Wide-sense Stationary Process Research Articles

An evaluation of sensory noise in the human visual system

It is assumed that the activity of a visual channel may be represented as V(t) = g(t) + xi(t), where g(t) is the deterministic response of the channel due to the presentation of a stimulus and xi(t) is the trajectory of a wide-sense stationary Gauss process. The stimulus is detected if the event (V(t) greater than S for at least one t epsilon[0, T]) occurs. Two approximations for the probability of this event are proposed, and it is demonstrated how they may be employed to estimate (i) the value of the second spectral moment lambda 2 of the noise process xi t, where lambda 2 reflects the speed of the fluctuations of the trajectories xi(t), and (ii) the value of the internal threshold S. The commonly made assumption of peak--detection is shown to serve as a very good first approximation in particular if the channel is of transient type or--in case of detection by a channel of sustained type--if the stimulus durations are not too long.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Solving ill-posed problems with artificial neural networks

With many physical problems, measurement of spectral distribution, cosmic radiation, aerial and satellite imaging indirect sensing/recording devices are used. In many of these cases, the recording systems can be modeled by a Fredholm integral equation of the first kind. An inversion of the kernel representing a system, in the presence of noise, is an ill-posed problem. The direct inversion often yields an unacceptable solution. In this paper, we suggest an artificial neural network (ANN) architecture to solve certain kinds of ill-posed problems. The weights in the model are initialized using eigen-vectors and eigen-values of the kernel matrix that characterize the recording system. We assume the solution to be a sample function of a wide sense stationary process with a known auto-correlation function. As an illustration, we consider two problems: a well known Phillips' problem and problem of restoration of a degraded image.

Non-full-rank causal approximations of full-rank multivariate stationary processes with rational spectrum

Let Y = ( Y n ) n ϵ ζ be a q-variate wide-sense stationary process with full-rank rational spectral density. We characterize the q-variate stationary processes ( Y ̂ n) n ϵ ζ , causally subordinated to Y, whose spectral densities have rank p < q almost everywhere, and for which E ∥ Y n − Y ̂ n ∥ 2 is minimum.

A note on sampling of bandlimited stochastic processes

It is pointed out that an irregular sampling theorem of the author (Siam J. Appl. Math., vol.47, no.5, p.1112-16, 1987) can be used to generalize the results of a paper by Z.A. Piranashvili (Theory Prob. Appl., vol.12, p.647-57, 1967) on interpolation of stochastic processes. Only the important case of wide sense stationary processes is considered here.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An analytical comparative study of the Haar transform in the context of image processing

For a class of covariance models including the covariance matrix of a wide-sense stationary process, the Haar transform (HT) is proven to be inferior to the Walsh-Hadamard transform (WHT) analytically. For positive correlation of a first-order stationary Markov process the lower bound of the performance of the HT is found. For high correlation of the process, the HT is shown to perform very close to the WHT in contrast to other superior transforms such as the slant transform or the discrete linear basis transform and proved to be superior to even the discrete Fourier transform, a member of the sinusoidal family.

Representation of strongly harmonizable periodically correlated processes and their covariances

This paper addresses the representation of continuous-time strongly harmonizable periodically correlated processes and their covariance functions. We show that the support of the 2-dimensional spectral measure is constrained to a set of equally spaced lines parallel to the diagonal. Our main result is that any harmonizable periodically correlated process may be represented in quadratic mean as a Fourier series whose coefficients are a family of unique jointly wide sense stationary processes; the corresponding family of cross spectral distribution functions may be simply identified from the two-dimensional spectral measure resulting from the assumption of strong harmonizability.

Nonstationary noise effects in the Abel inversion

A method for calculating the autocorrelation function of the output noise resulting from the Abel inversion of a wide-sense stationary random process is presented. An internal formula that expresses the autocorrelation in terms of the power spectral density of the input noise and explicitly as a function of radial position is derived. Asymptotic approximations for the variance are found for the case of bandlimited white input noise.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimal control in wide-sense stationary continuous-time stochastic models

This paper provides a rigorous treatment of the optimal control problem for a continuous-time stochastic model, with an infinite-horizon quadratic cost function, under weaker assumptions concerning the white-noise innovations than have been made in the literature on this subject. Since it is not assumed that the innovations are generated by Brownian motion, or even that the sample paths of the state variables are integrable, the Ito calculus is inapplicable. The solution depends on a more recently developed theory of stochastic differential equations, based on random measure theory, and on the ergodic theorem for a wide-sense stationary process.

Estimation - Correlation, Modeling and Identification in Adaptive Array Processors

This paper studies the system modeling and identification issues that arise in implementation of maximum likelihood detectors and adaptive array processors of signals that have propagated over randomly time-varying propagation and scattering channels. A multidimensional estimator correlator processor is used to combine the ideas of system modeling and identification with detection theory and adaptive array processing. Part of the estimator structure can be obtained from received data by the usual adaptive array processing techniques (estimation and inversion of noise covariance matrices), but the channel output estimator requires prior knowledge of the signal part of the channel output covariance matrix. This requires modeling and identification of deterministic and stochastic propagation and scattering operators. A number of convenient matrix representations for these operators have been developed. Identification of stochastic operators is based on the vector Karhunen-Loeve expansion. It is shown that in the special case of wide-sense stationary stochastic processes and periodic signal vector, orthonormal basis and eigenvalues for channel modeling and identification can be determined by discrete Fourier transform and singular value decomposition.

Minimum information spectral analysis

The problem of obtaining an adequate spectral estimate based on the information provided by only a finite number of observations regarded as an incomplete realization of a discrete wide sense stationary stochastic process is considered. From an a priori choice in favour of either the univariate autoregressive (AR), moving average (MA) or the mixed ARMA parameterization and the availability of some estimated autocovariance lags, both recursive and non-recursive estimators are obtained by maximizing the spectral entropy. These estimators preserve minimum information subject to the unknown autocovariance lags. Moreover, the approach of maximizing the spectral entropy is shown to be equivalent to the application of a rectangular window to the cepstrum.

On the computation of the stationary capacitor voltage variance matrix of an active RC network driven by white noise

A procedure is derived for numerically computing the variance matrix of the wide sense stationary capacitor voltage processes of a linear active RC network driven by white noise. In contrast to the state-space approach also mentioned, this procedure realizes a transfer function approach which simplifies the formulation of the system equations for the network. Our application of the procedure is in computer-aided noise analysis of switched-capacitor circuits.

Walsh power spectrum for wide-sense stationary stochastic processes (Corresp.)

Existing concepts of Walsh power spectra for wide-sense stationary stochastic processes are restricted to the case of autopower spectra because they are based on real Walsh functions. In this paper a Walsh power spectrum is developed which is based on a system of complex Walsh functions and thus applies to auto and cross power spectra as well. It is shown that this Walsh power spectrum is related to the Fourier power spectrum by a linear transformation. This fact makes it possible to calculate Fourier power spectrum estimates from corresponding Walsh power spectrum estimates. An estimation algorithm for the Walsh power spectrum is given in the paper.

Spectral factorization of wide sense stationary processes on [formula omitted

The problem of prediction of wide sense stationary processes on Z 2 with respect to the column-by-column or row-by-row lexicographic order is studied. A theorem giving an explicit realization of the corresponding canonical factorization of the spectral density is proved and conditions are given for a process to have a quarter-plane representation in terms of its innovations.

An analysis of the temporal fluctuations of cw acoustic propagation in the marginal ice zone

Continuous wave (cw) acoustic transmission data from MIZEX 84 (marginal ice zone experiments) were transmitted between two ships separated by approximately 100 km and propagated via a partially ice-covered path. The signals were stepped in frequency between 25 and 200 Hz for 1 h. Both vessels were drifting freely which resulted in a Doppler shift in the received multipath signal. The measured Doppler compares favorably with available navigational data. The quadrature demodulated received signal is modeled as a (locally) wide-sense stationary process. Accordingly, we can exploit the Gaussian correlation structure and estimate the moments of the power spectrum using the covariance method. This method is computationally efficient and relatively unexploited for this purpose. We relate the rms spectral width directly to the variance of the phase rate (ν2). The value of ν2 is a function of oceanic processes which dynamically perturb the sound field. Estimates of ν2 from the MIZEX data are compared with similar experiments carried out in the more quiescent Central Arctic, and in midlatitude regions.

Robust prediction of band-limited signals from past samples (Corresp.)

For a complex-valued deterministic signal of finite energy band-limited to the normalized frequency band <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|w| \leq \pi</tex> explicit coefficients <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{a_{kn}\}</tex> are found such that for any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</tex> satisfying <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0 &lt; T \leq 1/2</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\left| f(t)-\sum^{2n}_{k=1}a_{kn}f(t - kT)\right| \leq E_{f}\cdot \beta^{n}</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{f}</tex> is the signal energy and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta \doteq 0.6863</tex> . Thus the estimate of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2n</tex> past samples taken at a rate equal to or in excess of twice the Nyquist rate converges uniformly at a geometric rate to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(t)</tex> on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(- \infty , \infty)</tex> . The suboptimal coefficients <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{a_{kn}\}</tex> have the desirable property of being pure numbers independent of both the particular band-limited signal and of the selected sampling rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/T</tex> . It is also shown that these same coefficients can be used to estimate the value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> of a wide-sense stationary random process in terms of past samples.

Spectral analysis of&amp;lt;tex&amp;gt;Q&amp;lt;/tex&amp;gt;-ary digital signals encoded by&amp;lt;tex&amp;gt;P&amp;lt;/tex&amp;gt;-ary convolutional codes (Corresp.)

A method for spectral calculation of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> -ary digital signals encoded by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> -ary convolutional codes is developed. We have assumed a parallel encoding process such that for a code with rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/n</tex> and constraint length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> , the encoder can be modeled as a Mealy machine of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> inputs, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> outputs, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(m - 1) k</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> -ary memory cells. From this model and from the generator matrix of the code, the characterizing matrices of the Mealy machine are found in a straightforward way. Two assumptions are made on the statistics of the input signal. 1) The input sequence is a wide-sense stationary vector process of dimension <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> . 2) The input vectors are statistically independent. With these assumptions and from the characterizing matrices of the Mealy machine, the method developed by Cariolaro and Tronca can be readily used. The continuous and discrete components of the spectrum of the coded signal are expressed in matrix form, which is particularly appropriate for computer programming.

On the prediction of a band-limited signal from past samples

An alternate proof is given of a recent result due to A. Papoulis on predicting the current value of a wide-sense stationary band-limited random process in terms of its past samples. The possibility of such prediction is shown to be equivalent to the completeness of a certain set of complex exponentials on an interval, a property that also allows extension to the prediction of deterministic band-limited signals.

Multichannel arma spectral estimation by the modified Yule-Walker method

Abstract This paper proposes an algorithm for estimating the power spectra of multichannel wide-sense stationary processes. The processes are modeled as the output of a multivariate linear system driven by white noise. The transfer function of the system is given by a numerator matrix of polynomials divided by a scalar denominator polynomial. The denominator polynomial is estimated first, using the overdetermined, order over-estimated, modified Yule-Walker method. Modal decomposition is used to eliminate superfluous mode to reduce the order of the transfer function. Finally, the numerator of the spectral density matrix is estimated.

On the approximation of the discrete Karhunen-Loeve transform for stationary processes

Suboptimal fast transforms are useful substitutes to the optimal Karhunen-Loève transform (KLT). The selection of an efficient approximation for the KLT must be done with respect to some performance criterion that might differ from one application to another. A general class of criterion functions including most of the commonly used performance measures is introduced. They are shown to be optimized by the KLT. Various properties of the eigenvectors of the symmetric Toeplitz covariance matrix of a wide sense stationary process are reviewed. Several transforms such as the complex or real, odd and even Fourier transforms (DFT, DOFT, DREFT, DROFT), the cosine and even sine transforms (DCT, DEST) are obtained from the decomposition of a symmetric Toeplitz matrix in the sum of a circulant and a skew circulant matrix. These transforms are compared on the basis of a general performance criterion and appear to be good substitutes for the optimal KLT. Finally, it is shown that these transforms are asymptotically equivalent in performances to the KLT of an arbitrary wide sense stationary process.