Showing papers on "Spectral density estimation published in 1975"

A new algorithm in spectral analysis and band-limited extrapolation

Abstract: If only a segment of a function f (t) is given, then its Fourier spectrum F(\omega) is estimated either as the transform of the product of f(t) with a time-limited window w(t) , or by certain techniques based on various a priori assumptions. In the following, a new algorithm is proposed for computing the transform of a band-limited function. The algorithm is a simple iteration involving only the fast Fourier transform (FFT). The effect of noise and the error due to aliasing are determined and it is shown that they can be controlled by early termination of the iteration. The proposed method can also be used to extrapolate bandlimited functions.

Fourier analysis with unequally-spaced data*

Abstract: The general problems of Fourier and spectral analysis are discussed. A discrete Fourier transformF N (v) of a functionf(t) is presented which (i) is defined for arbitrary data spacing; (ii) is equal to the convolution of the true Fourier transform off(t) with a spectral window. It is shown that the ‘pathology’ of the data spacing, including aliasing and related effects, is all contained in the spectral window, and the properties of the spectral windows are examined for various kinds of data spacing. The results are applicable to power spectrum analysis of stochastic functions as well as to ordinary Fourier analysis of periodic or quasiperiodic functions.

Digital signal analysis

Abstract: Review of least squares, orthogonality and the Fourier series review of continuous transforms transfer functions and convolution sampling and measurement of signals the discrete Fourier transform the fast Fourier transform the z-transform non-recursive digital systems digital and continuous systems simulation of continuous systems analogue and digital filter design review of random functions correlation and power spectra least-squares system design random sequences and spectral estimation.

Numerical Comparison of Five Mean Frequency Estimators

Abstract: Five techniques of estimating power spectrum mean frequency are examined. Performance is given in terms of estimate bias, accuracy, and noise immunity. Techniques examined are: 1) fast Fourier transform, 2) covariance argument approximation, 3) vector phase change, 4) scalar phase change, and 5) time derivative form of covariance. Estimator evaluation is made from numerical results obtained with a computer-simulated signal having a Gaussian spectral density which serves as the population with known parameters in the statistical analysis, and 2) real data from a pulsed Doppler radar. Both data sets consist of uniformly time-spaced digital samples of a complex signal. Absolute and relative performance of each estimator are noted, and numerical results are compared with theoretical calculations made by other investigators. Insofar as the pulsed Doppler meteorological return is represented by the signal type examined (narrow, symmetrical spectral densities), the covariance technique of mean frequency...

Periodic splines for spectral density estimation: the use of cross validation for determining the degree of smoothing

Abstract: The cross validation mean square error technique is shown to be appropriate for choosing the smoothing or “bandwidth” parameter, in estimating the log spectral density with periodic splines.

Composite wavefront decomposition via multidimensional digital filtering of array data

Abstract: A solution is obtained to the problem of estimating the number, vector velocity, and waveshape of overlapping planewaves in the presence of interfering planewaves and channel noise, where previous solutions have assumed one or more of these quantities as known. A general optimum solution is not found; instead, a heuristic solution is presented along with a complete working implementation program for large scale computers. For the case where the number of waves and the vector velocities are known, the solution is optimum. The detection of waves and the estimation of their bearing, velocity, and waveshape is accomplished via digital filtering of the frequencywavenumber power spectrum, which is computed via an efficient estimator, of the array sensed data. A new approach to the multiwave estimation problem is to reduce it to a succession of single wave problems using especially developed frequency-wavenumber filters. Special attention is given throughout the study to computationally efficient approaches. The results of the paper are placed in perspective by showing how the historically important approaches to the processing of array data such as delay and sum, weighted delay and sum, array prewhitening, beam forming, inverse filtering, least mean-square estimation, and maximum likelihood estimation are related via the spatio-temporal filtering of the frequency-wavenumber spectrum. The spectral estimation, digital filtering, and the multiwave maximum likelihood estimator developments are demonstrated by the processing of a set of simulated planewaves of various bearings, velocities, and frequencies, as well as by processing electroencephalographic (brain wave) data monitored via an array of scalp electrodes.

Deconvolution and spectral estimation using final prediction error

Abstract: Least‐squares, zero‐lag inverse filters may be used for predictive deconvolution of stationary time series and for obtaining autoregressive or maximum entropy spectral estimates. The greatest problem in finding such an inverse filter is determining the optimum operator length for a given finite length of data. The identical problem of determining the correct order of an autoregressive model for the data has been solved by Akaike, whose final prediction error (FPE) statistic is a minimum for the optimum length model. This minimum FPE criterion may be applied to both single and multiple time series. The FPE procedure has been used successfully on simultaneous three‐component seismometer and hydrophone data for the detection of refracted arrivals from explosions up to 1350 km away and for estimation of spectra of microseismic noise observed at the time of each shot. The data were recorded with an ocean bottom seismometer.

Spectral Analysis and Cross-Correlation Techniques for Photon Counting Measurements on Fluid Flows

Abstract: Photon counting techniques for the measurement of turbulent fluid flows are analyzed, and it is shown that considerable errors can result if conventional Fourier methods are used to transform count correlation records from LDV systems onto the frequency domain. Two alternative schemes are presented that overcome this difficulty. The first involves the use of high resolution spectral techniques to transform count autocorrelation records, and the second makes use of the count cross-correlation between signals from two detectors. A theoretical analysis is presented for the count cross-correlation process, and experiments in air flow show turbulence levels predicted by the two methods to be in close agreement.

System identification using pseudorandom signals and the discrete Fourier transform

Abstract: The theory of estimation of the frequency response of a system as the ratio of the discrete Fourier transforms of its sampled output and input, when the input is a pseudorandom signal, is developed. The principal sources of error are identified, their effects on the estimates are determined, and methods of error correction and reduction are described. Properties of the discrete Fourier transforms of pseudorandom sequences derived from binary and ternary m sequences are obtained, and the suitability of the corresponding pseudorandom signals for use as test signals in this application is established. The use of fast Fourier-transform techniques for the reduction of computation time is discussed, and the relative performance of these techniques and the crosscorrelation method for the estimation of both frequency and impulse responses of systems is evaluated.

A 500-point fourier transform using charge-coupled devices

Abstract: A CCD transversal filter chip, which performs a 500-point discrete Fourier transform using the chirp z-transform algorithm, will be described. Performance characteristics will be demonstrated, new operational modes presented, and system applications discussed.

A Parallel Radix-4 Fast Fourier Transform Computer

Abstract: The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

Generalized harmonic analysis for distributions

Abstract: Generalized harmonic analysis in the sense of Wiener is extended to the framework of Schwartz distributions. The approach seems mathematically and physically more transparent than the classical scheme, since every distribution possesses a Fourier transform so that the use of integrated Fourier transforms is avoided. A generalized Wiener-Khintchine representation is given which agrees well with the intuitive concept of the power spectrum. The latter is shown to be a tempered measure, in general, whose support is contained in the support of the Fourier transform of the signal. The correlation functional and power spectrum of filtered distributional signals is derived for a class of generalized filter impulse responses, which includes those that have bounded support or correspond to stable rational transfer functions. As an illustration, the form of the correlation functional and power spectrum for periodic and almost-periodic distributions and for delta-pulse trains occurring in sampled-data systems is given, and a deterministic white noise signal is constructed.

Smoothing of irradiance spectra with finite-bandwidth windows, with application to particle-size analysis

Abstract: We investigate the problem of obtaining minimum-bias spectral estimates under the constraint of finite spectral-window bandwidth. The problem is meaningful for spectral estimation with a coherent optical system in which the power-spectrum estimate, i.e., the Fourier irradiance, is physically available for smoothing. An example that deals with the smoothing of statistically unstable optical data is furnished in connection with the problem of particle-size estimation with an optical-digital computer. We show that the smoothing of the irradiance spectrum prior to particle-size estimation furnishes more satisfactory results than operating with unsmoothed data.

Spectral analysis using Fourier Transform techniques

Abstract: The present study is intended as a guide for those who wish to undertake spectral analyses using Discrete Fast Fourier Transforms. Points of particular difficulty in using Fourier Transforms are derived in some detail. Experimental results are offered to illustrate the mathematical derivations. Finally the case of a signal with discontinuities along the time-axis is discussed.

Pitch extraction by trigonometric curve fitting

Abstract: An algorithm is proposed for extracting the pitch of voiced speech. The method is based on approximating a given segment of the speech waveform in a least-squares sense by a finite Fourier series. In the approximation the fundamental frequency of the Fourier series, as well as its coefficients, is considered variable.

Minimum-bias spectral estimation with a coherent optical spectrum analyzer

Abstract: We investigate the problem of obtaining minimum-bias high-resolution spectral estimators with a coherent optical system. Circular symmetry and isotropic statistics are assumed. We consider the class of spectral smoothing windows W(ρ) > 0 and derive a window function that ensures both minimum-bias and high-resolution spectral estimation. The effect of aberrations, as summarized by the optical transfer function of the system, on the spectral estimate is also considered. In the general case it is shown that minimum-bias, direct frequency-plane smoothing is not physically realizable.

High Resolution Spectral Estimates

Abstract: : The Maximum Likelihood Method and the Maximum Entropy Method of spectral estimation are described and interpreted in terms of the innovation filter concept. Algorithms are developed for mapping an observed data sequence into the spectral estimates. The resolution performance of these spectral estimation algorithms is compared by applying each to simulated data containing two tones in white noise.

The maximum entropy spectrum and the Burg technique. Technical report no. 1: Advanced signal processing

Abstract: : This tutorial paper describes the maximum entropy spectrum and the Burg technique for computing the prediction error power and prediction error filter coefficients in the associated spectral estimation formula. The maximum entropy spectrum is identical to the autoregressive spectral estimator. Also included in this paper is a discussion of the K-line spectrum, which is the wavenumber analogue of the frequency-domain maximum entropy spectrum, and the Burg technique modifications necessary for its implementation. The purpose of this paper is to provide a complete and self-contained account of the main features of the maximum entropy spectrum. Since many of the relevant mathematical derivations are not found in the formal published literature, they are incorporated in this paper. Supporting material and various sidelights of the maximum entropy spectrum appear in the appendices.

Composite wavefront decomposition via multidimensional digital filtering of array data

Abstract: A solution is obtained to the problem of estimating the number, vector velocity, and waveshape of overlapping planewaves in the presence of interfering planewaves and channel noise, where previous solutions have assumed one or more of these quantities as known. A general optimum solution is not found; instead, a heuristic solution is presented along with a complete working implementation program for large scale computers. For the case where the number of waves and the vector velocities are known, the solution is optimum. The detection of waves and the estimation of their bearing, velocity, and waveshape is accomplished via digital filtering of the frequencywavenumber power spectrum, which is computed via an efficient estimator, of the array sensed data. A new approach to the multiwave estimation problem is to reduce it to a succession of single wave problems using especially developed frequency-wavenumber filters. Special attention is given throughout the study to computationally efficient approaches. The results of the paper are placed in perspective by showing how the historically important approaches to the processing of array data such as delay and sum, weighted delay and sum, array prewhitening, beam forming, inverse filtering, least mean-square estimation, and maximum likelihood estimation are related via the spatio-temporal filtering of the frequency-wavenumber spectrum. The spectral estimation, digital filtering, and the multiwave maximum likelihood estimator developments are demonstrated by the processing of a set of simulated planewaves of various bearings, velocities, and frequencies, as well as by processing electroencephalographic (brain wave) data monitored via an array of scalp electrodes.

Applications of the partitioned and modified chirp z-transform to oceanographic measurements

Abstract: The partitioned modified chirp Z-transform (PAM-CZT) is a signal processing technique which aids in the analysis of acoustically propagated ocean noises. The technique utilizes the fast Fourier transform (FFT) to allow real time computation in which long time data are processed in short-time-ordered sequences, thus providing spectral analysis to any frequency resolution.