Showing papers on "Spectral density estimation published in 1976"

Spectral estimation techniques for the spectral calibration of a color image scanner

Abstract: Several methods of spectral sensitivity estimation for color image scanners are developed and evaluated. The estimation procedure involves response measurements of the scanner through a set of spectrally selective filters. These measurements form the observations for generalized inverse, smoothing, and Wiener estimation processes. Wiener estimation is found to provide the best results.

Spectral Analysis of a Univariate Process with Bad Data Points, via Maximum Entropy and Linear Predictive Techniques

Abstract: : A comparison of several methods for spectral estimation of a univariate process with equi-spaced samples, including maximum entropy, linear predictive, and autoregressive techniques, is made. The comparison is conducted via simulation for situations both with and without bad (or missing) data points. The case of bad data points required extensions of existing techniques in the literature and is documented fully here in the form of processing equations and FORTRAN programs. It is concluded that the maximum entropy (Burg) technique is as good as any of the methods considered, for the univariate case. The methods considered are particularly advantageous for short data segments. This report also reviews several available techniques for spectral analysis under different states of knowledge and presents the interrelationships of the various approaches in a consistent notation. Hopefully, this non-rigorous presentation will clarify this method of spectral analysis for readers who are nonexpert in the field.

Confidence intervals for regression (MEM) spectral estimates

Abstract: The probability density and confidence intervals for the maximum entropy (or regression) method (MEM) of spectral estimation are derived using a Wishart model for the estimated covariance. It is found that the density for the estimated transfer function of the regression filter may be interpreted as a generalization of the student's t distribution. Asymptotic expressions are derived which are the same as those of Akaike. These expressions allow a direct comparison between the performance of the maximum entropy (regression) and maximum likelihood methods under these asymptotic conditions. Confidence intervals are calculated for an example consisting of several closely space tones in a background of white noise. These intervals are compared with those for the maximum likelihood method (MLM). It is demonstrated that, although the MEM has higher peak to background ratios than the MLM, the confidence intervals are correspondingly larger. Generalizations are introduced for frequency wavenumber spectral estimation and for the joint density at different frequencies.

An empirical investigation of the properties of the autoregressive spectral estimator

Abstract: The autoregressive (AR) spectral estimator is used to make high resolution spectral estimates based on short data records. Measures of a frequency averaged normalized bias and normalized variance of the spectral estimates are introduced. A large number of spectra are generated. Based on the above mentioned measures and visual inspection of the estimates of the generated spectra, the AR and the conventional tapered and transformed (TT) spectral estimates are compared. It is shown that the AR spectral estimator is as stable as that given by its asymptotic variance. It is also shown that the AR spectral estimator is most powerful in estimating narrow spectral peaks with a high signal-to-interference ratio in the signal bandwidth.

Fast Fourier transform in the analysis of biomedical data.

Abstract: The fast Fourier transform (f.f.t.) is a powerful technique which facilitates analysis of signals in the frequency domain. This paper reviews some of the important features of the fast Fourier transform which are relevant to its increasing application to biomedical data. A distinction is made between the power spectrum of ergodic signals, computed from the autocorrelation function, and the frequency spectrum of nonstationary biomedical signals. The major practical pitfalls that are encountered in applying the f.f.t. technique to biomedical data are discussed, and practical hints for avoiding such pitfalls are suggested.

Calculation of Fourier Transforms by the Backus-Gilbert Method

Abstract: Summary The linear inverse theory of Backus & Gilbert has been applied to the problem of calculating the Fourier transform of digitized data with the objective of assessing the effects of missingportions of the data series and of contamination of the signal by ' noise '. When ' noise ' in the data is of concern this method achieves a maximum decrease in the variance of the Fourier transform estimate for a minimum sacrifice in resolution, thereby optimizing the trade-off between resolution and accuracy. The effects of data gaps are easily treated and it is shown that it may sometimes be desirable to interpolate these gaps even though a large variance must be ascribed to the fabricated data. We also apply the Backus-Gilbert technique to the calculation of the reverse Fourier transform, and an application to the downward continuation of potential field data is given.

Discrete-time spectral estimation of continuous-parameter processes -- A new consistent estimate

Abstract: This paper presents a new estimation scheme for the spectral density function of a stationary time series from observations taken at discrete instants of time. The sampling instants are determined by a Poisson point process on the positive real line. Under weak smoothness conditions on the spectral density, asymptotic expressions for the bias and Variance are derived, and it is shown that the estimate is mean-square consistent for all positive values of the average sampling rate. The new estimate compares favorably with the classical continuous-time spectral estimates.

Derivation of discrete Fourier transform components of a time dependent signal

Abstract: An automatic equalizer for calculating the equalization transfer function and applying same to equalize received signals. The initial calculation as well as the equalization proper are conducted entirely within the frequency domain. Overlapping moving window samplings are employed together with the discrete Fourier transformation and a sparse inverse discrete Fourier transformation to provide the equalized time domain output signals.

Frequency variant optical signal analysis.

Abstract: A class of coherent optical spectrum analyzers for one-dimensional signals is described that is characterized by a frequency variant response to the spectral components of the input signal. Operations performed exploit the second degree of freedom inherent in the optical systems. One example considered is a constant proportional bandwidth, log-frequency spectrum analyzer. Experimental and analytical results are presented.

Noise and Finite Register Effects in Infrared Fourier Transform Spectroscopy

Abstract: Finite registers used in computations act as additional noise sources in infrared Fourier transform spectroscopy The relationship between these noise sources and classical noise sources is examined The impact of finite register lengths on data acquisition, computation of the fast Fourier transform (FFT), and post-FFT spectral manipulations leads to the conclusion that the minimum recommended register length is 27 bits

Algorithmic Measurement of Digital Instantaneous Frequency

Abstract: This paper presents an algorithmic method for measuring the instantaneous frequency of a uniformly sampled FM signal. The measured parameter, termed digital instantaneous frequency, is defined in a manner similar to that used to describe frequency-modulated, continuous-time signals. The measurements are derived from an adaptive linear prediction spectral estimates. The proposed algorithm is utilized in the development of a digital processor for FM demodulation which operates on a uniformly sampled FM signal, and its output is a sampled sequence of the estimated demodulated message. The performance of the digital processor is demonstrated and compared with that of a conventional FM discriminator.

Power spectra obtained from exponentially increasing spacings of sampling positions and frequencies

Abstract: An estimate of the spectrum is based on the Laplace transform which is approximated at exponentially spaced samples and analysis frequencies. In this approximation the ratio of the intervals between pairs of adjacent sampling positions is a constant greater than one. The choice of this constant is influenced by the desired analysis bandwidth and by sampling effects. If analysis frequencies are spaced the same as sampling positions, this approximation becomes a discrete correlation. which can be computed by a fast Fourier transform (FFT) or a number theoretic transform. Except at low-analysis frequencies, the analysis bandwidth is "constant-Q," i.e., it is proportional to the analysis frequency. With a white noise input the noise in the computed spectrum is roughly constant at each analysis frequency. The numbers of samples and computations required for exponential spacing of samples and frequencies can be less than those required for equidistant spacing. Better performance at some (but not all) analysis frequencies is provided by a two-sided sampling arrangement consisting of a juxtaposition of the basic one-sided sampling arrangement and its mirror image.

The Autoregressive Method: A Method of Approximating and Estimating Positive Functions

Abstract: : The Fourier transform was considered for a positive function f(.) (or its sample Fourier transform) as a possibly complex covariance function of a hypothetical stationary complex-valued time series. This model time series by an autoregressive process of order p whose spectral density approximates (or estimates) the function f(.). The equivalence of this interpretation with the theory of orthogonal polynomials on the unit circle was studied; also the consistency of the autoregressive estimator as p increases with the sample size.

Signal processing using surface acousto-optic interaction in LiNbO3

Abstract: Signal processing functions such as convolution, correlation, and Fourier Transformation have been obtained in realtime using the efficient diffraction of laser light from acoustic surface waves progagating on LiNbO3. Different device configurations and detection schemes are discussed. Results are presented for the usual delay-line transducer configuration, as well as for an improved scheme which eliminates the problem of the reflection signal which is associated with this configuration. A discussion is given indicating the extension of the acousto-optical convolver to the generation of ambiguity functions and the correlation of a light amplitude distribution with an acoustic signal.

An assessment of prewhitening in estimating power spectra of atmospheric turbulence at long wavelengths

Abstract: Synthetic time histories were generated and used to assess the effects of prewhitening on the long wavelength portion of power spectra of atmospheric turbulence. Prewhitening is not recommended when using the narrow spectral windows required for determining power spectral estimates below the 'knee' frequency, that is, at very long wavelengths.

Methods of Computing the Power Spectrum for Equally Spaced Time Series of Finite Length

Abstract: Two conventional methods of computing the power spectrum, via the autocovariance function or via the fast Fourier transform (referred to as the lagged product method and the FFT method respectively for simplicity), have been examined analytically and numerically for equally spaced time series of finite length. It is found that the two methods are equivalent to each other, and that the only difference between them lies in regard to the spectral window. Spectral windows for the FFT method are superior to those for the lagged product method in that they do not show any negative values and that their influence is band-limited in frequency domain. There is little difference in spectral estimates between the two methods. In many cases the FFT method is economical in computation time, but for the case of large data points and small maximum lag the lagged product method is the more economical. It is proved that in the strict sense the power spectrum for higher frequencies than the Nyquist frequency is no...

Power spectral representation of nonstationary random processes defined over semi‐infinite intervals

Abstract: A new sequence of locally time‐averaged power spectra is defined that describes the time evolution of the frequency content of nonstationary random processes and linear system impulse response functions. The exact input‐response relations for these spectral sequences are shown to be finite discrete convolutions of the input and system spectral sequences. Expressions for the coefficients of a Laguerre function expansion of the time‐varying mean square response are derived from the response spectral sequence, and the time resolution and convergence of the expansion are discussed. Simple formulas are derived for generation of the spectra from recorded sample functions using Fourier transform computational algorithms. The relationship of the spectra to the Laplace transform is also developed. Physical interpretation of the spectra is discussed in detail and related to the uncertainty principle for Fourier transforms.Subject Classification: [43]45.40; [43]60.20; [43]20.50; [43]30.30; [43]40.35.

The Time-Frequency Error Product in Time-Resolved Spectroscopy

Abstract: Optical events often produce spectra which change in time. Because time and frequency are conjugate variables, they are limited by Fourier analysis to a minimum time-frequency error product.

Spectral density in time‐dependent perturbation theory

Abstract: The spectral density of radiation is defined within the context of the random phase approximation. This gives an expression for the radiation density in terms of the Fourier components of the field amplitudes. The result is applied to the case of a molecule in interaction with light. The approach with Fourier integrals overcomes some difficulties associated with an approach which uses a summation of discrete waves.

Log Spectral Estimation for Stationary and Nonstationary Processes

Abstract: : This research is concerned with two log spectral estimators in the context of both stationary and nonstationary signals. They differ because in one smoothing is realized before the logarithmic transformation, while the other is smoothed in the logarithmic domain. It is shown that for stationary signals the two estimators are similar, differing in expected value by only a universal constant. The first estimator, however, is smoother. For nonstationary signals, the estimators are biased by different amounts dependent upon the nonstationarity. The difference between the estimators is shown to be a sensitive test for nonstationarity. The estimators are used in the analysis and implementation of two solutions to the problem of blind deconvolution. It is found that the methods are equivalent for stationary signals, but differ markedly for nonstationary signals in the presence of stationary background noise. Recommendations are made for the practical digital implementation of the log spectral estimators.

Role of the stationarity equation in linear least‐squares estimation, spectral analysis and wave propagation

Abstract: We develop the properties of the innovations representation for continuous‐time stationary processes and consider applications to spectral estimation and the description of wave propagation in a lossless nonuniform medium. Both applications are based on the Levinson recursion for the causal least‐squares estimation filter and the continuous‐time counterpart of this recursion, the stationarity equation. In the first application, we develop an unconstrained representation of positive definite extensions by observing that the stationarity equation completely characterizes the least‐squares estimation filter for stationary processes. This representation is incorporated in a noval approach to spectral estimation whereby the maximum entropy method is generalized so as to include arbitrary positive definite extensions of the estimated covariance. In the second application, we observe that the stationarity equation is descriptive of wave propagation in a lossless nonuniform medium and use this fact to obtain an interesting time‐domain solution of the nonuniform wave equation. The solution is an extension of discrete‐time results in seismology concerning acoustical wave propagation in a layered medium.

Frequency-variant optical systems for signal analysis and processing

Abstract: Coherent optical systems have been employed in many configurations for the spectral analysis of signal waveforms. With some systems, signal time-bandwidth products approach 107, and real time or near real time analysis is now common. We describe here methods for obtaining frequency-variant time and frequency resolution with a coherent optical spectrum analyzer and for displaying signal spectral content vs. arbitrary functions of frequency. These methods greatly enhance the applicability of optical systems to specific spectral analysis problems, A constant proportional bandwidth-log frequency system is considered as an example. Applications to frequency-mapping signal processing are also discussed.

Synthesis of linear stochastic signals in identification problems

Abstract: Stationary stochastic inputs are generated from linear processes of the autoregressive moving average type. Since the spectral density of such a process is nonzero everywhere, this belongs to the class of admissible signals satisfying identifiability requirements. We obtain a characterization of the optimal signals in terms of their spectral densities using the results on asymptotic eigenvalue distribution of Toeplitz matrices. These signals belong to the general class of random inputs that can be generated using standard instrumentation consisting of delay lines and white noise generator.