Rational Spectrum Research Articles

Spectrum manipulation for improved resolution

A novel approach to improving the frequency resolution of parametric power spectrum estimation for signals with rational spectra is discussed. Specifically, the effect of errors in the calculated autocorrelation values is considered. Analysis and numerical simulation reveal that the error sensitivity becomes acute when the poles are closely spaced, demonstrating one of the difficulties encountered when high-resolution spectrum estimates are required. To remedy this problem, two modified spectrum-estimation procedures are proposed. The first applies to situations where discrete-time techniques are used to estimate the spectra of continuous-time processes. It involves selecting the autocorrelation function sampling period to separate the poles. The second procedure involves desampling an autocorrelation sequence to separate closely spaced poles artificially. Numerical examples verify improved resolution for both procedures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the Structure of the Smallest Splitting $\sigma $-Algebra for a Generalized Gaussian Field with Rational Spectrum

On the Structure of the Smallest Splitting $\sigma $-Algebra for a Generalized Gaussian Field with Rational Spectrum

Parameter estimation of continuous-time stationary Gaussian processes with rational spectra

This paper considers the problem of estimating the parameters of continuous-time stationary Gaussian processes with rational spectra, from uniformly sampled measurements. The sampled process is shown to be an autoregressive moving-average process, and explicit relationships between the parameters of the continuous-time and the sampled processes are derived. These relationships are then used to derive a lower bound on the variances of biased estimates of the continuous-time parameters, and on the generalized variance of such estimates. It is shown by some examples that the bound on the generalized variance depends on the sampling interval in a nonmonotonic manner. In particular, for each specific set of parameters there exists a sampling interval for which the lower bound is minimized.

Non-Gaussian stochastic response of nonlinear compliant platforms

A moment function method is presented to estimate the stochastic response of compliant offshore platforms with nonlinearity in stiffness based on non-Gaussian closure. For guyed towers with clump weight, the nonlinearity in stiffness is of the softening type. The random wave loading is expressed in terms of a rational spectrum, making the system Markovian. Using Ito's rule for stochastic differentiation, differential equations for moments up to the fourth order are developed. The system of equations is closed by considering the fifth and sixth cumulants to be zero. For stationary response, differential equations become algebraic equations. The moments are obtained by solving the system of nonlinear algebraic equations. It is observed that the Gaussian closure method is inadequate for defining the complete probabilistic characteristics of the response.

State-space modeling and estimation of time differences of arrival

We propose a new method for time differences of arrival (TDOA's) estimation of multiple sources whose spectra are unknown. In the first part, we show how state-space techniques can be used to easily arrive at an exact discrete global model of the transfer between the source inputs and sensor outputs for sources having rational spectra and for integer and noninteger delays. This part is of interest by itself-e.g., for simulation purposes-but it also leads in a natural way to an original TDOA's estimation procedure presented in the second part of the paper. A condensed parametric model of the covariance information is the unique input to what can be seen as a generalized parametric covariance method. The difficulties inherent to these methods in the presence of several sources are easily turned around by a decoupling scheme which isolates the contributions of the individual, elementary sources. Simulation results are presented.

Monte Carlo turbulence simulation using rational approximations to von Karman spectra

Turbulence simulation is computationally much simpler using rational spectra, but turbulence falls off as f exp -5/3 in frequency ranges of interest to aircraft response and as predicted by von Karman's model. Rational approximations to von Karman spectra should satisfy three requirements: (1) the rational spectra should provide a good approximation to the von Karman spectra in the frequency range of interest; (2) for stability, the resulting rational transfer function should have all its poles in the left half-plane; and (3) at high frequencies, the rational spectra must fall off as an integer power of frequency, and since the -2 power is closest to the -5/3 power, the rational approximation should roll off as the -2 power at high frequencies. Rational approximations to von Karman spectra that satisfy these three criteria are presented, along with spectra from simulated turbulence. Agreement between the spectra of the simulated turbulence and von Karman spectra is excellent.

Canonical Correlations of Past and Future for Time Series: Definitions and Theory

The concepts of canonical correlations and canonical components are familiar ideas in multivariate statistics. In this paper we extend these notions to stationary time series with a view to determining the most predictable aspect of the future of a time series. We relate properties of the canonical description of a time series to well known structural properties of the series such as (i) rational spectra (i.e., ARMA series), (ii) strong mixing, (iii) absolute regularity, etc.

Structure of stationary finite observation records of discrete-time stochastic linear systems

We discuss the structural features of finite observation records from discrete-time stationary Gaussian stochastic processes with rational power spectra. The processes may be viewed as arising from discrete-time linear systems excited by white noise or as autoregressive-moving average processes. The latter parametrization is chosen for convenience, and the existence of nontrivial sufficient statistics is studied. It is shown that only autoregressive processes have sufficient statistics whose dimension is less that the number of observations. Some connections with the theory of stochastic realization are described.

A special family of ergodic flows and their $$\bar d - limits$$

A flow built under a step function with a multi-step Markov partition on the base is a direct product of a Bernoulli flow with a finite rotation. A $$\bar d - limit$$ of the flows in this family cannot have two irrationally related rotation factors. $$\bar d - closure$$ of this family is shown to consist of all direct products of Bernoulli flows and flows of rational pure point spectrum with respect to some number.

On the rational spectra of graphs with abelian singer groups

Let G be a finite abelian group. We investigate those graphs G admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra P ( G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in P ( G) for various G's, we relate subschemes of a related association scheme, subalgebras of P ( G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors.

Additive and multiplicative minimum-phase decompositions of 2-D rational power density spectra

This paper considers the multiplicative and additive minimum-phase decompositions of a two-dimensional (2-D) rational power density spectrum (commonly known as the spectral factorization and partial fraction expansion problems). It is shown that both of these decompositions generally involve filters which are ratios of 2-D minimum-phase filters, with finite support in one variable, and infinite support in the other variable. It is shown that a sufficient condition for a 2-D autoregressive (AR) spectrum to have a rational additive decomposition is for the minimum-phase whitening filter to have finite reflection coefficient support.

Lattice methods for spectral estimation

Lattice forms provide convenient parametrization of rational spectra of stationary processes. A comprehensive summary of lattice algorithms for estimating spectral parameters of AR, MA, and ARMA processes is presented. It is shown that various well-known spectral estimation techniques, such as the Maximum Entropy Method (MEM) and Maximum Likelihood Method (MLM), can be efficiently computed from lattice parameters. Algorithms are presented for the autocorrelation, pre-windowed, and covariance methods of forming the sample covariance matrix.

Covariance sequence approximation for parametric spectrum modeling

Parametric methods of spectrum analysis are founded on finite-dimensional models for covariance sequences. Rational spectrum approximants for continuous spectra are based on autoregressive (AR), moving average (MA), or autoregressive moving average (ARMA) models for covariance sequences. Line spectrum approximants to discrete spectra are based on cosinusoidal models for covariance sequences. In this paper we make the point that a wide variety of spectrum types admit to modal analysis wherein the modes are characterized by amplitudes, frequencies, and damping factors. The associated modal decomposition is appropriate for both continuous and discrete components of the spectrum. The domain of attraction for the decomposition includes ARMA sequences, harmonically or nonharmonically related sinusoids, damped sinusoids, white noise, and linear combinations of these. The parametric spectrum analysis problem now becomes one of identifying mode parameters. This we achieve by solving two modified least squares problems. Numerical results are presented to illustrate the identification of mode parameters and corresponding spectra from finite records of perfect and estimated covariance sequences. The results for sinusoids and sinusoids in white noise are interpreted in terms of in-phase and quadrature effects attributable to the finite record length.

Recursive Estimation Based on ARMA Models

A recursive estimate of the stochastic structure of a stationary time series is constructed based on the assumption that the true structure is ARMA, i.e., has a rational spectrum. The estimate is recursive in the sense that each successive estimate is obtained from the previous one by a relatively simple adjustment, that could be effected in a "real time" situation. The procedure is basically that of updating a regression when all variates involved are constructed from previous estimates of the parameter vector. The strong convergence of the estimate to the true value is established as well as a result relating to the rate of convergence.

Forwards and backwards models for finite-state Markov processes

The construction and properties of reversible and dynamically reversible models for finite-state Markov processes are studied. Certain results on approximating processes with rational power spectra with dynamically reversible finite-state models are also obtained.

BIBO stability of multidimensional filters with rational spectra

Classical conditions relating bounded input-bounded output (BIBO) stability to the behavior of the impulse response for one-dimensional discrete filters with rational spectra are reviewed and generalized to the multidimensional case. It is shown, in this manner, that these conditions are fundamentally predicated by rationality and not by the existence of isolated filter natural frequencies. Furthermore, the conditions are seen to lead to novel stability criteria applicable in the frequency-domain and minimum mean-square design of multi-dimensional filters.

The limits in $$\bar d$$ of multi-step Markov chainsof multi-step Markov chains

The\(\bar d\) of the class of multi-step Markov chains is shown to consist of all direct products of Bernoulli processes with processes of rational pure point spectrum. The class of processes that are approached in\(\bar d\) by their canonical multi-step Markov approximations is also studied. It is found to be strictly smaller than the former class, dense in it, and characterized within it by a certain (noninvariant) property of its rotation factors.

Generalized Partial Response for Equalized Channels with Rational Spectra

Harashima and Miyakawa [1] and Tomlinson [2] have described a generalized partial response technique which achieves the performance of the decision-feedback equalizer without the error propagation problem. We show here that when the equalized and baud-rate sampled channel assumes the special rational <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> transform <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(Z) =\frac{\sum\min{i=0}\max{M}{g_{i}Z^{i}}{1/L\sum\min{i=0}\max{N}{l_{i}Z^{i}</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l_{0} = L, g_{0} = 1</tex> , the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gi</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">li</tex> are integers, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> is a power of 2, the implementation can assume an especially simple form not requiring the storage of analog samples. The numerator polynomial can be chosen to achieve transmission zeros, as in ordinary partial response, while the denominator can be chosen to reduce the noise enhancement in equalization. This technique results in as much as a doubling of the peak transmitted voltage and, as in ordinary partial response, an increase in the number of received levels. It is shown that on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f^{1/2}</tex> channel characteristic of coaxial cable, most of the noise advantage of decision-feedback equalization can be achieved with a moderate number of received levels, and that some of this noise advantage can be traded for a reduced number of received levels. The greatest advantage accrues in multilevel transmission because of the lower peak transmitted voltage penalty.

Visual evaluation and computer analysis of the EEG—A comparison

Spectral parameter analysis (SPA) of the EEG provides a description of the distribution of spectral power in the EEG signal in the form of a rational spectrum with not more than 8 parameters. The spectrum is divided into 1–3 components described by frequency and power parameters: bandwidth (σ), peak frequency (f) and power (G). These spectral parameters are determined with the aid of a computer. The character of the EEG signal decides whether 1, 2 or 3 components (delta, alpha, beta) are needed to describe the spectrum. To test its practical value, the result of SPA was compared with that of ordinary visual evaluation of the EEG of 65 healthy men between the ages of 18 and 22 years. 20 sec sections from different leads were analysed and evaluated visually. Each EEG section was graded according to the amount of visually evaluated slow activity (VESA). To investigate the relation between the degree of VESA and the SPA result, statistical calculations (variance and regression analyses) were carried out, both for single SPA parameters and for the general type of spectrum, i.e., the number of components composing the spectrum. The SPA results from sections with artefacts were treated separately and compared statistically with results from artefact-free sections. In records with a high degree of VESA, all the leads analysed showed a tendency to have a power spectrum of low order, i.e., with few components. In most leads there was a linear regression between the degree of VESA and the bandwidth and power of the delta and alpha components. In several cases this relation was an expression of a significant linear change of the SPA parameter as a function of the degree of VESA. On the other hand the parameters of the beta component showed no relation to the degree of VESA. It was found that muscle activity could influence any spectral component thereby providing it with a strongly increased bandwidth. This is probably due to the fact that muscle activity resembles white noise in this particular frequency range. Low frequency artefacts affected only the delta component the bandwidth of which was significantly smaller than in the artefact-free sections.

Spectral factorization of a finite-dimensional nonstationary matrix covariance

For a given nonstationary matrix covariance with a finite-dimensionality property that is the time-varying generalization of the rational power spectrum matrix property, we show how to find a linear finite-dimensional system driven by white noise with output covariance equal to the prescribed covariance.