Showing papers on "Spectral density estimation published in 1973"

Estimation of the magnitude-squared coherence function via overlapped fast Fourier transform processing

Abstract: A method for estimating the magnitude-squared coherence function for two zero-mean wide-sense-stationary random processes is presented. The estimation technique utilizes the weighted overlapped segmentation fast Fourier transform approach. Analytical and empirical results for statistics of the estimator are presented. The analytical expressions are limited to the nonoverlapped case; empirical results show a decrease in bias and variance of the estimator with increasing overlap and suggest a 50-percent overlap as being highly desirable when cosine (Hanning) weighting is used.

Characterization of Frequency Stability: Uncertainty due to the Finite Number of Measurements

Abstract: We consider the characterization of the frequency stability of signal generators by means of the Allan variance of their fractional frequency fluctuations. As the number m of measurements is always finite, one can only get an estimate of this variance, which is a random function of m. We calculate the variance of the Allan variance for commonly encountered spectral densities of frequency fluctuations and deduce the relative uncertainty in the evaluation of frequency stability, as a function of m. The theory is experimentally checked for white noise of phase and flicker noise of frequency. The main hypothesis of the calculations, i.e., the Gaussian character of frequency fluctuations is verified. The m dependence of the uncertainty (one sigma) in the evaluation of frequency stability agrees with the predicted value.

Estimation of power spectra with finite-order autoregressive models

Abstract: Results of an empirical study of the application of Akaike's final predictor error (FPE) criterion to the estimation of the order of finite autoregressive models to infinite autoregressive model scheme data and the subsequent application of those models to spectral estimation are given.

Fully digital spectrum analyzer using time compression and discrete fourier transform techniques

Abstract: A fully digital spectrum analyzer accepting as an input either an analog signal or a series of digital numbers and using time compression and DFT (Discrete Fourier Transform) techniques to provide the spectral component values of the input signal. Novel techniques and means are used in obtaining the power values for selected spectral lines and in averaging these power values. Statistically controlled noise is added to the input of the spectrum analyzer to enhance its resolutin beyond the resolution which would be otherwise available. Advanced and efficient techniques are used for generating and applying trigonometric functions in the course of finding the real and imaginary part of Fourier transforms, and for providing running averages of the power spectra.

Fast Equalization System

Abstract: A discrete frequency domain equalization system is disclosed for utilization in a high-speed synchronous data transmission system where, in a preferred embodiment, samples of an input signal in the time domain are transformed by a discrete fast Fourier transform device into samples in the frequency domain. Reciprocal values of these frequency domain samples are derived from a reciprocal circuit and then transformed by an inverse discrete fast Fourier transform device into time domain samples which are the desired tap gains that are applied to a transversal equalizer in order to minimize the errors in a received signal caused by intersymbol interference and noise.

Functions with a spectral gap

Abstract: Introduction. In harmonic analysis, it is important to know how various properties of a function on R reflect themselves as restrictions on its spectrum, i.e., the support of its (distributional) Fourier transform. Thus, according to Paley and Wiener, a compact spectrum is characteristic of entire functions of exponential type. In this note we consider a milder restriction : it is only required of the spectrum that it be smaller than the whole rc-space. Our results extend those of Levinson, Logan, Ehrenpreis and Malliavin ; cf. also Boas [1]. Here we give only bare outlines of proofs ; we employ standard vector notations : t = (tl9..., tn) and x = (xu ... xn) are points of R and (t, x) denotes £ \" tjXji \\t\\ = (t, t), and at denotes Haar measure on R. 1. A gap in a distribution on R is a nonvoid open ball disjoint from its support. A spectral gap in a tempered distribution is a gap in its Fourier transform. In particular, an L function ƒ has a spectral gap if its Fourier transform ƒ (x) vanishes on some nonvoid open set. Such ƒ cannot decay too rapidly, by virtue of the following result of N. Levinson.

Spectral Density Analysis: Frequency Domain Specification and Measurement of Signal Stability

Abstract: Stability in the frequency domain is commonly specified in terms of spectral densities. The spectral density concept is simple, elegant, and very useful, but care must be exercised in its use. There are several different but closely related spectral densities, which are relevant to the specification and measurement of stability of the frequency, phase, period, amplitude, and power of signals. Concise, tutorial descriptions of useful spectral densities are given in this survey. These include the spectral densities of fluctuations of (a) phase, (b) frequency, (c) fractional frequency, (d) amplitude, (e) time interval, (f) angular frequency, and (g) voltage. Also included are the spectral densities of radio frequency power and its two normalized components, Script X(f) and Script %(f), the phase modulation and amplitude modulation portions, respectively. Some of the simple, often-needed relationships among these various spectral densities are given. The use of one-sided spectral densities is recommended. The relationship to two-sided spectral densities is explained. The concepts of cross-spectral densities. spectral densities of time-dependent spectral densities, and smoothed spectral densities are discussed.

Time, frequency, sequency, and their uncertainty relations (Corresp.)

Abstract: We study the form assumed by the classical time-frequency uncertainty relations in discrete as well as nontrigonometric spectral analysis. In particular we find that if an N -sample time signal is to contain a fraction \gamma of its energy in T consecutive samples, then the minimum number of frequency components containing that same energy fraction must be greater than N/T(2\gamma - 1)^2 . It is also found that the discrete Walsh transform permits greater energy concentration (less uncertainty) than the discrete Fourier transform.

Effects of binary quantisation when using the fast Fourier transform to compute power spectral estimates

Abstract: The effects of severely quantising the multiplication coefficients in the fast Fourier transform have been studied. Rounding these off to the nearest power of 2 produces power-spectrum errors of as little as 0.08% (maximum) for a pure sinewave sampled at 64 points.

Spectral modes of the generalized transform

Abstract: Spectral modes for the generalized transform are developed. These spectra are invariant to cyclic shift of the input sequence and possess sequency resolution in their squared terms. They cannot, however, be called energy spectra as there are cross terms, some of which are negative.

Partitioned and modified chirp Z-transform: Signal processing technique for simultaneous multifrequency evaluation of the surface reradiated spectrum in slowly varying environments

Abstract: The partitioned and modified chirp Z-transform (PAM-CZT) is a signal processing technique for computing the Z-transform of a sequence at equally spaced samples on a sector of the unit circle in the Z-plane. Unlike the computationally cumbersome classical technique (4), the PAM-CZT utilizes the fast Fourier transform (FFT) that allows real time implementation with state of the art minicomputers. Another advantage of the PAM-CZT is its partitioning capability; long sequences can be processed in short-time-ordered sequences, thereby providing spectral analysis to any frequency resolution by allowing large time records. This signal processing technique will aid in the analysis of acoustically propagated noises in the ocean. Classical experiments utilize a monochromatic probe set at different frequencies for each of several orthogonal time records. However, the quasi-stationarity of the ocean inhibits reaching conclusions as to the frequency dependence of the surface reradiated spectrum' because the measurements are corrupted by the slowly varying characteristics of the ocean. An alternative approach is to simultaneously probe the ocean surface with several sinusoids and perform a PAM-CZT analysis of the results.

Estimation of the Average Frequency of a Random Process

Abstract: : The purpose of the report is to develop a means of estimating, in real time, the average frequency of a random process, given only a time-limited sample of that process. The average frequency of a random process is defined in terms of the power spectral density. Previous estimators of average frequency were mathematically based on estimating the power spectral density. In this work, the average frequency is expressed in terms of the autocorrelation and cross-correlation functions, and a new estimator is obtained which is based on estimates of these correlation functions. Two estimators presented in the literature and the proposed estimator are compared, and expressions are obtained for the power spectral density and autocorrelation function at each point in the systems, exclusive of the outputs. Approximate expressions are obtained for the bias and variance of the proposed estimator and one of the referenced estimators. A lab model of the proposed device was constructed and tested under several sets of operating conditions. It is found that, under typical operating conditions, the proposed estimator performs almost as well as those proposed earlier, and is simpler in form. For very short samples of the random process, however, the difference in variance becomes more noticeable.

Redundant processing sensitivity

Abstract: The computational speed of special-purpose Fourier transformers has made redundant processing of data with very narrow resolution bandwidths feasible. One prominent application of this type of equipment is its use as an optimum receiver for detection of narrow-band signals with random phase and unknown frequency. As shown in this paper, the redundantly processed data can be usefully employed to improve the minimum detectable signal over that attainable with independent data. The relationships for processing gain with both time redundancy and frequency redundancy are developed. The values for the processing gains are presented as a function of the amount of redundancy in the data.

Minimum-Bias Windows for Spectral Estimation by Means of Overlapped Fast Fourier Transform Processing

Abstract: : The time-limited nonnegative data windows that minimize the bias in auto- and cross-spectral estimation of stationary random processes by means of overlapped Fast Fourier Transform (FFT) processing are derived for a variety of constraints. The author constrains the time duration of the data window, or constrain the bandwidth in various ways, such as the equivalent-noise bandwidth, the half-power bandwidth, or the root-mean-square bandwidth. In the three bandwidth-constrained cases, the window duration is adjusted to meet the constraint. We find that the Hanning window is a reasonable compromise for achieving minimum bias, because in addition to being the optimum for one bandwidth constraint, it is very close to the optima for two other bandwidth constraints. The relative merits of the spectral characteristics of the windows are also discussed.

Computer Implementation of Digital Techniques and Spectral Estimation

Abstract: In this paper along with a brief theoretical analysis of the design aspects of recursive and non-recursive digital filters, derivation of design algorithms and their performance characteristics are studied with the help of general-purpose digital computer, Honeywell-400. The basic mathematical tools used are the well-known z-transformation calculus and the bilinear z-transformation methods. We have presented in this paper in some detail the digital filtering with sampled spectrum frequency shift technique and its application to spectral estimation and the design procedure of a bank of filters using the methods along with the performance results.

Concepts of Consistency in Spectral Estimation for Multivariate Time Series

Abstract: Publisher Summary As the probabilistic theory of multivariate stationary time series is a straightforward and natural extension of that of univariate stationary time series, most of the basic problems of the statistical inference concerning multivariate processes are almost exactly the ones familiar from the one-dimensional case. The methods of inference actually in use are often based on personal beliefs, prior experience, or computational advantages rather than on generally accepted statistically desirable properties. This chapter focuses on the basic property of consistency and also focuses on the alternative of estimating the spectral distribution function or the spectral density function. Statistical analysis of numerical time series data in many cases consists in answering primarily two principally different questions. The first one is to find a satisfactory probabilistic model, while the second one is to specify its parameters.

Holographic storage of multiple images using spatial division in Fourier plane

Abstract: A new holographic method for multiple storage of images has been suggested in which the spectra of low frequency signals are made to spread widely in the Fourier transform plane, by modulating them with a high frequency random diffuser. The Fourier transform plane is divided into a number of non-overlapping areas and with each of them is associated a signal. Due to symmetry of revolution about the optical axis, the spatial division of the frequency plane is made by using annular filters of different diameters. The directly transmitted beam corresponding to zero spatial frequency provides the reference beam. The method can be used to for multicolor wavefront reconstructions. The storage capacity of the method depends on the spread of the spectrum in the Fourier transform plane. This can, however, be increased by incorparating an additional periodic modulation in the Fourier transform plane.