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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.

One main drawback of using time difference of arrival (TDOA) methods for source localization and navigation is that they require centralization of multiple copies of a signal. This paper considers blindly estimating the location of a cyclic prefix (CP) in an orthogonal frequency division multiplexing (OFDM) signal, enabling distributed TDOA computation up to an integer ambiguity. This ambiguity can be resolved using integer least-squares methods, if enough TDOAs are available, requiring only minimal cooperation between receivers. The contributions of this paper are development of an algorithm for simultaneously resolving the integer ambiguities and obtaining a position estimate; and derivation of the Cramer-Rao lower bound (CRLB) on locating the CP, and hence, on the underlying source localization and navigation problems.

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.

With the advent of high-throughput genomic and proteomic technologies, in conjunction with the difficulty in obtaining even moderately sized samples, small-sample classifier design has become a major issue in the biological and medical communities. With small samples, training-data error estimation becomes mandatory. Yet none of the popular error estimation techniques have been rigorously designed based on statistical inference and optimization. In this investigation, we place classifier error estimation into the framework of optimal mean-square error (MSE) signal estimation in the presence of uncertainty, which results in a Bayesian approach to error estimation based on a parameterized family of feature-label distributions with the prior distribution of the parameters governing the choice of feature-label distribution. These Bayesian error estimators are optimal when averaged over a given family of distributions, unbiased when averaged over a given family and all samples, and analytically address a trade-off between robustness (modeling assumptions) and accuracy (minimum mean-square error). In this paper, Part I of a two-part study, we define the minimum mean-square error (MMSE) error estimator, discuss its basic properties, provide closed-form analytic estimator representation for discrete classifiers with both non-informative and informative prior distributions, and examine the performance and robustness of the MMSE error estimator via simulations. In Part II of this paper, in this same issue of IEEE Transactions on Signal Processing, we address all of these issues, in particular, closed-form representation for linear classification in the Gaussian model with known and unknown covariance matrices. For both the discrete and Gaussian cases, the MMSE error estimator has especially good performance for distributions having moderate true errors.

In a companion paper, we have studied the zero-forcing (ZF) transceiver with decision feedback equalizer (DFE) over slowly time-varying narrowband multiinput multioutput (MIMO) channels. The space-time generalized triangular decomposition (ST-GTD) was used for the design of ZF-DFE transceivers. The space-time geometric mean decomposition (ST-GMD) ZF transceiver minimizes both the arithmetic mean square error (MSE) at the feedback detector and the average uncoded bit error rate (BER) in moderate high signal-to-noise ratio (SNR). This paper addresses the design problem of DFE transceiver without zero-forcing constraint. In the first part, a channel independent temporal precoder is superimposed on the conventional block-wise GMD-based minimum mean square error (MMSE) DFE transceiver to take advantage of the temporal diversity. In the second part, ST-GTD is applied for the design of MMSE DFE transceivers. With accurate channel prediction and space-time powerloading, the proposed ST-GMD MMSE transceiver minimizes the arithmetic MSE at the feedack detector, and maximizes Gaussian mutual information. For practical applications, the ST-GTD MMSE transceiver which does not require channel prediction but shares the same asymptotic BER performance with the ST-GMD MMSE system is also developed. In the convex region, our analysis shows that the proposed MMSE transceivers has better BER performance than the conventional GMD-based MMSE transceiver; the average BERs of the proposed systems are nonincreasing functions of the ST-block size. The superior performance of ST-GMD MMSE transceiver over the ST-GMD ZF transceiver is also verified analytically.

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.

The Laplacian pyramid (LP) is a multiresolution representation introduced originally for images, and it has been used in many applications. A major shortcoming of the LP representation is that it is oversampled. The dependency among the LP coefficients is studied in this paper. It is shown that whenever the LP compression filter is interpolatory, the redundancy in the LP coefficients can be removed effortlessly by merely discarding some of the LP coefficients. Furthermore, it turns out that the remaining, now critically sampled, LP coefficients are actually the coefficients of a wavelet filter bank. As a result, a new algorithm for designing a nonredundant wavelet filter bank from non-biorthogonal lowpass filters is obtained. Our methodology presented in this paper does not depend on the spatial dimension of the data or the dilation matrix for sampling.

This paper analyzes the convergence of the constant modulus algorithm (CMA) in a decision feedback equalizer using only a feedback filter. Several works had already observed that the CMA presented a better performance than decision directed algorithm in the adaptation of the decision feedback equalizer, but theoretical analysis always showed to be difficult specially due to the analytical difficulties presented by the constant modulus criterion. In this paper, we surmount such obstacle by using a recent result concerning the CM analysis, first obtained in a linear finite impulse response context with the objective of comparing its solutions to the ones obtained through the Wiener criterion. The theoretical analysis presented here confirms the robustness of the CMA when applied to the adaptation of the decision feedback equalizer and also defines a class of channels for which the algorithm will suffer from ill-convergence when initialized at the origin.

In this paper, we design robust precoders, under the minimum mean square error (MMSE) criterion, for different types of channel state information (CSI) in multiple-input multiple-output (MIMO) channels. We consider low-complexity pre-fixed receivers that may adapt to the channel but are oblivious to the existence of a precoder at the transmitter. In particular, three types of CSI are taken into account: i) perfect CSI, ii) statistical CSI in the form of mean feedback, and iii) deterministic imperfect CSI assuming that the actual channel is within the neighborhood of a nominal channel, which leads to the worst-case robust design that is the focus of this paper. Interestingly, it is found that, under some mild conditions, the optimal transmit directions, i.e., the left singular vectors of the precoder, are equal to the right singular vectors of the channel, the channel mean, and the nominal channel for perfect CSI, statistical CSI, and the worst-case design, respectively. Consequently, the matrix-valued problems can be simplified to scalar power allocation problems that either admit closed-form solutions or can be efficiently solved by the proposed algorithm.

Sparse graphical models have proven to be a flexible class of multivariate probability models for approximating high-dimensional distributions. In this paper, we propose techniques to exploit this modeling ability for binary classification by discriminatively learning such models from labeled training data, i.e., using both positive and negative samples to optimize for the structures of the two models. We motivate why it is difficult to adapt existing generative methods, and propose an alternative method consisting of two parts. First, we develop a novel method to learn tree-structured graphical models which optimizes an approximation of the log-likelihood ratio. We also formulate a joint objective to learn a nested sequence of optimal forests-structured models. Second, we construct a classifier by using ideas from boosting to learn a set of discriminative trees. The final classifier can interpreted as a likelihood ratio test between two models with a larger set of pairwise features. We use cross-validation to determine the optimal number of edges in the final model. The algorithm presented in this paper also provides a method to identify a subset of the edges that are most salient for discrimination. Experiments show that the proposed procedure outperforms generative methods such as Tree Augmented Naive Bayes and Chow-Liu as well as their boosted counterparts.

Limited feedback improves link reliability with a small amount of feedback from the receiver back to the transmitter. In cellular systems, the performance of limited feedback will be degraded in the presence of other cell interference, when the base stations have limited or no coordination. This paper establishes the degradation in sum rate of users in a cellular system, due to uncoordinated other cell interference and delay on the feedback channel. A goodput metric is defined as the rate when the bits are successfully received at the mobile station, and used to derive an upper bound on the performance of limited feedback systems with delay. This paper shows that the goodput gained from having delayed limited feedback decreases doubly exponentially as the delay increases. The analysis is extended to precoded spatial multiplexing systems where it is shown that the same upper bound can be used to evaluate the decay in the achievable sum rate. To reduce the effects of interference, zero forcing interference cancellation is applied at the receiver, where it is shown that the effect of the interference on the achievable sum rate can be suppressed by nulling out the interferer. Numerical results show that the decay rate of the goodput decreases when the codebook quantization size increases and when the doppler shift in the channel decreases., Comment: 30 pages, submitted to IEEE transactions on Signal Processing

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.

This paper considers the statistical performance of the MUSIC method under the condition that two closely spaced sources impinging on an array of sensors are effectively resolved, i.e., the spectrum exhibits two peaks in the neighborhood of the true directions-of-arrival (DOA). The MUSIC algorithm is known to have an infinite resolution power in theory. However, in the presence of modeling errors, sources can not be resolved with certainty, even if the array correlation matrix is perfectly known. The focus of this paper is to predict the bias and variance of the DOA estimates taking into account the possible resolution failure of MUSIC. This performance prediction, based on our recent mathematical investigation, is new to the best of our knowledge. A general mathematical framework to derive closed form expressions of the bias and variance versus the model mismatch, conditioned on a general statistical resolution test is proposed. In order to illustrate our mathematical approach, statistical tests with one and two conditions, respectively, are investigated. The accuracy of the performance prediction is illustrated in a simulation study. It is found that the proposed approach outperforms the “classical” technique, which ignores the possible resolution failure of the MUSIC algorithm. Therefore, our results provide better tools for determining the necessary antenna calibration accuracy to achieve some targeted specifications on the estimator performance.

This paper deals with the Bayesian signal denoising problem, assuming a prior based on a sparse representation modeling over a unitary dictionary. It is well known that the maximum a posteriori probability (MAP) estimator in such a case has a closed-form solution based on a simple shrinkage. The focus in this paper is on the better performing and less familiar minimum-mean-squared-error (MMSE) estimator. We show that this estimator also leads to a simple formula, in the form of a plain recursive expression for evaluating the contribution of every atom in the solution. An extension of the model to real-world signals is also offered, considering heteroscedastic nonzero entries in the representation, and allowing varying probabilities for the chosen atoms and the overall cardinality of the sparse representation. The MAP and MMSE estimators are redeveloped for this extended model, again resulting in closed-form simple algorithms. Finally, the superiority of the MMSE estimator is demonstrated both on synthetically generated signals and on real-world signals (image patches).

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 is first in a series of papers studying a system-theoretic approach to the problem of reconstructing an analog signal from its samples. The idea, borrowed from earlier treatments in the control literature, is to address the problem as a hybrid model-matching problem in which performance is measured by system norms. In this paper we present the paradigm and revise underlying technical tools, such as the lifting technique and some topics of the operator theory. This material facilitates a systematic and unified treatment of a wide range of sampling and reconstruction problems, recovering many hitherto considered different solutions and leading to new results. Some of these applications are discussed in the second part.

This paper considers the blind maximum-likelihood (ML) detection problem for orthogonal space-time block codes (OSTBCs) in multiple-input multiple-output flat-fading channels. While the blind ML detection problem for general space-time codes is difficult to solve, it has been shown that for OSTBCs with constant modulus constellations, the blind ML detection problem can be formulated as a discrete quadratic program, and then handled by a powerful convex approximation technique known as semidefinite relaxation (SDR). In this paper, we turn our attention to the case of higher order QAM OSTBCs. Due to the nonconstant modulus nature of higher order QAM signals, the blind ML detection problem turns out to be a discrete Rayleigh quotient maximization problem, and as a result the current SDR technique is no longer directly applicable. We propose a linear fractional SDR (LFSDR) approach to this problem. This approach first relaxes the higher order QAM blind ML detection problem into a quasi-convex problem, followed by a simple solution approximation procedure. In general, quasi-convex problems are computationally more complex to solve than convex problems, but we show that an optimum solution of our quasi-convex problem can be efficiently obtained by solving a convex semidefinite program. The approximation accuracy of the proposed approach relative to other possible relaxation approaches is also studied. Simulation results are presented to demonstrate that the proposed LFSDR-based blind ML detector outperforms some existing suboptimal detectors and can yield promising performance even with a small to moderate number of code blocks.

In this paper, the robust H∞ filtering problem is studied for a class of uncertain nonlinear networked systems with both multiple stochastic time-varying communication delays and multiple packet dropouts. A sequence of random variables, all of which are mutually independent but obey Bernoulli distribution, are introduced to account for the randomly occurred communication delays. The packet dropout phenomenon occurs in a random way and the occurrence probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval. The discrete-time system under consideration is also subject to parameter uncertainties, state-dependent stochastic disturbances and sector-bounded nonlinearities. We aim to design a linear full-order filter such that the estimation error converges to zero exponentially in the mean square while the disturbance rejection attenuation is constrained to a give level by means of the H∞ performance index. Intensive stochastic analysis is carried out to obtain sufficient conditions for ensuring the exponential stability as well as prescribed H∞ performance for the overall filtering error dynamics, in the presence of random delays, random dropouts, nonlinearities, and the parameter uncertainties. These conditions are characterized in terms of the feasibility of a set of linear matrix inequalities (LMIs), and then the explicit expression is given for the desired filter parameters. Simulation results are employed to demonstrate the effectiveness of the proposed filter design technique in this paper.

Group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two-channel perfect reconstruction finite-impulse-response (FIR) filter banks. The lifting factorizations generated by a group lifting structure are characterized by Abelian groups of lower and upper triangular lifting matrices, an Abelian group of unimodular gain scaling matrices, and a set of base filter banks. Examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two-channel linear phase FIR filter banks, the whole- and half-sample symmetric classes, including both the reversible and irreversible cases. This covers the lifting specifications for whole-sample symmetric filter banks in Parts 1 and 2 of the ISO/IEC JPEG 2000 still image coding standard. The theory is used to address the uniqueness of lifting factorizations. With no constraints on the lifting process, it is shown that lifting factorizations are highly nonunique. When certain hypotheses developed in the paper are satisfied, however, lifting factorizations generated by a group lifting structure are shown to be unique. A companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole- and half-sample symmetric filter banks., 19 pages, 3 figures

Time synchronization and localization are two important issues in wireless sensor networks. Although these two problems share many aspects in common, they are traditionally treated separately. In this paper, we present a unified framework to jointly solve time synchronization and localization problems at the same time. Furthermore, since the accuracy of synchronization and localization is very sensitive to the accuracy of anchor timings and locations, the joint time synchronization and localization problem with inaccurate anchors is also considered in this paper. For the case with accurate anchors, the joint maximum likelihood estimator and a more computationally efficient least squares (LS) estimator are proposed. When the anchor timings and locations are inaccurate, a generalized total least squares (GTLS) scheme is proposed. Crame?r-Rao lower bounds (CRLBs) and the analytical mean square error (MSE) expressions of the LS based estimators are derived for both accurate and inaccurate anchor cases. Results show that the proposed joint estimators exhibit performances close to their respective CRLBs and outperform the separate time synchronization and localization approach. Furthermore, the derived analytical MSE expressions predict the performances of the proposed joint estimators very well.

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 studies the problem of distributed average consensus in sensor networks with quantized data and random link failures. To achieve consensus, dither (small noise) is added to the sensor states before quantization. When the quantizer range is unbounded (countable number of quantizer levels), stochastic approximation shows that consensus is asymptotically achieved with probability one and in mean square to a finite random variable. We show that the meansquared error (m.s.e.) can be made arbitrarily small by tuning the link weight sequence, at a cost of the convergence rate of the algorithm. To study dithered consensus with random links when the range of the quantizer is bounded, we establish uniform boundedness of the sample paths of the unbounded quantizer. This requires characterization of the statistical properties of the supremum taken over the sample paths of the state of the quantizer. This is accomplished by splitting the state vector of the quantizer in two components: one along the consensus subspace and the other along the subspace orthogonal to the consensus subspace. The proofs use maximal inequalities for submartingale and supermartingale sequences. From these, we derive probability bounds on the excursions of the two subsequences, from which probability bounds on the excursions of the quantizer state vector follow. The paper shows how to use these probability bounds to design the quantizer parameters and to explore tradeoffs among the number of quantizer levels, the size of the quantization steps, the desired probability of saturation, and the desired level of accuracy $\epsilon$ away from consensus. Finally, the paper illustrates the quantizer design with a numerical study., Comment: 31 pages, 2 figures. Original version submitted Nov, 2007. Revised on Aug, 2008. Re-revised on Sept. 2009. Accepted fpr piblication in the IEEE Transactions on Signal Processing with mandatory revisions

International audience; Blind recognition of communication parameters is a research topic of high importance for both military and civilian communication systems. Numerous studies about carrier frequency estimation, modulation recognition as well as channel identification are available in literature. This paper deals with the blind recognition of the space–time block coding (STBC) scheme used in multiple input–multiple-output (MIMO) communication systems. Assuming there is perfect synchronization at the receiver side, this paper proposes three maximum-likelihood (ML)-based approaches for STBC classification: the optimal classifier, the second-order statistic (SOS) classifier, and the code parameter (CP) classifier. While the optimal and the SOS approaches require ideal conditions, the CP classifier is well suited for the blind context where the communication parameters are unknown at the receiver side. Our simulations show that this blind classifier is more easily implemented and yields better performance than those available in literature.

This paper investigates the impulse response estimation of linear time-invariant (LTI) systems when only noisy finite-length input-output data of the system is available. The competing parametric candidates are the least square impulse response estimates of possibly different lengths. It is known that the presence of noise prohibits using model sets with large number of parameters as the resulting parameter estimation error can be quite large. Model selection methods acknowledge this problem, hence, they provide metrics to compare estimates in different model classes. Such metrics typically involve a combination of the available least-square output error, which decreases as the number of parameters increases, and a function that penalizes the size of the model. In this paper, we approach the model class selection problem from a different perspective that is closely related to the involved denoising problem. The method primarily focuses on estimating the parameter error in a given model class of finite order using the available least-square output error. We show that such an estimate, which is provided in terms of upper and lower bounds with certain level of confidence, contains the appropriate tradeoffs between the bias and variance of the estimation error. Consequently, these measures can be used as the basis for model comparison and model selection. Furthermore, we demonstrate how this approach reduces to the celebrated AIC method for a specific confidence level. The performance of the method as the noise variance and/or the data length varies is explored, and consistency of the approach as the data length grows is analyzed.

This paper proposes a Discrete TFD definition that satisfies all desirable properties and is (approximately) equal to samples of the continuous WVD convolved with a time-frequency kernel. Also, the proposed definition—unlike others—contains a DWVD definition that satisfies important DWVD properties such as Moyal’s formula and signal recovery. The proposed definition can be computed by simple and efficient algorithms. Matlab and Octave code to accompany this paper is available to download at http://www.ieeexplore.ieee.org/. (Additional details can be found in the comprehensive book on Time-Frequency Signal Analysis and Processing (see http://www.elsevier.com/locate/isbn/0080443354). In addition, the most recent upgrade of the original software package that calculates Time-Frequency Distributions and Instantaneous Frequency estimators can be downloaded from the web site: www.time-frequency.net. This was the first software developed in the field, and it was first released publicly in 1987 at the 1st ISSPA conference held in Brisbane, Australia, and then continuously updated). Computation of a time-frequency distribution (TFD) requires a discrete version of the continuous distribution. This discrete TFD (DTFD) should be free from aliasing and conserve all the important mathematical properties of the continuous distribution. Existing DTFD definitions, however, poorly approximate this ideal. One popular definition, the generalized DTFD (GDTFD), is alias free but does not retain all the desirable properties from the continuous distribution. Another definition, the so-called alias-free GDTFD (AF-GDTFD), retains most properties yet is not always alias free. We propose a new DTFD definition, based on the GDTFD, that retains all desirable properties and is always alias free.

The decoding of a class of discrete cosine transform (DCT) and discrete sine transform (DST) codes that are maximum distance separable codes (MDS) is considered in this paper. These class of codes are considered for error correction over real fields. All the existing algebraic decoding algorithms are capable of decoding only a subclass of these codes [which can be characterized into the Bose-Chaudhuri-Hocquenghem (BCH) form], and fails to decode the remaining even though they are MDS. In this paper, we propose a new generic algorithm along the lines of coding theoretic and subspace methods to decode the entire class of MDS DCT and DST codes. The proposed subspace approaches are similar to popular ESPRIT and MUSIC algorithms. The proposed algorithms also perform significantly better than the existing algorithms on the BCH-like subclass. A perturbation analysis is also presented to study the effect of various parameters on the error localization due to the quantization noise. Simulation results are presented to demonstrate the capability of proposed algorithms to decode the entire class and to perform significantly better on the BCH-like subclass than the existing algorithm under the influence of quantization noise.

Information theory, and particularly the mutual information (MI), has provided fundamental guidance for communications research. In Bell's 1993 paper, the MI was first applied to radar waveform design. Similar to its communications counterpart, the solution comes in a water-filling form. However, the practical meaning of MI in the sensing context remains unclear to date. Recently, Yang and Blum's 2007 paper shows that under the white noise assumption, the optimum water-filling scheme simultaneously maximizes the MI and minimizes the estimation minimum mean square error (MMSE). Such an equivalence, however, does not hold when the target parameter statistics are not perfectly known as shown in Yang and Blum's subsequent work. To further the understanding of the practical meaning of MI and to establish a connection between the MI and commonly adopted MSE measures for sensing, this paper takes a fresh look at the target estimation problem. We consider the general colored noise, incorporate the normalized MSE (NMSE), and develop joint robust designs for both the transmitter (waveforms) and the receiver (estimator) under various target and noise uncertainty models. Our results show that i) the optimum waveform designs resulted from the MI, MMSE and NMSE criteria are all different and ii) compared to MMSE, the NMSE-based designs share more similarities with the MI-based ones, especially when the target and noise statistics are not perfectly known.

In this paper, we address the problem of redundancy reduction of high-dimensional noisy signals that may contain anomaly (rare) vectors, which we wish to preserve. Since anomaly data vectors contribute weakly to the l2-norm of the signal as compared to the noise, l2 -based criteria are unsatisfactory for obtaining a good representation of these vectors. As a remedy, a new approach, named Min-Max-SVD (MX-SVD) was recently proposed for signal-subspace estimation by attempting to minimize the maximum of data-residual l2-norms, denoted as l2,l and designed to represent well both abundant and anomaly measurements. However, the MX-SVD algorithm is greedy and only approximately minimizes the proposed l2,l-norm of the residuals. In this paper we develop an optimal algorithm for the minization of the l2,l-norm of data misrepresentation residuals, which we call Maximum Orthogonal complements Optimal Subspace Estimation (MOOSE). The optimization is performed via a natural conjugate gradient learning approach carried out on the set of n dimensional subspaces in IRm, m ≫ n, which is a Grassmann manifold. The results of applying MOOSE, MX-SVD, and l2- based approaches are demonstrated both on simulated and real hyperspectral data.

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 deals with the problem of uncertainties in the periodicities of linear almost-periodically time-variant (LAPTV) filters. These filters are usually implemented as a set of branches, each consisting of a frequency shifter followed by a linear time-invariant (LTI) filter. This implementation is also known as FRESH filters. This paper is motivated by the fact that, when there exist errors in the frequency shifts, the optimum set of LTI filters is obtained by canceling the outputs of the corresponding branches. The purpose of this paper is to analyze the nonstationary behavior of adaptive filters in order to mitigate this problem. Our results show that an adaptive filter can offset the errors in the frequency shifts. The reason is that the coefficients of the adaptive filter are updated so that the filter actually performs as a linear periodically time-variant filter for each branch. This allows to track the errors in the frequency shifts when the rate of convergence of the adaptive algorithm is suitably selected. An analytical study of the convergence is presented, which allows to compute the optimal rate of convergence and the mean squared-error attained by the adaptive filter.

Due to their linear-phase property, symmetric filters are an interesting class of finite-impulse-response (FIR) filters. Moreover, symmetric FIR filters allow an efficient implementation. In this paper we extend the classical definition of Hermitian symmetry to a more general symmetry that is also applicable to complex filters. This symmetry is called generalized-Hermitian symmetry. We show the usefulness of this definition as it allows for a unified treatment of even and odd-length filters. Central in this paper is a theorem on the reduction of generalized-Hermitian-symmetric filters to Hermitian-symmetric filters, both with finite precision coefficients. A constructive proof of this theorem is presented and an associated procedure for reducing generalized-Hermitian-symmetric filters is derived. Two of the examples show the application of the reduction procedure and the achieved savings on arithmetic costs. Finally, all three examples show that a special instance of the generalized-Hermitian-symmetric filters with finite precision coefficients, may have lower arithmetic costs than the Hermitian-symmetric filter from which it is derived.

Low complexity optimal (or nearly optimal) decoders for space-time codes have recently been under intensive investigation. For example, recent works by Sirianunpiboon and others show that the Silver code and the Golden code can be decoded optimally (or nearly optimally) with quadratic decoding complexity. Fast decodability makes them very attractive in practice. In implementing these decoders, floating-point to fixed-point conversion (FFC) needs to be carefully undertaken to minimize hardware cost while retaining decoding performance. The process of quantization for fixed-point representations is often ignored by research community and lacks investigation, and so FFC is often conducted heuristically based on simulations. This paper studies the effects of quantization to space-time coded systems from an information theoretic perspective. It shows the analytical relationship between quantization error and decoding performance deterioration. This paper also proposes a general finite precision implementation methodology including two FFC criteria for space-time coded systems within an integer optimization framework. As a particular example, this paper examines the finite precision implementation of the quadratic optimal decoding algorithm of the Silver code. However, our methodology and techniques can be applied to general space-time codes.

This paper considers the problem of time difference-of-arrival (TDOA) source localization when the TDOA measurements from multiple disjoint sources are subject to the same sensor position displacements from the available sensor positions. This is a challenging problem and closed-form solution with good localization accuracy has yet to be found. This paper proposes an estimator that can achieve this purpose. The proposed algorithm jointly estimates the unknown source and sensor positions to take the advantage that the TDOAs from different sources have the same sensor position displacements. The joint estimation is a highly nonlinear problem due to the coupling of source and sensor positions in the measurement equations. We introduce the novel idea of hypothesized source locations in the algorithm development to enable the formulation of psuedolinear equations, thereby leading to the establishment of closed-form solution for source location estimates. Besides the advantage of closed-form, the newly developed algorithm is shown analytically, under the condition that the TDOA measurement noise and the sensor position errors are sufficiently small, to reach the CRLB accuracy. For clarity, the localization of two disjoint sources is used in the algorithm development. The developed algorithm is then examined under the special case of a single source and extended to the more general case of more than two unknown sources. The theoretical developments are supported by simulations.

This paper considers the problem of recovering a sparse signal representation according to a signal dictionary. This problem could be formalized as a penalized least-squares problem in which sparsity is usually induced by a lscr1-norm penalty on the coefficients. Such an approach known as the Lasso or Basis Pursuit Denoising has been shown to perform reasonably well in some situations. However, it was also proved that nonconvex penalties like the pseudo lscrq-norm with q < 1 or smoothly clipped absolute deviation (SCAD) penalty are able to recover sparsity in a more efficient way than the Lasso. Several algorithms have been proposed for solving the resulting nonconvex least-squares problem. This paper proposes a generic algorithm to address such a sparsity recovery problem for some class of nonconvex penalties. Our main contribution is that the proposed methodology is based on an iterative algorithm which solves at each iteration a convex weighted Lasso problem. It relies on the family of nonconvex penalties which can be decomposed as a difference of convex functions (DC). This allows us to apply DC programming which is a generic and principled way for solving nonsmooth and nonconvex optimization problem. We also show that several algorithms in the literature dealing with nonconvex penalties are particular instances of our algorithm. Experimental results demonstrate the effectiveness of the proposed generic framework compared to existing algorithms, including iterative reweighted least-squares methods.

This paper introduces Bayes risk (expected loss) as a criterion for linear equalization. Since the probability of error is equal to the Bayes risk (BR) for a particular binary loss function, this work is a natural generalization of previous works on minimum probability of error (PE) equalizers. Adaptive equalization algorithms are developed that minimize the BR. Like the minimum PE equalizers, the BR algorithms have low computational complexity which is comparable to that of the LMS algorithm. The advantage of the BR criterion is that the loss function can be specified in a manner that accelerates adaptive equalizer convergence relative to the minimum PE adaptive algorithm as illustrated in simulation examples. Besides introducing a new criterion, this paper provides another independent contribution to the field of PE minimizing equalization. While most prior works focus on M-ary QAM type modulations with rectangular decision regions, this paper uses upper bounds on the probabilities of certain events to yield tractable mathematics that apply to two-dimensional constellations with arbitrarily shaped decision regions. The resulting adaptive algorithm use the full information available in the phase of the error signal, whereas previous algorithms use a quantized version of this error phase.

In this paper, we study the invertibility of M-variate Laurent polynomial N times P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multiple-input multiple-output systems, and multirate systems. Given an N times P Laurent polynomial matrix H(z1, ..., zM) of degree at most k, we want to find a P times N Laurent polynomial left inverse matrix G(z) of H(z) such that G(z)H(z) = J. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. The main result of this paper is to prove that H(z) is generically invertible when N - P ges M; whereas when N - P < M, then H(z) is generically noninvertible. As a result, we propose an algorithm to find a particular inverse of a Laurent polynomial matrix that is faster than current algorithms known to us.

An algorithm that is capable not only of tracking multiple targets but also of ldquotrack managementrdquo-meaning that it does not need to know the number of targets as a user input-is of considerable interest. In this paper we devise a recursive track-managed filter via a quantized state-space (ldquobinrdquo) model. In the limit, as the discretization implied by the bins becomes as refined as possible (infinitesimal bins) we find that the filter equations are identical to Mahler's probability hypothesis density (PHD) filter, a novel track-managed filtering scheme that is attracting increasing attention. Thus, one contribution of this paper is an interpretation of, if not the PHD itself, at least what the PHD is doing. This does offer some intuitive appeal, but has some practical use as well: with this model it is possible to identify the PHD's ldquotarget-deathrdquo problem, and also the statistical inference structures of the PHD filters. To obviate the target death problem, PHD originator Mahler developed a new ldquocardinalizedrdquo version of PHD (CPHD). The second contribution of this paper is to extend the ldquobin-occupancyrdquo model such that the resulting recursive filter is identical to the cardinalized PHD filter.

This paper presents a kernelized version of the extended recursive least squares (EX-KRLS) algorithm which implements for the first time a general linear state model in reproducing kernel Hilbert spaces (RKHS), or equivalently a general nonlinear state model in the input space. The center piece of this development is a reformulation of the well known extended recursive least squares (EX-RLS) algorithm in RKHS which only requires inner product operations between input vectors, thus enabling the application of the kernel property (commonly known as the kernel trick). The first part of the paper presents a set of theorems that shows the generality of the approach. The EX-KRLS is preferable to 1) a standard kernel recursive least squares (KRLS) in applications that require tracking the state-vector of general linear state-space models in the kernel space, or 2) an EX-RLS when the application requires a nonlinear observation and state models. The second part of the paper compares the EX-KRLS in nonlinear Rayleigh multipath channel tracking and in Lorenz system modeling problem. We show that the proposed algorithm is able to outperform the standard KRLS and EX-RLS in both simulations.

Constrained least-squares design and constrained Chebyshev design of one- and two-dimensional nonlinear-phase FIR filters with prescribed phase error are considered in this paper by a unified semi-infinite positive-definite quadratic programming approach. In order to obtain unique optimal solutions, we propose to impose constraints on the complex approximation error and the phase error. By introducing a sigmoid phase-error constraint bound function, the group-delay error can be greatly reduced. A Goldfarb-Idnani based algorithm is presented to solve the semi-infinite positive-definite quadratic program resulting from the constrained least-squares design problem, and then applied after some modifications to the constrained Chebyshev design problem, which is proved in this paper to be equivalent also to a semi-infinite positive-definite quadratic program. Through design examples, the proposed method is compared with several existing methods. Simulation results demonstrate the effectiveness and efficiency of the proposed method.

This paper presents an analysis of the steady-state mean-square error (MSE) of the set-membership normalized least-mean square (SM-NLMS) algorithm with relaxation and regularization parameters. These parameters are introduced for the purpose of deriving in a unified way the steady-state MSE performances of the epsiv-normalized least mean square (epsiv-NLMS) algorithm and a special case of the adaptive parallel subgradient projection (PSP) algorithm. The approach of the paper is to employ the energy conservation relation as a starting point of our analysis. This relation enables us to avoid the transient analysis of the SM-NLMS algorithm, which is in general hard due to the nonlinearity of the SM-NLMS algorithm. As a result, a few nonlinear equations whose solutions are theoretical steady-state MSEs are derived, where two types of reasonable assumptions are introduced to overcome the nonlinearity of the SM-NLMS algorithm. Our results are generalizations of well-known results of the steady-state MSE of the epsiv-NLMS. Extensive simulations show the close agreement between our theories and experiments.

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.

Fractional delay (FD) filters are an important class of digital filters and are useful in various signal processing applications. This paper discusses a design problem of FD infinite-impulse-response (IIR) filters with the maxflat frequency response in frequency domain. First, a flatness condition of FD filters at an arbitrarily specified frequency point is described, and then a system of linear equations is derived from the flatness condition. Therefore, a set of filter coefficients can be easily obtained by solving this system of linear equations. For a special case in which the frequency response is required to be maxflat at omega = 0 or pi , a closed-form expression for its filter coefficients is derived by solving a linear system of Vandermonde equations. It is also shown that the existing maxflat FD finite-impulse-response (FIR) and IIR filters are special cases of the FD IIR filters proposed in this paper. Finally, some examples are presented to demonstrate the effectiveness of the proposed filters.

This paper proposes a novel evolution strategy for a genetic algorithm (GA). This new algorithm is then applied to design robust D(alpha,r)-stable infinite-impulse-response (IIR) filters. Unlike existing research on designing IIR filters by using GA, in which the stability of IIR filters is tested by trial and error after the evolution of each generation of a GA, the stability criterion in this paper is embedded within the evolution of each generation. Consequently, the stability of this system can be guaranteed without the need for any other checks of the stability criterion in the evolution of each generation. Numerical experimental results are discussed to illustrate the soundness of the proposed evolution strategy. The robustness of the IIR filters is achieved by ensuring that all poles of the filters are located inside a disk D(alpha,r) contained in the unit circle, in which alpha is the center, r is the radius of the disk and IalphaI +r < 1 . So, in this paper, a D(alpha,r)-stability criterion will be first derived and then embedded in the GA for the design of robust IIR filters. Finally, two examples will be presented to show that the designed filters remain D(alpha,r)-stable during the evolution of the GA and will provide satisfactory results.

The oversampled Gabor transform is more effective than the critically sampled one in many applications. The biorthogonality relationship between the analysis window and the synthesis window of the Gabor transform represents the completeness condition. However, the traditional discrete cosine transform (DCT)-based real-valued discrete Gabor transform (RGDT) is available only in the critically sampled case and its biorthogonality relationship for the transform has not been unveiled. To bridge these important gaps, this paper proposes a novel DCT-based RDGT, which can be applied in both the critically sampled case and the oversampled case, and their biorthogonality relationships can be derived. The proposed DCT-based RDGT involves only real operations and can utilize fast DCT algorithms for computation, which facilitates computation and implementation by hardware or software as compared to that of the traditional complex-valued discrete Gabor transform. This paper also develops block time-recursive algorithms for the efficient and fast computation of the RDGT and its inverse transform. Unified parallel lattice structures for the implementation of these algorithms are presented. Computational complexity analysis and comparisons have shown that the proposed algorithms provide a more efficient and faster approach for discrete Gabor transforms as compared to those of the existing discrete Gabor transform algorithms. In addition, an application in the noise reduction of the nuclear magnetic resonance free induction decay signals is presented to show the efficiency of the proposed RDGT for time-frequency analysis.

In order to find sparse approximations of signals, an appropriate generative model for the signal class has to be known. If the model is unknown, it can be adapted using a set of training samples. This paper presents a novel method for dictionary learning and extends the learning problem by introducing different constraints on the dictionary. The convergence of the proposed method to a fixed point is guaranteed, unless the accumulation points form a continuum. This holds for different sparsity measures. The majorization method is an optimization method that substitutes the original objective function with a surrogate function that is updated in each optimization step. This method has been used successfully in sparse approximation and statistical estimation [ e.g., expectation-maximization (EM)] problems. This paper shows that the majorization method can be used for the dictionary learning problem too. The proposed method is compared with other methods on both synthetic and real data and different constraints on the dictionary are compared. Simulations show the advantages of the proposed method over other currently available dictionary learning methods not only in terms of average performance but also in terms of computation time.

This is the second part of our paper. In this paper, we propose, model, and analyze the performance of a detector array for localizing far-field particle-emitting sources, which is inspired by but generalizes the compound eye of insects. The array consists of multiple eyelets, each having a conical module with a lens on its top and an inner subarray containing multiple particle detectors. Using a parametric measurement model introduced for the array in Part I, in this part we analytically and numerically analyze the statistical performance of the array. First, we compute the statistical Cramer-Rao bounds (CRBs) on the errors in estimating the direction of arrival of the incident particles; then we derive a lower bound on the mean-square angular error (MSAE) of source localization for any specific array configuration; thirdly, we consider two source-direction estimators, the maximum likelihood estimator (MLE) and the weighted direction estimator (WDE), and analyze their MSAE performance. In the numerical examples, we quantitatively compare the performance of the proposed array with the biological compound eye; show the array performance as a function of the array configuration variables; optimally design the array configuration; illustrate that the MLE asymptotically attains the performance bound, whereas the WDE is nearly optimal for sufficiently large SNR; and analyze the hardware efficiency by comparing the two MSAE bounds. Potential applications of this work include artificial vision in medicine or robotics, astronomy assisted, security, and particle communications.

This paper proposes an approach to deducing the floor plan of a building using only a set of noisy room dimension measurements that have been obtained indirectly through radar probes. It is not assumed that the set of measurements is complete. The key feature of this approach is a method of relating, via a graph-theoretical construction, a given floor plan topology to a model that depends linearly on a minimal set of parameters. Given a list of floor-plan topologies and a system of realizability constraints, it is possible to formulate this problem as a constrained least-squares optimization problem. An algorithm for generating all floor plan topologies with a specified number of rooms is presented, as well as an algorithm for deciding if two floor plan representations are isomorphic. The techniques in this paper are intended to serve as an exploration into the theoretical performance bounds on the floor plan estimation problem.