Wide-sense Stationary Process Research Articles

Signal models for TDOA/FDOA estimation

Much research has been done in the area of estimating time-difference-of-arrival (TDOA) and frequency-difference-of-arrival (FDOA) and their use in locating a radiating source. Early work in this area was focused on locating acoustic sources using passive sonar processing. Only later was TDOA/FDOA-based location considered for the case of passively locating electromagnetic sources. As a result of this, it is tempting to use results derived for the acoustic case when answering questions about the electromagnetic case. This correspondence shows that such borrowing can lead to incorrect results. The key factor that drives the significant differences between these two cases is the difference between the signal model assumptions for the two cases: wide-sense stationary (WSS) Gaussian process in the acoustic case and a deterministic signal in the electromagnetic case. Although the received signal equations may look identical (showing delay and Doppler shift), the resulting Fisher information, Cramer-Rao bound (CRB), and maximum likelihood estimator (MLE) are fundamentally different for the two signal scenarios.

A New Nonstationarity Detector

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A new test to determine the stationarity length of a locally wide sense stationary Gaussian random process is proposed. Based on the modeling of the process as a time-varying autoregressive process, the time-varying model parameters are tested using a Rao test. The use of a Rao test avoids the necessity of obtaining the maximum likelihood estimator of the model parameters under the alternative hypothesis, which is intractable. Computer simulation results are given to demonstrate its effectiveness and to verify the asymptotic theoretical performance of the test. Applications are to spectral analysis, noise estimation, and time series modeling. </para>

Gabor discretization of the Weyl product for modulation spaces and filtering of nonstationary stochastic processes

We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed, the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sjöstrand space M ω ∞ , 1 , then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on L 2 , the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time–frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter.

Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Dual-tree wavelet decompositions have recently gained much popularity, mainly due to their ability to provide an accurate directional analysis of images combined with a reduced redundancy. When the decomposition of a random process is performed -- which occurs in particular when an additive noise is corrupting the signal to be analyzed -- it is useful to characterize the statistical properties of the dual-tree wavelet coefficients of this process. As dual-tree decompositions constitute overcomplete frame expansions, correlation structures are introduced among the coefficients, even when a white noise is analyzed. In this paper, we show that it is possible to provide an accurate description of the covariance properties of the dual-tree coefficients of a wide-sense stationary process. The expressions of the (cross-)covariance sequences of the coefficients are derived in the one and two-dimensional cases. Asymptotic results are also provided, allowing to predict the behaviour of the second-order moments for large lag values or at coarse resolution. In addition, the cross-correlations between the primal and dual wavelets, which play a primary role in our theoretical analysis, are calculated for a number of classical wavelet families. Simulation results are finally provided to validate these results.

On the Relationship Between Correlation Length and Rate of Fluctuation of Random Fields

We show that the correlation length of a random field, modelled as a real wide-sense stationary stochastic process, defines a lower limit on the average squared rate of fluctuation of the field. For a sample-continuous process, the uncertainty relation between the correlation length and the spectral bandwidth exhibits a lower limit that depends explicitly on the selected correlation threshold level and implicitly on the functional form of the correlation function. We discuss implications of this result on how a selected correlation threshold relates to the estimated levels of field distortion and maximum amplitude of mode-stirred reverberation fields.

Compression Efficiency and Delay Tradeoffs for Hierarchical B-Pictures and Pulsed-Quality Frames

Real-time video applications require tight bounds on end-to-end delay. Hierarchical bidirectional prediction requires buffering frames in the encoder input buffer, thereby contributing to encoder input delay. Long-term frame prediction with pulsed quality requires buffering at the encoder output, increasing the output buffer delay. Both hierarchical B-pictures and pulsed-quality coders involve uneven bit-rate allocation. Both the encoder and decoder buffering requirements depend on the rate allocation. We derive an efficient rate allocation for hierarchical B-pictures using the power spectral density of a wide-sense stationary process. In addition, we discuss important aspects of hierarchical predictive coding, such as the effect of the temporal prediction distance and delay tradeoffs for prediction branch truncation. Finally, we investigate experimentally the tradeoff between delay and compression efficiency.

TH-C-L100F-10: Monte Carlo Coupled Modeling of a Computed Radiography System

Purpose: To develop a model that allows for understanding the physical properties and the effect of readout parameters on objective image quality metrics of a ‘flying spot’ computed radiography (CR) system. Method and Materials: A Monte Carlo simulation method [Fasbender et al., Nucl Instrum Methods Phys Res A 512 (2003): 610–8] that describes the optical transport in a ‘flying spot’ readout was implemented by adapting the code of Jacques [Photochem Photobiol 67 (1998), 23–32]. This program was coupled with a ‘parallel-cascades’ approach [Yao and Cunningham, Med Phys 28 (2000): 2020–38] to predict system MTF and DQE. Grain size was assumed to follow a normal distribution in the imaging plate (IP) and granular noise was determined using the works of Nutting and Siedentopf. Additional assumptions include, orthogonal x-ray incidence, no scatter, no glare, no defects in the IP, a constant discharge fraction, and a linear readout amplifier were assumed for a linear and wide-sense stationary process. X-ray spectra representative of radiography and mammography were used in the simulations. Published or estimated parameters that correspond to two common IPs were compiled from multiple sources. Results: For radiography with 0.2-mm sampling, model results showed good agreement with published literature values for system MTF (r>0.99) and moderate agreement for DQE (r = 0.75 to 0.99) over four orders of incident exposure (0.03 to 30-mR). For mammography with 0.1-mm sampling, model results showed good agreement with published literature values for system MTF (r>0.99) and for DQE (r = 0.95 to 0.99) over two orders of incident exposure (∼1 to 100-mR). Conclusion: The proposed model facilitates identifying factors that could improve CR performance. Accurate estimates of IP parameters and model improvements such as incorporation of Lubberts' effect could further improve model accuracy. Supported in part by NIH R01EB004015 and Georgia Cancer Coalition award.

The Generalized Wiener Process for Colored Noise

We define the generalized Wiener process as the output of a first-order differential equation when the input is an arbitrary stochastic input. This is in contrast to the standard Wiener process, where the input is white Gaussian noise. We obtain a simple explicit result for any input wide sense stationary random process, namely, that the Wigner spectrum of the output random process is the product of the power spectrum of the input process times a simple universal function of time and frequency. We also obtain the impulse response function wherein the output of the generalized process is expressed as the impulse response function integrated with the input random process. Various limiting values are derived. We apply the method to the case where the driving stochastic process has an exponentially decaying autocorrelation function

Nonparametric functionals of spectral distributions and their applications to time series analysis

There is a close analogy between empirical distributions of i.i.d. random variables and normalized spectral distributions of wide-sense stationary processes. Herein we make use of this analogy to develop nonparametric comparisons of two spectral distributions and nonparametric tests of stationarity versus change-point alternatives via spectral analysis of a time series.

On the prediction of vector-valued random fields and the spectral distribution of their evanescent component

Let ( X m , n ) ( m , n ) ∈ Z 2 be a C p -valued wide sense stationary process. We study the prediction theory of such processes according to different total orders on Z 2 . In the case of a “rational order”, we give the spectral distribution of the resulting evanescent component and prove that for two different rational orders, the resulting evanescent components are mutually orthogonal.

Performance of multicarrier DS/SSMA over fast Rayleigh fading channels with Doppler diversity

The significant loss in performance of multicarrier direct-sequence spread-spectrum multiple-access systems over fast-fading channels is investigated. First, the channel model for individual subchannels is obtained, using a canonical decomposition to a wide-sense stationary (WSS), uncorrelated scattering channel. Next, a receiver structure, which features both Doppler diversity and frequency diversity, is presented and analyzed. It is found that large Doppler spreads cause the average magnitude of the desired signals to diminish dramatically. Expressions for the bit-error rate are derived for both uncoded and coded systems by applying the standard Gaussian approximation. Numerical results, which show that Doppler diversity is preferable over frequency diversity in fast-fading channels, are provided for different combinations of the diversity orders. Numerical results, which show that the intersymbol and intersubchannel interference can be effectively suppressed, and the multiple-access interference tends to degenerate into a WSS process with the statistical periodicity destroyed by the fast-fading channels, are also provided.

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.

Multisine Transformation–Properties and Applications

Multisine time-series are sums of discrete-time sines with deterministic amplitudes and phase shifts chosen in a special way. In this paper a new recursive approach to synthesis and simulation of multisine time-series with predefined spectral properties, is presented. On the basis of the power spectral density function and phase shifts chosen with some well-defined properties, the finite discrete Fourier transform of multisine time-series is synthesised and the corresponding multisine time-series is simulated by performing an inverse discrete Fourier transform of the synthesised spectrum. In the approach the phase shifts needed to synthesise and simulate N-sample multisine random time-series are calculated using $$\frac{N}{2}$$ -sample multisine time-series obtained in the previous iteration. A transformation of the phase shifts into the corresponding N-sample multisine time-series is called a multisine transformation. Its properties are discussed. Applications of the multisine transformation to generation of uniformly distributed random numbers as well as to simulation of realisations for multisine approximations of wide-sense stationary random processes given by their power spectral densities are included.

Relative information of multi-rate sensors

Fusion of the information provided by distributed sensors is important in the future pervasive computing systems. The objective of this paper is to quantify the relative amount of information delivered by individual sensors in a distributed sensor system. To formalize our approach, we consider the problem of fusing the statistical information obtained using low-rate sensor recordings v i ( n ) in order to find the power spectrum of a high-rate random signal x ( n ). It turns out that this problem is inherently ill-posed since, in general, low-rate observations are not sufficient for specifying a unique solution. We use the Maximum Entropy principle to resolve this issue. It is known that a complete statistical description of a Gaussian wide-sense stationary process x ( n ) is provided by its power spectrum. This leads us to a measure of informativity for multirate sensors based on the qualities of the maximum entropy solution. We study the main properties of the proposed measure and provide computational methods for its calculation. Finally, we illustrate the concepts and methods discussed in the paper using a detailed simulation example.

Desempenho do Algoritimo LMS na Identificação de Canais Variantes no Tempo e seu Emprego em Esquemas de Recepção MJSE-PSP

An analysis of the steady-state mean square error (MSE) performance of the Least Mean Square (LMS) algorithm in the identification of time-varying channels is presented. It is based on a set hypotheses usually adopted in the mobile communications context. The channel impulse response is modelled as time-varying transversal filter whose coefficients are wide-sense stationary stochastic processes with known power (Doppler) spectra. A generic expression of the steady state MSE as a function of the LMS step-size parameter is obtained. Two specific Doppler spectrum models are considered for which the optimization of the LMS step-size parameter in the sense of MSE minimization is also addressed. Close match between numerical and analytical results is shown in several examples. Besides, the impact of the LMS step-size parameter optimization on the performance of an adaptive MLSE (Maximum Likelihood Sequence Estimation) receiver is also addressed by Monte Carlo simulation.

Reproducing Kernel Hilbert Space Methods for wide-sense self-similar Processes

It has recently been observed that wide-sense self-similar processes have a rich linear structure analogous to that of wide-sense stationary processes. In this paper, a reproducing kernel Hilbert space (RKHS) approach is used to characterize this structure. The RKHS associated with a selfsimilar process on a variety of simple index sets has a straightforward description, provided that the scale-spectrum of the process can be factored. This RKHS description makes use of the Mellin transform and linear self-similar systems in much the same way that Laplace transforms and linear time-invariant systems are used to study stationary processes. The RKHS results are applied to solve linear problems including projection, polynomial signal detection and polynomial amplitude estimation, for general wide-sense self-similar processes. These solutions are applied specifically to fractional Brownian motion (fBm). Minimum variance unbiased estimators are given for the amplitudes of polynomial trends in fBm, and two new innovations representations for fBm are presented. 1. Introduction. This paper is concerned with the linear problems associated with wide-sense self-similar processes, that is, with processes whose first and second moments are essentially scale-invariant. In the linear problems we are referring to, the solution is constrained to be a linear functional of an observed random process, and the criterion of optimality is such that the solution is determined by the first and second moments of the observed process. Examples include linear projection with the mean-square error metric, maximum signal-to-noise ratio signal detection and minimum-variance unbiased linear estimation. For Gaussian processes, these linear solutions are still optimal when the linearity constraint is removed. The main finding is that wide-sense self-similar processes have essentially the same structure as wide-sense stationary processes, and that concepts such as autocorrelation and spectral density, used to study stationary processes, have simple analogs useful in the study of self-similar processes. The connection between these two classes was first made by Lamperti (1962), who pointed out a simple invertible transformation which connects any self-similar process with a stationary counterpart, and which is a central idea in this paper. Other recent workmak ing use of Lamperti’s transformation includes Albin (1998),

Bayesian robust optimal linear filters

Qualitatively, a robust filter maintains acceptable performance for signals statistically close to those for which it has been designed. For a parameterized family of signal models, we measure the robustness as the increase in error from applying the optimal filter for one state to the model for a different state. A Bayesian approach results from assuming that the state space possesses a probability distribution. In this case, the mean robustness for a state is the expected value of the error increase when the optimal filter for the given state is applied over all states of nature. A maximally robust state is one whose mean robustness is minimal. This paper treats robustness for optimal linear filters, in which case optimality depends on second-order statistics. Formulation of robustness is achieved by placing the matter into the context of canonical representation of random functions. Specifically, we use the representation of the optimal linear filter in terms of the cross-correlation between the signal to be estimated and the white-noise expansion of the observed signal. The general robustness formulation is reduced in particular cases, such as for linear degradation models and wide-sense stationary processes. For wide-sense stationary processes, robustness becomes a function of the power spectral densities. A maximally robust state depends on both statistical characteristics of the model and the distribution of the state vector. By incorporating the characteristics and state distribution into filter design, one can define a global filter that has good performance across all states.

Using Variability Related to Families of Spectral Estimators for Mixed Random Processes

Traditionally, characterization of spectral information for wide sense stationary processes has been addressed by identifying a single best spectral estimator from a given family. If one were to observe significant variability in neighboring spectral estimators then the level of confidence in the chosen estimator would naturally be lessened. Such variability naturally occurs in the case of a mixed random process, since the influence of the point spectrum in a spectral density characterization arises in the form of approximations of Dirac delta functions. In this work we investigate the nature of the variability of the point spectrum related to three families of spectral estimators: Fourier transform of the truncated unbiased correlation estimator, the truncated periodogram, and the autoregressive estimator. We show that tones are a significant source of bias and variability. This is done in the context of Dirichlet and Fejer kernels, and with respect to order rates. We offer some expressions for estimating statistical and arithmetic variability. Finally, we include an example concerning helicopter vibration. These results are especially pertinent to mechanical systems settings wherein harmonic content is prevalent.

On the linear minimum-mean-squared-error estimation of an undersampled wide-sense stationary random process

We consider the problem of linearly estimating, in the sense of minimum-mean-squared error, a wide-sense stationary process in noise given uniformly spaced samples where the sampling interval is such that significant aliasing occurs. We derive the corresponding aliased Wiener filter and provide a technique for determining a closed form for the necessary power spectral density functions. We conclude with an example where both signal and noise are modeled using a second-order innovations representation.

Optimal subband filter banks for multiple description coding

Multiple description coding refers to the encoding of an information source into multiple bit streams, in a manner that each bit stream independently represents a coarse description of the source, while multiple descriptions jointly convey a higher fidelity source representation. Thus multiple description coding allows enhanced resilience to loss of individual descriptions, and can be viewed as a form of source-based diversity coding. This paper addresses the design of subband filter banks to optimize the rate-distortion performance of multiple description subband coding. The problem is formulated in a Lagrangian optimization framework, and is solved directly in the frequency domain to produce the optimal filter spectral responses. By varying the Lagrangian parameter, our design obtains all points on the redundancy-distortion curve for Gaussian wide-sense stationary input processes. We consider the cases of both orthonormal and more general biorthogonal filter banks. We also analyze the connections between these two solutions, and quantify the gap in their optimal performance. Application of the proposed methods to the popular first-order autoregressive (AR(1)) model yields very interesting and insightful results. In the two extreme cases of maximum and minimum redundancy, our solutions degenerate to previously known results. Finally, comparisons with popularly deployed filter banks reveal large gaps from the theoretical performance bound.