Minimum-variance deconvolution and maximum-likelihood deconvolutionfor nonwhite Bernoulli-Gaussian processes with a Joseph spectrum

Todoeschuck and Jensen (1988) recently reported that the reflectivity sequences, denoted p(k), calculated from some sonic logs are not white and have a power spectral density approximately proportional to frequency, called a Joseph spectrum. It is shown here how to compute the minimum-variance estimate and maximum-likelihood estimate for a μ(k) modeled as a nonwhite Bernoulli-Gaussian (B-G) process with a Joseph spectrum. Also presented are the corresponding estimates for a statistically equivalent white B-G process μ*(k) which mimics μ(k). Some conclusions regarding the acceptability of these estimates are drawn     

Blind deconvolution of symmetric noncausal impulse responses usingtwo-sided linear prediction

Considers the problem of estimating the time-symmetric, noncausal impulse response of a linear time-invariant system from measurements of the response of the system to an unknown input signal, which is assumed to be a realization of a white random process. The symmetric impulse response is modeled by a two-sided AR or ARMA system model. The two-sided AR coefficients are estimated using a two-step procedure. First, an estimate of an unconstrained parameter vector is computed by solving a close-to-Toeplitz-plus-Hankel system of equations using previously developed fast algorithms. Then, the polynomial square root of the result is obtained by solving a constrained least-squares problem which has a simple solution. Unlike previous methods, this approach requires no iterative procedure. However, it may lead to an unstable model in some extreme cases. Simulation results illustrate the performance of the proposed methods     

Minimum-variance time-frequency distribution kernels

When dealing with random processes, reduced spectrum estimate variance becomes an important property that augments the list of desirable time-frequency (t-f) distribution properties. In this correspondence, we derive the t-f kernel that satisfies the t-f constraints and provides the minimum variance for the power spectrum estimate for Gaussian white noise processes     

基于UMVE算法的恒虚警检测器

基于无偏最小方差估计（UMVE）算法，提出了一种新的恒虚警检测器（UMVEM-CFAR）。它的前沿和后沿滑窗均采用UMVE算法来产生局部估计，再对两者求和得到背景功率水平估计。在SwerlingⅡ型目标假设下，推导出UMVEM-CFAR在均匀背景下虚警概率Pfa和检测概率Pd及多目标环境下检测概率只的解析表达式，与OS—CFAR相比，UMVEM在均匀背景和多目标环境下均具有最好的检测性能，并且它的处理时间只有OS的一半。     

Sample-by-sample approach to recursive noncausal filtering

A sample-by-sample processing technique is proposed for recursive noncausal filter implementation. Compared with the conventional block-processing approach, the proposed method has a small basic group delay and small memory size requirements.     

Two-dimensional spectrum estimation using noncausal autoregressive models

Two-dimensional (2-D) spectrum estimation from raw data is of interest in signal and image processing. A parametric technique for spectrum estimation using 2-D noncausal autoregressive (NCAR) models is given. The NCAR models characterize the statistical dependency of the observation at location s on its neighbors in all directions. This modeling assumption reduces the spectrum estimation problem to two subproblems: the choice of appropriate structure of the NCAR model and the estimation of parameters in NCAR models. By assuming that the true structure of the NCAR model is known, we first analyze the existence and uniqueness of Gaussian maximum likelihood (GML) estimates of NCAR model parameters. Due to the noncausal nature of the models, the computation of GML estimates is burdensome. By assuming specific boundary conditions, computationally tractable expressions are obtained for the likelihood function. Expressions for the asymptotic covariance matrix of the GML estimates as well as the simultaneous confidence bands for the estimated spectrum using GML estimates are derived. Finally, the usefulness of the method is illustrated by computer simulation results.     

Optimal and robust noncausal filter formulations

The paper describes an optimal minimum-variance noncausal filter or fixed-interval smoother. The optimal solution involves a cascade of a Kalman predictor and an adjoint Kalman predictor. A robust smoother involving H/sub /spl infin// predictors is also described. Filter asymptotes are developed for output estimation and input estimation problems which yield bounds on the spectrum of the estimation error. These bounds lead to a priori estimates for the scalar /spl gamma/ in the H/sub /spl infin// filter and smoother design. The results of simulation studies are presented, which demonstrate that optimal, robust, and extended Kalman smoothers can provide performance benefits.     

Lagrange wavelets for signal processing

This paper deals with the design of interpolating wavelets based on a variety of Lagrange functions, combined with novel signal processing techniques for digital imaging. Halfband Lagrange wavelets, B-spline Lagrange wavelets and Gaussian Lagrange (Lagrange distributed approximating functional (DAF)) wavelets are presented as specific examples of the generalized Lagrange wavelets. Our approach combines the perceptually dependent visual group normalization (VGN) technique and a softer logic masking (SLM) method. These are utilized to rescale the wavelet coefficients, remove perceptual redundancy and obtain good visual performance for digital image processing.     

Analysis of identified 2-D noncausal models

There are two approaches to the identification of noncausal autoregressive systems in two dimensions differing in the assumed noise model. For both approaches, the maximum likelihood estimator formulated in the frequency domain is presented. The Fisher information matrix is evaluated and found to be the sum of a block-Toeplitz and a block-Hankel matrix. The variance of the parameters, however, cannot be used for comparison of the two approaches, so the variance in the frequency domain is evaluated, assuming that the true system in each case can be described by a model of that type, possibly high-order. In particular, the variance of the spectrum estimate is derived. If the number of parameters tends to infinity, it is shown that the two approaches give the same spectrum estimate variance. The question of which set of true spectra can be described by the respective approaches is discussed     

Nonminimum phase channel equalization using noncausal filters

The Viterbi algorithm is the optimum method for detection of a data sequence in the presence of intersymbol interference and additive white Gaussian noise. Since its computational complexity is very large, several simplifications and alternative methods have been proposed, most of which are more effective when dealing with minimum phase channels. We present a novel technique for the equalization of nonminimum phase channels that employs noncausal all-pass filters operating in reversed time. The impulse response of the equalized channel approximates a minimum phase sequence with higher energy concentration at its left-hand end than at the right-hand end. The method can be modified to obtain a desired impulse response with few nonzero samples with only minor variations in noise level, providing significant complexity reduction in the Viterbi algorithm for detection. In addition, a twopass decoding strategy is developed, leading to significant improvement in performance with little increase in computational cost. Simulation results are included to verify the advantages of the proposed techniques     

Parameter estimation for noncausal ARMA models of non-Gaussiansignals via cumulant matching

We consider the problem of estimating the parameters of a stable (stationary), scalar ARMA(p,q) signal model driven by an i.i.d. non-Gaussian sequence. The driving noise sequence is not observed. The signal is allowed to be nonminimum phase and/or noncausal (i.e., poles may lie both inside as well as outside the unit circle). We address the problem of parameter identifiability given the higher order cumulants of the signal on a finite set of lags. The sufficient set of lags required to achieve parameter identifiability is the smallest to date. The sufficient conditions for parameter identifiability are also the least restrictive to date. We also propose a frequency-domain approach for time-domain, nonlinear optimization of a quadratic cumulant matching criterion. Illustrative computer simulation results are presented     

Relative optimization for blind deconvolution

We propose a relative optimization framework for quasi-maximum likelihood (QML) blind deconvolution and the relative Newton method as its particular instance. Special Hessian structure allows fast Newton system construction and solution, resulting in a fast-convergent algorithm with iteration complexity comparable to that of gradient methods. We also propose the use of rational infinite impulse response (IIR) restoration kernels, which constitute a richer family of filters than the traditionally used finite impulse response (FIR) kernels. We discuss different choices of nonlinear functions that are suitable for deconvolution of super- and sub-Gaussian sources and formulate the conditions under which the QML estimation is stable. Simulation results demonstrate the efficiency of the proposed methods.     

Bayesian techniques for blind deconvolution

This paper introduces extended Bayesian filters (EBFs), a new family of blind deconvolution filters for digital communications. The blind deconvolution problem is formulated as a nonlinear and non-Gaussian fixed-lag minimum mean square error filtering problem, and the EBF is derived as a suboptimal recursive estimator. The model-based setting makes extensive use of the transmitted symbol and noise distributions. A key feature of the EBF is that the filter lag can be chosen to be larger than the channel length, while the complexity is exponential in a parameter which is typically chosen to be smaller than both the channel length and the filter lag. Extensive simulations characterizing the performance of EBFs in severe intersymbol interference channels are presented. The fast convergence and robust equalization of the EBFs are demonstrated for uncoded linearly modulated signals [e.g., differentially encoded quaternary phase shift keying (QPSK)] transmitted over unknown channels. Comparisons are made to other blind symbol-by-symbol demodulation algorithms. The results show that the EBF provides much better performance (at increased complexity) compared to the constant modulus algorithm and the extended Kalman filter, and achieves a better performance-complexity trade-off than other Bayesian demodulation algorithms. The simulations also show that the EBF is applicable with large constellations and shaped modulations     

Super-exponential methods for blind deconvolution

A class of iterative methods for solving the blind deconvolution problem, i.e. for recovering the input of an unknown possibly nonminimum-phase linear system by observation of its output, is presented. These methods are universal do not require prior knowledge of the input distribution, are computationally efficient and statistically stable, and converge to the desired solution regardless of initialization at a very fast rate. The effects of finite length of the data, finite length of the equalizer, and additive noise in the system on the attainable performance (intersymbol interference) are analyzed. It is shown that in many cases of practical interest the performance of the proposed methods is far superior to linear prediction methods even for minimum phase systems. Recursive and sequential algorithms are also developed, which allow real-time implementation and adaptive equalization of time-varying systems     

Gradient-driven update lifting for adaptive wavelets

Over the past few years, wavelets have become extremely popular in signal and image processing applications. The classical linear wavelet transform, however, performs a homogeneous smoothing of the signal contents which, in some cases, is not desirable. This has led to a growing interest in (nonlinear) wavelet representations that can preserve discontinuities, such as transitions and edges.

In this paper, we present the construction of adaptive wavelets by means of an extension of the lifting scheme. The basic idea is to choose the update filters according to some decision criterion which depends on the local characteristics of the input signal. We show that these adaptive schemes yield lower entropies than schemes with fixed update filters, a property that is highly relevant in the context of compression. Moreover, we analyze the effect of a scalar uniform quantization and the stability in such adaptive wavelet decompositions.     

Bussgang blind deconvolution for impulsive signals

Many blind deconvolution algorithms have been designed to extract digital communications signals corrupted by intersymbol interference (ISI). Such algorithms generally fail when applied to signals with impulsive characteristics, such as acoustic signals. While it is possible to stabilize such procedures in many cases by imposing unit-norm constraints on the adaptive equalizer coefficient vector, these modifications require costly divide and square-root operations. In this paper, we provide a theoretical analysis and explanation as to why unconstrained Bussgang-type algorithms are generally unsuitable for deconvolving impulsive signals. We then propose a novel modification of one such algorithm (the Sato algorithm) to enable it to deconvolve such signals. Our approach maintains the algorithmic simplicity of the Sato algorithm, requiring only additional multiplies and adds to implement. Sufficient conditions on the source signal distribution to guarantee local stability of the modified Sato algorithm about a deconvolving solution are derived. Computer simulations show the efficiency of the proposed approach as compared with various constrained and unconstrained blind deconvolution algorithms when deconvolving impulsive signals.     

Least-squares time-domain deconvolution for transversal-filter equalisers

The N tap weights for a transversal-filter equaliser are calculated efficiently by solving a Toeplitz set of linear equations which represents the discrete-time convolution of filter and system impulse responses. Computation time is proportional to N2, approximately. Discrete Fourier transforms are not used. For many practical designs, the procedure is suited to small, desk-top programmable calculators, since it requires only multiplications and additions of real numbers and about 5N data storage registers.     

A new method for optimal deconvolution

Using an innovation analysis method in the time domain, the problem of optimal input estimation is considered, Two algorithms for calculating optimal deconvolution estimators are presented. A new tool for obtaining the estimators is described. It is based on the projection method and innovation theory. The approach covers input prediction, filtering, and smoothing problems. The solution is also applied to unstable linear systems, disturbances, or input models     

Complex spatial filters for image deconvolution

The history and principles of image distortion correction by means of holographic spatial filtering are reviewed. Both predetection and postdetection filtering are considered. These and other principle applications of complex spatial filtering are assessed. The major problem areas are discussed, along with possible solutions.     

Phase-shifting for nonseparable 2-D Haar wavelets

In this paper, we present a novel and efficient solution to phase-shifting 2-D nonseparable Haar wavelet coefficients. While other methods either modify existing wavelets or introduce new ones to handle the lack of shift-invariance, we derive the explicit relationships between the coefficients of the shifted signal and those of the unshifted one. We then establish their computational complexity, and compare and demonstrate the superior performance of the proposed approach against classical interpolation tools in terms of accumulation of errors under successive shifting.