier to estimate the frequency than the distortion phase. It will be seen that the CRLBs differ widely from one modulation order to another in the medium SNR range. The newly derived expressions corroborate previous attempts to numerically or empirically compute the considered CRLBs as well as their asymptotical expressions derived only in special SNR regions.

Constant modulus transforms like discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used over several decades in several engineering applications, including discrete multi-tone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communications systems. In this paper, we present a generalized framework for DFT called Generalized DFT (GDFT) with nonlinear phase by exploiting the phase space. We show that GDFT offers sizable correlation improvements over DFT, Walsh, Oppermann and Gold codes, leading to better performance in all multi-carrier communications scenarios investigated. We also highlight how known constant modulus orthogonal transforms are special solutions of the proposed GDFT framework. Moreover, we introduce practical design methods offering computationally efficient implementations of GDFT as enhancements to DFT. We conclude the paper with examples of communications applications where GDFT is shown to outperform DFT and other known constant modulus bases.

The theory of autoregressive (AR) modeling, also known as linear prediction, has been established by the Fourier analysis of infinite discrete-time sequences or continuous-time signals. Nevertheless, for various finite-length discrete trigonometric transforms (DTTs), including the discrete cosine and sine transforms of different types, the theory is not well established. Several DTTs have been used in current audio coding, and the AR modeling method can be applied to reduce coding artifacts or exploit data redundancies. This paper systematically develops the AR modeling fundamentals of temporal and spectral envelopes for the sixteen members of the DTTs. This paper first considers the AR modeling in the generalized discrete Fourier transforms (GDFTs). Then, we derive the modeling to all the DTTs by introducing the analytic transforms which convert the real-valued vectors into complex-valued ones. Through the process, we build the compact matrix representations for the AR modeling of the DTTs in both time domain and DTT domain. These compact forms also illustrate that the AR modeling for the envelopes can be performed through the Hilbert envelope and the power envelope. These compact forms can be used to develop new coding technologies or examine the possible defects in the existing AR modeling methods for DTTs, We apply the forms to analyze the current temporal noise shaping (TNS) tool in MPEG-2/4 advanced audio coding (AAC).

This paper proposes a method to convert a fixed instant interpolator for band-limited signals into a variable instant one, which has the form of a barycentric interpolator. This interpolator makes it possible to approximate the signal and its derivatives with minimal complexity in a range that surrounds the initial instant, and is applicable to uniform and nonuniform sampling grids. The barycentric form of the interpolator is derived using two different procedures, first by the repeated use of Bernstein's inequality and Taylor's theorem, and second by truncating a Lagrange-type series. These procedures show that the proposed method can be applied to existing fixed instant interpolators, and that it can be accurate in long time intervals. Finally, an evaluation procedure for barycentric interpolators and their derivatives is presented that minimizes the number of divisions, solving at the same time the numerical problems associated with small denominators. This paper includes several numerical examples.

The paper derives the reference prior for complex covariance matrices. The reference prior is a noninformative prior that circumvents some of the weaknesses of common alternatives in multidimensional settings. As a consequence, inference based on this prior renders well-behaving solutions that in many cases outperform traditionally used approaches. The main obstacle is that inference based on this prior require integration over high-dimensional spaces which have no closed form solutions. A focus of the paper is therefore to discuss efficient implementation strategies based on Markov chain Monte Carlo methods. It is identified that certain structures can be treated analytically both for the case where the parameter of interest is the covariance matrix itself but also for cases in which the covariance matrix is a nuisance parameter that characterizes noise color. Evaluation in both these settings also verify the superior performance obtained by using the proposed prior as compared to traditional techniques to treat unknown covariance matrices.

This paper is about the identification of discrete-time Wiener systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum-likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramér-Rao lower bound is calculated. A two-step procedure for generating high-quality initial estimates is presented as well. The paper includes the illustration of the method on a simulation example.

In this paper, the problem of spectral search-free direction-of-arrival (DOA) estimation in arbitrary nonuniform sensor arrays is addressed. In the first part of the paper, we present a finite-sample performance analysis of the well-known manifold separation (MS) based root-MUSIC technique. Then, we propose a new class of search-free DOA estimation methods applicable to arrays of arbitrary geometry and establish their relationship to the MS approach. Our first technique is referred to as Fourier-domain (FD) root-MUSIC and is based on the fact that the spectral MUSIC function is periodic in angle. It uses the Fourier series to expand this function and reformulate the underlying DOA estimation problem as an equivalent polynomial rooting problem. Our second approach applies the zero-padded inverse Fourier transform to the FD root-MUSIC polynomial to avoid the polynomial rooting step and replace it with a simple line search. Our third technique refines the FD root-MUSIC approach by using weighted least-squares approximation to compute the polynomial coefficients. The proposed techniques are shown to offer substantially improved performance-to-complexity tradeoffs as compared to the MS technique.

A modification of the conventional Lagrange interpolator is proposed in this paper, that allows one to approximate a band-limited signal from its own nonuniform samples with high accuracy. The modification consists in applying the Lagrange method to the signal, but pre-multiplied by a fixed function, and then solving for the desired signal value. Its efficiency lies in the fact that the fixed function is independent of the sampling instants. It is shown in this paper that the function can be selected so that the interpolation error decreases exponentially with the number of samples, for the case in which the sampling instants have a maximum deviation from a uniform grid. This paper includes a low-complexity recursive implementation of the method. Its accuracy is validated in the numerical examples by comparison with several interpolators in the literature, and by deriving upper and lower bounds for its maximum error.

In Part I of this paper [ldquoTheory of the Stochastic Resonance Effect in Signal Detection: Part I-Fixed Detectors,rdquo IEEE Transactions on Signal Processing, vol. 55, no. 7, pt. 1, pp. 3172-3184], the mechanism of the stochastic resonance (SR) effect for a fixed detector has been examined. This paper analyzes the stochastic resonance (SR) effect under the condition that the detector structure or its parameters can also be changed. The detector optimization problem with SR noise under both Neyman-Pearson and Bayesian criteria is examined. In the Bayesian approach when the prior probabilities are unknown, the minimax approach is adopted. The form of the optimal noise pdf along with the corresponding detector as well as the maximum achievable performance are determined. The developed theory is then applied to a general class of weak signal detection problems. Under the assumptions that the sample size N is large enough and the test statistics satisfies the conditions of central limit theorem, the optimal SR noise is shown to be a constant vector and independent of the signal strength for both Neyman-Pearson and Bayesian criteria. Illustrative examples are presented where performance comparisons are made between the original detector and the optimal SR noise modified detector for different types of SR noise.

Random fields are useful models of spatially variable quantities, such as those occurring in environmental processes and medical imaging. The fluctuations obtained in most natural data sets are typically anisotropic. The parameters of anisotropy are often determined from the data by means of empirical methods or the computationally expensive method of maximum likelihood. In this paper, we propose a systematic method for the identification of geometric (elliptic) anisotropy parameters of scalar fields. The proposed method is computationally efficient, nonparametric, noniterative, and it applies to differentiable random fields with normal or lognormal probability density functions. Our approach uses sample-based estimates of the random field spatial derivatives that we relate through closed form expressions to the anisotropy parameters. This paper focuses on two spatial dimensions. We investigate the performance of the method on synthetic samples with Gaussian and Matern correlations, both on regular and irregular lattices. The systematic anisotropy detection provides an important preprocessing stage of the data. Knowledge of the anisotropy parameters, followed by suitable rotation and rescaling transformations restores isotropy thus allowing classical interpolation and signal processing methods to be applied.

International audience; In this paper, we investigate the existence and the algorithm analysis of an adaptive scheme that has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non-Gaussian noise. This latter has been modeled by spherically invariant random vector (SIRV), which is the product c of the square root of a positive unknown random variable tau and an independent Gaussian vector x,c=radic(tau) x. A similar line of work was undertaken in the context of compound Gaussian noise, and this paper extends the previous results in the case of SIRV modeled noise. More precisely, the fixed-point estimate to be studied verifies a nonlinear algebraic equation (E)x=f(x). The aim of this paper is twofold. First, we prove that (E) admits a unique solution x; secondly, we show that the corresponding iterative algorithm xn+1=f(xn) converges to x for every admissible initial condition.