Showing papers on "Spectral density published in 1974"

Consistent Autoregressive Spectral Estimates

Abstract: We consider an autoregressive linear process $\{x_t\}$, a one-sided moving average, with summable coefficients, of independent identically distributed variables $\{e_t\}$ with zero mean and fourth moment, such that $\{e_t\}$ is expressible in terms of past values of $\{x_t\}$. The spectral density of $\{x_t\}$ is assumed bounded and bounded away from zero. Using data $x_1,\cdots, x_n$ from the process, we fit an autoregression of order $k$, where $k^3/n \rightarrow 0$ as $n \rightarrow \infty$. Assuming the order $k$ is asymptotically sufficient to overcome bias, the autoregression yields a consistent estimator of the spectral density of $\{x_t\}$. Furthermore, assuming $k$ goes to infinity so that the bias from using a finite autoregression vanishes at a sufficient rate, the autoregressive spectral estimates are asymptotically normal, uncorrelated at different fixed frequencies. The asymptotic variance is the same as for spectral estimates based on a truncated periodogram.

On the spectral dissipation of ocean waves due to white capping

Abstract: The effect of white capping on the spectral energy balance of surface waves is investigated by expressing the white-cap interactions in terms of an equivalent ensemble of random pressure pulses. It is shown first that the source function for any non-expansible interaction process which is weak-in-the-mean is quasi-linear. In the case of white capping, the damping coefficient is then shown to be proportional to the square of the frequency, provided the wave scales are large compared with the white-cap dimensions. The remaining free factor is determined indirectly from consideration of the spectral energy balance. The proposed white-capping dissipation function is consistent with the structure of the energy balance derived from JONSWAP, and the existence of a δ−5 spectrum governed by a non-local energy balance between the atmospheric input, the nonlinear energy transfer and dissipation. However, closure of the energy balance involves hypotheses regarding the structure of the atmospheric input function which need to be tested by further measurements. The proposed set of source functions may nevertheless be useful for numerical wave-prediction. According to the model, nearly all the momentum transferred across the air-sea interface enters the wave field. For fetchlimited and fully developed spectra in a stationary, uniform wind field, the drag coefficient remains approximately constant. However, for more general wind conditions, this will not be the case and the wave spectrum should be included in an accurate parameterisation of the air-sea momentum transfer.

On Large-Sample Estimation for the Mean of a Stationary Random Sequence

Abstract: For a wide class of stationary random sequences possessing a spectral density function, the variance of the best linear unbiased estimator for the mean is seen to depend asymptotically only on the behavior of the spectral density near the origin. Asymptotically efficient estimators based only on this behavior may be chosen. For spectral densities behaving like $\lambda^ u$ at the origin, $ u > -1$ a constant, the minimum variance decreases like $n^{- u-1}$, where $n$ is the sample size. Asymptotically efficient estimators depending on $ u$ are given. Finally, the consequences of over- or under-estimating the value of $ u$ in choosing an estimator are considered.

Weyl's theory applied to the Stark effect in the hydrogen atom

Abstract: A short review of Hermann Weyl's theory for singular second-order differential equations is given and its numerical aspects are discussed. It is pointed out that this method is suitable for the treatment of perturbations which make the spectrum continuous. The Stark effect on the ground state of the hydrogen atom is taken as an example. The spectral density, the imaginary part of Weyl's "$m$ function," is calculated numerically using Runge-Kutta integration and Airy integrals for the asymptotic region. Showing $\ensuremath{\delta}$-function-like behavior with poles of $m$ on the real axis for the discrete levels, the spectral density involves approximate Lorentzians for the metastable states of the continuous spectrum, corresponding to poles of $m$ in the complex plane. Trajectories of these poles for electric fields up to 0.25 a.u. are shown for the one-dimensional as well as for the full three-dimensional problem.

Fast-Fourier-Transform Spectral-Analysis Techniques as a Plasma Fluctuation Diagnostic Tool

Abstract: A review is given of the key ideas involved in computing power spectral density functions from fast-Fourier-transformed plasma fluctuation data. Next a model is described which enables one to determine, from the computer-generated power spectra, the amplitude, frequency, and wavenumber of each of several waves present in the plasma. The potential of digitally implemented spectral analysis techniques as a plasma diagnostic tool is demonstrated in an experimental study of the evolution, and ultimate turbulent destruction, of a multi-mode drift wave spectrum.

The high frequency spectrum of wind-generated waves

Abstract: The high frequency part (10 Hz∼50 Hz) of the one-dimensional wave spectrum was measured in a wind-wave channel under accurately controlled conditions. The results are compared with the spectral forms for the capillary range that have been proposed recently byPierson andStacy (1973) andToba (1973). In a general sense, fairly good agreement is found between the present results and those ofPierson andStacy (1973) and ofToba (1973). The spectrum in the capillary range is clearly wind speed dependent, and the spectral density in that range increases with increasing wind speed.

Coarse frequency estimation using the discrete Fourier transform (Corresp.)

Abstract: The discrete Fourier transform (DFT) is applied as a coarse estimator of the frequency of a sine wave in Gaussian noise. Probability of anomaly and the variance of the estimation error are determined by computer simulation for several DFT block sizes as a function of signal energy-to-noise density ratio \mathcal{E}/N_0 . Several data windows are considered, but uniform weighting gives the best performance.

Properties of mobile radio propagation above 400 MHz

Abstract: This paper begins with a discussion of multipath interference. The spatial description of the field impinging upon a mobile radio antenna is derived and the power spectrum and other properties of the signal envelope are considered. Next, large scale variations of the average signal are discussed. Measurements of observed attenuation on mobile paths over both smooth and irregular terrain are summarized. The paper concludes with a discussion of methods of predicting the area of coverage from a base station.

The spectrum of Earth strain from 10^−8 to 10 ^2Hz

Abstract: We have measured the power spectrum of the earth strain fluctuations over 10 decades in frequency from 10−8 to 102 Hz using data from three strain observatories. Although the strain meters were widely separated and of different design, they produced records whose power spectra are in close agreement with each other. We find that the composite power spectrum shows an approximate inverse square dependence on frequency over the entire band investigated.

Characterization of Frequency Stability: A Transfer Function Approach and Its Application to Measurements via Filtering of Phase Noise

Abstract: Frequency stability of high-quality signal sources is characterized in the Fourier frequency domain by the spectral density Sy(f) of the fractional instantaneous frequency deviation y(t), and in the time domain by the Allan variance ?y2(?). Two well-known types of measuring apparatus used to evaluate these parameters are analog spectrum analyzers and digital electronic counters, respectively. A detailed analysis of the structure of the relation between ?y2(?) and Sy(f) shows that it is possible to define a variance, i.e., a time-domain measure, by its transfer function in the Fourier frequency domain, even when no corresponding measurement sequence exists in the time domain. Two different kinds of variance are then defined, which possess different properties for white and flicker phase noises. One of these variances is an estimate of the Allan variance. These variances may be measured by a suitable filtering of phase noise at the output of a phase detector.

Real time analysis of voiced sounds

Abstract: A power spectrum analysis of the harmonic content of a voiced sound signal is conducted in real time by phase-lock-loop tracking of the fundamental frequency, fo, of the signal and successive harmonics hl through hn of the fundamental frequency, measuring the quadrature power and phase of each frequency tracked, differentiating the power measurements of the harmonics in adjacent pairs and analyzing successive differentials to determine peak power points in the power spectrum for display or use in analysis of voiced sound, such as for voice recognition.

Interpretation of time records and power spectra of scattered ultrasonic pulses in solids

Abstract: Experiments were conducted to investigate the scattering of a wide bandwidth pulse by a fluid‐filled cylindrical cavity in an elastic solid. The pulse amplitude‐time record and the corresponding Fourier power spectra of the scattered pulses are analyzed. The sequence of pulses received by a transducer is identified by applying the theory of geometric acoustics and the pulse arrival times are related to the cavity diameter and the wave speed in the fluid. The power spectrum is interpreted in terms of the resonance of the fluid inclusion, the natural frequencies of which also depend on the size and property of the inclusion. Investigations show that both pulse‐time record and power spectrum can be utilized to detect the size and the wave speed of a fluid inclusion in a solid.

Phase effects in a three‐component signal

Abstract: Phase effects in a particular three‐component signal, consisting of the 9th, 10th, and 11th harmonic of 200 Hz were studied. The main interest was the effect of phase changes on the prominence of the residue pitch evoked by this signal. The prominence of residue pitch appears to be correlated to the acoustical power spectrum of the signal, modified by the internally generated combination tones. An estimate of this “internal spectrum” was obtained by means of a cancellation technique. Because of an interaction of combination tones and acoustical components, this “internal spectrum” is phase‐dependent. In this context de Boer's phase rule is reconsidered. The residue pitch seems to be most prominent if there are prominent lower harmonics in the “internal spectrum.” In general, subjective phase effects in this signal, which could be described as changes in timbre, seem to be dependent on changes in the “internal spectrum.”

The effects of spline interpolation on power spectral density

Abstract: This paper discusses the power spectral effects of spline interpolators. A general technique is given for finding the steady-state spectral effects of splines of all orders, when applied following uniform sampling of the input function. The following observations are made: 1) the even order splines that were examined (second and fourth order) possessed divergent steady-state frequency transfer functions, 2) the degree of preservation of the power spectral density of the input process increased with the order of the (odd order) spline used for interpolation, and 3) the reconstruction of a stationary random process over a finite record length will, on the average, have less power than indicated by the steady-state transfer function.

Spectra of digital phase modulation by matrix methods

Abstract: We derive the spectral density of a sinusoidal carrier phase modulated by a random baseband pulse train in which the signaling pulse duration is finite and the signaling pulses may have different shapes. The spectral density is expressed as a compact Hermitian form in which the Hermitian matrix is a function of only the symbol probability distribution, and the associated column vector is a function of only the signal pulse shapes. If the baseband pulse duration is longer than one signaling interval, we assume that the symbols transmitted during different time slots are statistically independent. The applicability of the method to compute the spectral density is illustrated by examples for binary, quaternary, octonary, and 16-ary PSK systems with different pulse overlap. Similar methods yield the spectral density of the output of a nonlinear device whose input is a random baseband pulse train with overlapping pulses.

Measurement of plasma wave spectral density from the cross-power density spectrum

Abstract: Determination of the spectral density of plasma waves from the cross‐power density spectrum is shown to be simpler than the alternative determination from the cross‐correlation function. A device for measuring the cross‐power spectral density is described in detail. A model with a bandwidth of 30 MHz has been constructed using integrated circuits, and applied to the study of ion acoustic wave propagation on a plasma positive column. Since it is an analog device, it should be possible to extend the bandwidth for operation into the gigahertz range. Only a fixed delay is required to measure the quadrature component of the cross‐power spectral density. These features are in contrast to other correlators and cross‐power spectrum analyzers described to date, which are either limited to the low megahertz range, or require a variable delay line.

Asymptotic Behavior of Digital FM Spectra

Abstract: An asymptotic form for the spectral density of a continuous-phase digital FM signal is derived. This expression is valid at frequencies far from the carrier and shows how the rate at which the spectral density falls away depends on the baseband pulse shape.

Loop fault locater

Abstract: Faults on a telephone cable pair are located by a frequencydomain detection system. The derivative, with respect to frequency, of the phase angle of the complex input impedance of the pair is measured periodically across a predetermined frequency band to produce a periodic time function. The power spectrum of this function is determined and the frequencies of the spectrum maxima are used to estimate the distances to all of the faults from the measurement point.

Time-amplitude correlation effect between large Barkhausen discontinuities (in the magnetization noise)

Abstract: A theory of the power spectrum of the Barkhausen noise is presented, accounting for the frequency behaviour of the experimental spectra in the whole range of analysis frequency between 1 Hz and 20kHz, when the magnetizing frequency is kept relatively low (below 0.1 Hz). The theory assumes the existence of a correlation between the number of clustered pulses in a large Barkhausen discontinuity and the time interval from such a discontinuity to the preceding one: the longer this time interval, the larger the number of clustered pulses. An experimental check of the dependence of the noise power spectrum on the magnetizing frequency in rings of Fe−Si 4% is given. It is shown that the theoretical results fit the experimental data without the introduction of any arbitrary constant.

Statistical ground excitation models for high speed vehicle dynamic analysis

Abstract: The method used to statistically describe a digitally defined ground roughness profile, based on measured data, is outlined and illustrated by reduction of the survey data of the Tracked Levitated Research Vehicle's (TLRV) Initial guideway section. The power spectral density (PSD) is the statistic of prime interest. Its use as the ground roughness excitation function in linear vehicle dynamic response analyses provides, the impetus. The PSD is not unique; therefore, when calculated for use in such analyses, the vehicle speed and suspension natural frequencies become critical parameters. The statistical estimates of the PSD and autocorrelation (AC) functions based on this TLRV data indicate a reasonable fit with mathematical models which consist of Fourier transform pairs. These PSD models contain less energy at low frequencies than the model presently used. A first-order vehicle dynamic response analysis incorporating one of these models results in substantial reductions of suspension stroke response. The roughness coefficient (A) of this section of thw TLRV guideway is approximately 1.2 x 10 to the minus 6th power feet-radians.

Spectrum of a binary signal block coded for DC suppression

Abstract: This paper analyzes a block-coding scheme designed to suppress spectral energy near f = 0 for any binary message sequence. In this scheme, the polarity of each block is either maintained or reversed, depending on which decision drives the accumulated digit sum toward zero. The polarity of the block's last digit informs the decoder as to which decision was made. Our objective is to derive the average power spectrum of the coded signal when the message is a random sequence of + 1's and −1's and the block length (M) is odd. The derivation uses a mixture of theoretical analysis and computer simulation. The theoretical analysis leads to a spectrum description in terms of a set of correlation coefficients, {p q } = q = 1, 2, etc., with the p q 's functions of M. The computer simulation uses FFT algorithms to estimate the power spectrum and autocorrelation function of the block-coded signal. From these results, {p q } is estimated for various M. A mathematical approximation to p q in terms of q and M is then found which permits a closed-form evaluation of the power spectrum. Comparisons between the final formula and simulation results indicate an accuracy of ±5 percent (±0.2 dB) or better. The block-coding scheme treated here is of particular interest because of its practical simplicity and relative efficiency. The methods used to analyze it can be applied to other block-coding schemes as well, some of which are discussed here for purposes of comparison.

Measurement of averaged squared modulus of atmospheric-lens modulation transfer function

Abstract: The theory of speckle interferometry shows that the averaged squared modulus of the atmospheric-lens modulation transfer function becomes proportional, for high frequencies, to the lens modulation transfer function. The experimental curve obtained by comparison of the power spectrum of a turbulence-degraded image and that of a nondegraded image of the solar granulation is consistent for high frequencies with the theoretical curve. We propose to use it to correct power spectra obtained with a large ground-based telescope, at frequencies up to the diffraction limit.

A theoretical analysis of laser Doppler flowmeters

Abstract: A theoretical analysis is given of single and dual beam scattering flowmeters. All propagating beams are treated rigorously, taking their Gaussian characteristics into account. An exact derivation of the Doppler frequency shows how it depends on the position of the scattering particle within the scattering volume. The spatial and spectral resolutions of both Doppler systems are given. It is shown in what way these depend on the geometry of the optical arrangement. The average power spectrum of the detector signal is derived as a function of the particle concentration. The signal-to-noise ratios of both systems are calculated and compared. Some inherent differences between the two systems will be pointed out.

Spectrum analyzer using delta modulation encoding

Abstract: Disclosed are a method and apparatus for computing the power spectrum of an applied analog signal at a chosen frequency, by converting the analog signal into a binary sequence, by separately multiplying the binary sequence by a sequence of cosine and sine functions, by accumulating and squaring the resultant product signals, by adding the squared signals, and by multiplying the added signals with a constant signal which is a function of the chosen frequency.

Tests for periodic components in multiple time series

Abstract: SUMMARY The problem of testing for periodicities in multiple time series is considered. The test statistic proposed is based on the Euclidean norm of the matrix of periodograms. We first obtain the exact distribution of our test statistic under the hypothesis that observations are taken from an n-vector time series consisting of independent series of independent normal variables. We extend the theory to normal linear series with a specified spectral density matrix and then show that a suitable estimate can be used to replace this matrix in our test statistic. These results are asymptotic in nature. Extreme value theorems are stated for the case of normal variates and it is shown that the distributions for nonnormal series can be approximated adequately by those for normal series. An exact test for periodic components in an univariate time series was first proposed by Fisher (1929) who considered the null hypothesis of normal white noise. The test was based upon the maximum of a set of periodogram ordinates each divided by their sum, the division making it unnecessary to know in advance the variance of the components of the series. The use of the rth largest of these ordinates, r > 1, in testing for periodicities was also discussed by Fisher (1940). Whittle (1951, p. 101) showed how one could apply the Fisher tests to normal linear series provided the number of observations was large and one could specify the spectral density for the series. Hannan (1961) proposed a modification to this test that removed the necessity of knowing beforehand the spectral density function. Nicholls (1967) proposed a variation on Hannan's test statistic that improves its power under certain alternatives. Some extreme value results for periodogram ordinates were given by Walker (1965), who also demonstrated that dropping the normality condition has little effect on the large sample distributions, at least in the upper tails. In the sequel we consider the problem of testing for periodicities in multiple time series and pursue a course that parallels the historical development for univariate series as outlined in the paragraph above. Our tests are based on the Euclidean norm of the matrix of periodo- grams. In this connection Nicholls (1969) has discussed a test for discrete mass in a cross- spectrum. We consider real, discrete time parameter, n-vector, time series, Z(t) (t = 0, + 1, + 2, ...), where Z(t) = It+m(t) +X(t), with ,u+m(t) being a mean value function and X( ) being a weakly stationary, zero mean, time series possessing a spectral density matrix fxx(A) with tr{fxx(A)} <1 M < oo (JAI < 7r). We also assume the existence of a moving average representation 00 wo,t) , CO v - )

Visibility of unpredictably flickering lights.

Abstract: We have measured the sensitivity of the visual system to temporal modulation with unpredictable, aperiodic signals. We used three kinds of stimulation, (i) a band-limited gaussian random signal, (ii) a passband-limited gaussian random signal, (iii) a periodically quenched random signal. The sensitivity to stimulation with random signals can be predicted from the sensitivity of the visual system to periodic temporal signals. The sensitivity to random signals with narrow frequency bands at high frequencies is governed by the pseudoflash phenomenon. If the bandwidth is such that the signal contains less than two independent samples per second, the psychometric curve follows from the amplitude distribution of the random signal. If the signal contains a larger number of independent samples per second, the psychometric curves are as steep as they are for sine-wave stimulation. If the De Lange characteristic is the envelope of the sensitivity characteristics of independent channels, sensitive to specific frequency bands, then these experiments permit us to estimate the bandwidth of the most-sensitive channel.

The saturation spectrum of parametric instabilities

Abstract: Recent calculations on the nonlinear saturation spectrum of the parametric decay instability are described. The initial calculations did not use the correct expression for the spontaneous emission term and were aimed at obtaining the distribution of spectral energy in the unstable part of wave vector space. Results of those initial calculations are combined here with the correct expression for the spontaneous emission term to obtain the distribution of spectral energy in the stable part of wave vector space. These latter calculations are believed to be relevant to the interpretation of the so called plasma line spectra obtained in ionospheric heating experiments at Arecibo, Puerto Rico.

Reduction of Speckle in Coherent Imaging by Spatial Frequency Sampling. II. Random spatial frequency sampling

Abstract: The use of a moving, random aperture in the plane of a hologram has been recently suggested for reducing speckle in the reconstructed image of a diffusing object. A theoretical analysis shows that while the attenuation of the power spectrum of the speckle is proportional to the square of the average transmittance of the random aperture, the power spectrum of the signal is also attenuated by the same factor, resulting in a loss of image contrast. Because of this there is no overall gain. Some experimental results are presented which support these conclusions.

The effect of surface on the power spectrum of the Barkhausen noise in ferromagnetic materials

Abstract: The Barkhausen noise power spectrum was investigated on laminated polycrystalline toroids of Fe and 4% SiFe as a function of sample thickness; that is, of surface-to-volume ratio. Upon decreasing the thickness strong variations of spectrum shape are observed-the cut-off frequency increases and the spectral density drops in the very-low-frequency range. On the other hand, only negligible variations of static magnetic properties are observed. According to a recent theory, the results are explained assuming that a change of thickness directly affects the number of elementary pulses clustering into a large Barkhausen discontinuity. The calculated variations of the spectrum intensity and cut-off frequency are in good agreement with the experimental results.