Showing papers on "Spectral density estimation published in 1972"

Spectral errors resulting from random sampling-position errors in fourier transform spectroscopy.

Abstract: In the design and operation of a two-beam interferometer in Fourier spectroscopy, it is important to known how much the lack of precision in the setting of the optical path differences will affect the measured spectra. This problem has been considered by Surh, and by Sakai, who has given a relation for the stan­ dard deviation in the size of the \"ghost\" lines due to the random error in the sampling of the interferogram of a monochromatic spectral line. We will reformulate and extend the results to apply to a more general spectrum. Consider an interferogram P(x) which is sampled at optical path differences x = jΔx giving values Pj for N, conveniently an even integer, values of the integer j in the range — N/2 ≤ j < N/2. The discrete Fourier transform pair

Some recent advances in time series analysis

Abstract: : Research in time series analysis is conducted from many different viewpoints by researchers concerned with a wide range of concrete problems. Because researchers are spread so thinly over the field one may not perceive any feeling of progress in the theory of structure, analysis, and synthesis of time series models. It is the opinion of the author that the last few years have seen many highly applicable developments. The section headings are: (1) Time Series, Covariance, and Spectra; (2) Reproducing Kernel Hilbert Spaces; (3) Spectral Estimation; (4) Autoregressive Spectral Estimation; (5) Estimation of Parameters of Moving Average and Mixed Schemes; (6) Equivalences Among Time Series, Approximation, and Control; (7) Probability Density Estimation; (8) Time Series Modeling and Identification. (Author)

Anharmonic frequency analysis

Abstract: A new numerical method of frequency analysis is described, designed mainly to search for discrete frequencies in a time series. An integral transform is applied twice to the data for different reference times. A complex amplitude within a selected narrow frequency band is obtained for each transform. The frequency is then determined from the phase change of the complex amplitude over the difference of the two reference times. Very high precision is obtained, which is demonstrated in two examples. 1. Introduction. We consider a real time function X(t) and assume that it can be described either by a set of discrete frequencies and their complex amplitudes (at a given reference time) or by a continuous complex amplitude density function over some frequency range or a combination of both. The term frequency analysis is used here for a process which determines the frequencies, amplitudes and phases of the spectral components of the time function. There are three classical methods available which perform frequency analysis under specific conditions: the Fourier series, the Fourier integral and Prony's method. Each is restricted in its application, depending on the properties of the time function. Since all three methods are well known and described in standard textbooks, we will only repeat some of their essential properties. (a) Fourier series. If X(t) is periodic with a period T, it can be described by a constant term and a finite or infinite set of harmonic frequencies, where the basic frequency is equal to the reciprocal of the period T and all other frequencies are integer multiples of the basic frequency. The complex amplitude is obtained by the well-known integral transform of X(t) over the interval T. The integration extends only over the (finite) time interval T, since X(t) is periodic, and no additional in- formation is introduced into the process, if the integral transform were to be ex- tended over several periods. On the other hand, when a time function X(t) is given only for a time interval T and this integral transform is applied, it is automatically assumed that the time function is periodic with the period T, and no additional information is obtained by an extrapolation. Therefore, for example, the Fourier series has no real application in prediction problems. (b) Fourier integral. Here, a frequency continuum is provided for the analysis. The integral transform now determines the amplitude density as a function of fre- quency. The time function is aperiodic and must not contain discrete frequencies with finite amplitudes (equivalent to infinite amplitude density), otherwise the trans- form is not convergent. The integration has to be performed over the infinite time range from - a to + a, unless the time function is zero outside a certain time

Signal detectability performance of optimum Fourier receivers

Abstract: FFT processing systems have become important and extremely useful implementations of narrow-band signal detectors for signals with unknown phase and frequency This paper examines the Fourier filter and detector and shows it to be an optimum receiver for a very generalized signal model The general signal model covers a single sinusoidal pulse, pulse trains, and continuous signals The model provides for the finite bandwidth encountered in real signals and includes the affect of redundant processing which can be employed with high-speed FFT processors The paper provides a method for evaluating detection sensitivity for this class of signal detectors and gives experimental results from an FFT processor implementation

An application of correlation to radar systems

Abstract: This paper considers an application of correlation techniques to radar systems. One particular form of the system, the frequency modulated continuous wave radar, is described and it is shown that with this form of radar, correlation can be obtained by a combination of mixing and spectral analysis. A method of performing the spectral analysis using a digital fast Fourier transform spectral analyser is also described.

Method of an apparatus for processing of signals to transform information between the time domain and frequency domain

Abstract: A method of and apparatus for signal processing to transform information between the time domain and frequency domain (Fourier transform) wherein a time-varying electromagnetic signal to be analyzed and a series of radio frequency pulses are applied to a piezoelectric medium so as to interact at predetermined positions, and generate sampling pulses which are subsequently combined in the medium in a fashion to provide the discrete Fourier transform of the time-varying signal.

The Fourier transform of an unbounded spectral distribution

Abstract: The Fourier transform of an unbounded spectral distribution is studied: An explicit integral representation is obtained; connections are drawn to the associated generalized scalar operator. It is proved that every generalized pseudo-hermitian operator is the infinitesimal generator of a temperate CO group. Introduction. In [6] the author introduced a theory of unbounded spectral distributions in Banach spaces, and a corresponding theory of the generalized scalar operators which they represent. Properties of these objects were studied, culminating in a spectral mapping theorem [6, Theorem 4]. In this paper we study the Fourier transform of an unbounded spectral distribution, deriving an explicit integral representation, as well as growth estimates at infinity. The case of real support is considered in some detail, leading to the proof that every generalized pseudo-hermitian operator (i.e., generalized scalar with real spectrum) is the infinitesimal generator of a temperate group. This result generalizes the corresponding result for bounded operators proved in [5]. In this paper, all definitions are as in [6]. Integral representation of the Jourier transform. THEOREM I (EXTENSION OF SPECTRAL DISTRIBUTIONS). Let T be a spectral distribution [6, Definition 2] in a Banach space X. Then (a) for each fuinction ffromn the space (R2) of C' functions wt ith bounded lerivatives, and for each x E X, limit,, Tf pnx exists, where {c p} is a sequence of test flnctions as in [6, Definition 2(c)]. (b) If we define Tfx=Jimitn Tfp,,x for all x E X, then Tf is in the space Y(X) of all bounded linear operators on X, and the correspondence f--*Tf is a continuous linear mapping of X(R2) into f(X). (c) 7 T= Tf T, for al/f, g E .6(R2). (d) The operator Tf is independent of the sequence {p7J PROOF. (a) Define a double sequence of functions m-f(%-pm). Then {y form i X(R 2)-bounded subset of ?/ (R2) (cf. [3, p. 91 ] for the Received by the editors October 20, 1971. AAIS 1970 sub/jecr ch ssifications. Primary 47A65, 47A60, 47B40; Secondary 46F99. ? American Mathematical Society 1972

Frequency domain processing of digital microbarograph array data

Abstract: In this brief note we describe two frequency-domain processes suitable for signal detection and analysis of digital microbarograph array data. The processes are time-varying spectral estimation (sonograms) and estimation of frequency-wave-number spectra. When used together these processes can yield estimates of spectrum, phase velocity, group velocity, signal arrival azimuth, modal composition of signals, and multipath effects. We show examples of these processes applied to records from the large aperture microbarograph array in Montana.

Spectral Analysis of Unequally Spaced Data Samples

Abstract: : Most of the theory and practice of digital spectral analysis is based on use of the discrete Fourier transform, which by definition requires a set of data values measured at equally spaced times. In the paper the authors consider the analysis of data that are only approximately on an equally spaced grid and that in addition suffers from the problem of missing observations. In this case, the discrete Fourier transform is not directly applicable, and the authors consider several alternative approaches. The authors then consider in detail the measurement of the frequency of a single sinusiodal function in the presence of noise, using an approach based on minimizing the residual squared error after fitting. (Author)

Power spectral density estimation by spline smoothing in the frequency domain

Abstract: An approach, based on a global averaging procedure, is presented for estimating the power spectrum of a second order stationary zero-mean ergodic stochastic process from a finite length record. This estimate is derived by smoothing, with a cubic smoothing spline, the naive estimate of the spectrum obtained by applying FFT techniques to the raw data. By means of digital computer simulated results, a comparison is made between the features of the present approach and those of more classical techniques of spectral estimation.

Extended Array Evaluation Program. Special Report No. 2. Resolution and Stability of Wavenumber Spectral Estimates

Abstract: : The relative resolution and stability of conventional and new methods of spectral estimation are investigated. Techniques considered are conventional beamsteer, maximum likelihood, maximum entropy and principle components. These techniques are evaluated using both synthetic data and real seismic data from the TFO extended short-period array.