Model parameter estimation for reciprocal Gaussian random processes

The problem of estimating the model parameters of a discrete-index reciprocal Gaussian random process from a limited number of noisy observations is addressed. The general case of a first-order multivariate process is analyzed, stating its basic properties and deriving a linear equation set that relates the model parameters (including the unknown variance of the observation noise) to the (generally nonstationary) autocorrelation function of the observed process. It generalizes to the reciprocal processes the so-called `high-order Yule-Walker equations' for AR processes. Based on these results, a practical estimation algorithm is proposed     

Zero-mean proper complex Gaussian random processes through nonlinear channels

The statistical characterisation of the output of a memory less nonlinear device having, as an input, a zero mean proper complex Gaussian random process is studied. The closed form expression obtained for the high-order moments involved allowed for neat mathematical expressions for the autocorrelation function and the power spectral density of the signal at the nonlinearity output. Parts of the autocorrelation function and of the power spectral density that correspond to the desired signal and to the intermodulation products of different orders are easily identified.     

Theory of orthonormal M-band wavelet packets

In recent years, M-band orthonormal wavelet bases, due to their good characteristics, have attracted much attention. The ability of 2-band wavelet packets to decompose high frequency channels can be employed to improve the performance of wavelets for time-frequency localization, which makes more kinds of signals for analyzing by wavelets. Similar to the notations from the extension of 2-band wavelets to 2-band wavelet packets, the theoretic framework of M-band wavelet packets is developed, a generalization of the notations and properties of 2-band wavelet packets to that of M-band wavelet packets is made and the corresponding proofs are given.     

Time-varying lapped transforms and wavelet packets

The perfect reconstruction conditions for a time-varying lapped transform (paraunitary filter bank) are developed through the factorization of the transform matrix into sparse factors. A general formulation is presented allowing one to switch between two paraunitary filter banks. However, the extended lapped transform (ELT) is often used as an example. Furthermore, an adaptive wavelet packet is developed employing a time varying, tree association of ELTs. In all cases perfect reconstruction is inherently assured     

Local spectral analysis using wavelet packets

Wavelet packets are a useful extension of wavelets, which are of wide potential use in a statistical context. In this paper, an approach to the local spectral analysis of a stationary time series based on wavelet packet decomposition is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. Some asymptotic properties of the new estimate are provided. The technique is illustrated by simulated signals and its application to physiological data, and its potential use in studies of time-dependent spectral analysis is discussed.     

Time-frequency localisation of Shannon wavelet packets

It is well known that the 2/spl pi/ minimally supported frequency scaling function /spl phi//sup /spl alpha//(x) satisfying /spl phi//spl circ//sup /spl alpha//(/spl omega/)=/spl chi//sub (-/spl alpha/,2/spl pi/-/spl alpha/)/(/spl omega/), 0相似文献   

Some distributional properties of the continuous wavelet transformof random processes

Without finite moment conditions, some properties of random processes, such as stationarity and self-similarity, are characterized via corresponding properties of their wavelet transform. Anyone of these distributional properties of the wavelet transform characterizes the corresponding property of the increments of the random process, of order equal to the order of regularity of the analyzing wavelet. Extensions of these results to random fields are then indicated     

A matrix-valued wavelet KL-like expansion for wide-sense stationary random processes

Matrix-valued wavelet series expansions for wide-sense stationary processes are studied in this paper. The expansion coefficients a are uncorrelated matrix random process, which is a property similar to that of a matrix Karhunen-Loe/spl grave/ve (MKL) expansion. Unlike the MKL expansion, however, the matrix wavelet expansion does not require the solution of the eigen equation. This expansion also has advantages over the Fourier series, which is often used as an approximation to the MKL expansion in that it completely eliminates correlation. The basis functions of this expansion can be obtained easily from wavelets of the Matrix-valued Lemarie/spl acute/-Meyer type and the power-spectral density of the process.     

High-order wavelet packets and cumulant field analysis

In many applications it is necessary to characterize the statistical properties of the wavelet/wavelet packet coefficients of a stationary random signal. In particular, in a stationary non-Gaussian noise scenario it may be useful to determine the high-order statistics of the wavelet packet coefficients. In this work we prove that this task may be performed through multidimensional filter banks. In particular, we show how the cumulants of the M-band wavelet packet coefficients of a strictly stationary signal are derived from those of the signal and we provide scale-recursive decomposition and reconstruction formulae to compute these cumulants. High-order wavelet packets, associated with these multidimensional filter banks, are presented along with some of their properties. It is proved that under some conditions these high-order wavelet packets allow us to define frame multiresolution analyses. Finally, the asymptotic normality of the coefficients is studied by showing the geometric decay of their polyspectra/cumulants (of order greater than two) with respect to the resolution level     

基于Haar小波包的时域多分辨分析法

把基于Haar小波的时域多分辨分析(MRTD)推广到Haar小波包,扩展了基函数的选择范围,导出了一维小波包MRTD(WP-MRTD)的时间迭代格式,并给出WP-MRTD与传统FDTD的接口算法.计算结果显示,用小波包基作为电磁场展开函数和小波基相比可以获得能量更集中的展开系数,有利于进一步提高计算效率;对于给定级数的小波包二叉树,存在一个最佳小波包基,使得计算效率最高.该方法可直接推广到二维和三维问题.     

New aggressive way to search for the best base in wavelet packets

The binary-tree best base (BTBB) searching method developed by R. Coifman and M.V. Wickerhauser (1992) is well known and widely used in wavelet packet applications. However, the requirement that the base vectors be chosen from either a parent or its directly related children in the binary-tree structure is a limitation because it does not search all possible orthogonal bases and therefore may not provide an optimal result. We have recently found that the set of all possible orthogonal bases in a wavelet packet is much larger than the set searched by the BTBB method. Based on this observation, we have developed a tree-elimination based best orthogonal base (TBB) searching method, a new way to search the best base among a much larger set of orthogonal bases. We show that considerable improvements in signal compression, time-frequency analysis, and feature extraction may be achieved using the newly developed TBB method. Similar to the matching pursuit method (MP), TBB uses an aggressive searching method. However, its computation is faster than that of the orthogonal MP searching method.     

Synthesis of multivariate Gaussian random processes with a preassigned covariance (Corresp.)

Evaluation of complex systems in a laboratory environment requires the generation of inputs to the system sensors that are representative of the operational environment. It is therefore necessary to synthesize input test signals that reflect the mutual dependencies found in situ. For multivariate Gaussian inputs, algorithms are derived allowing 1) the transformation of dependent Gaussian random variables into independent variables; 2) the generation of jointly Gaussian random variables with a constant covariance matrix; and 3) the synthesis of stationary multivariate Gaussian random processes. These algorithms have simple electronic hardware and computer software implementations that will facilitate the laboratory evaluation and digital computer simulation of complex systems.     

Cramer-Rao bound for Gaussian random processes and applications toradar processing of atmospheric signals

Calculations of the exact Cramer-Rao bound (CRB) for unbiased estimates of the mean frequency, signal power, and spectral width of Doppler radar/lidar signals (a Gaussian random process) are presented. Approximate CRBs are derived using the discrete Fourier transform (DFT). These approximate results are equal to the exact CRB when the DFT coefficients are mutually uncorrelated. Previous high SNR limits for CRBs are shown to be inaccurate because the discrete summations cannot be approximated with integration. The performance of an approximate maximum likelihood estimator for mean frequency approaches the exact CRB for moderate SNR and moderate spectral width     

Image inpainting using directional wavelet packets originating from polynomial splines

The paper presents a new algorithm for the image inpainting problem. The algorithm uses a recently designed versatile library of quasi-analytic complex-valued wavelet packets (qWPs) which originate from polynomial splines of arbitrary orders. Tensor products of 1D qWPs provide a diversity of 2D qWPs oriented in multiple directions. For example, a set of the fourth-level qWPs comprises 62 different directions. The properties of these qWPs such as refined frequency resolution, directionality of waveforms with unlimited number of orientations, (anti-)symmetry of waveforms and windowed oscillating structure of waveforms with a variety of frequencies, make them efficient in image processing applications, in particular, in dealing with the inpainting problem addressed in the paper. The obtained results for this problem are quite competitive with the best state-of-the-art algorithms. The inpainting is implemented by an iterative scheme, which expands the Split Bregman Iteration (SBI) procedure by supplying it with an adaptive variable soft thresholding based on the Bivariate Shrinkage algorithm. In the inpainting experiments, performance comparison between the qWP-based methods and the state-of-the-art algorithms is presented.     

Efficient framework for model-based tomographic image reconstruction using wavelet packets

The use of model-based algorithms in tomographic imaging offers many advantages over analytical inversion methods. However, the relatively high computational complexity of model-based approaches often restricts their efficient implementation. In practice, many modern imaging modalities, such as computed-tomography, positron-emission tomography, or optoacoustic tomography, normally use a very large number of pixels/voxels for image reconstruction. Consequently, the size of the forward-model matrix hinders the use of many inversion algorithms. In this paper, we present a new framework for model-based tomographic reconstructions, which is based on a wavelet-packet representation of the imaged object and the acquired projection data. The frequency localization property of the wavelet-packet base leads to an approximately separable model matrix, for which reconstruction at each spatial frequency band is independent and requires only a fraction of the projection data. Thus, the large model matrix is effectively separated into a set of smaller matrices, facilitating the use of inversion schemes whose complexity is highly nonlinear with respect to matrix size. The performance of the new methodology is demonstrated for the case of 2-D optoacoustic tomography for both numerically generated and experimental data.     

Stationarizable random processes

The familiar notion of inducing stationarity into a cyclostationary process by random translation is extended through characterization of the class of all second-order continuous-parameter processes (with autocorrelation functions that possess a generalized Fourier transform) that are {em stationarizable} in the wide sense by random translation. This class includes the nested set of proper subclasses: {em almost cyclostationary} processes, {em quasi-cyclostationary} processes, and {em cyclostationary} processes. The random translations that induce stationarity are also characterized. The concept of stationarizability is extended to the concept of asymptotic stationarizability, and the class of {em asymptotically stationarizable} processes is characterized. These characterizations are employed to derive characterizations of optimum linear and nonlinear time-invariant filters for nonstationary processes. Relative to optimum time-varying filters, these time-invariant filters offer advantages of implementational simplicity and computational efficiency, but at the expense of increased filtering error which in some applications is quite modest. The uses of a random translation for inducing stationarity-of-order-n, for increasing the degree of local stationarity, and for inducing stationarity into discrete-parameter processes are briefly described.     

Information-singular random processes

Random processes that generate information slower than linearly with time are termed information-singular. The study of information-singularity contributes to a more thorough understanding of the mathematical nature of information generation. Specifically, it elucidates the manner in which generation of information by a time series is critically dependent on the detailed behavior of the sample functions of its spectral representation. The main theorem states that any random sequence whose spectral representation has stationary independent increments with no Brownian motion component is information-singular in the mean-squared sense. The concept of information-singularity can be construed as a means for discriminating between deterministic and nondeterministic processes. It is felt that information-singularity fulfills this discriminating function in a physically more satisfying manner than does the classical Hilbert space theory of linear and nonlinear prediction. The desire for a still more satisfying discriminant motivates investigation of the class of random processes that retain their information-singularity even when corrupted by additive noise. In the case of strictly stationary processes, the discussion focuses on the relationship between information-singularity and zero entropy. Lastly, some alternative definitions of information-singularity are considered and several open problems are identified.