Showing papers on "Spectral density published in 1973"

Time impulse response and time frequency response of optical pupils.:Experimental confirmations and applications

Abstract: The concepts and experiments discussed in this article are scarcely used in physical optics although they can introduce new ways of processing the information content of any optical pupil. They are related to diffraction phenomena in polychromatic light. One aspect deals with the well-known channelled spectra. The other consists in the definition of the time response function of an optical system - that is the result given by any pupil receiving a very short impulse of light. The time response function allows to generalize the explanation of the channelled spectra. It is also deduced from Fourier techniques and brings an argument to the parallelism between space and time domains. If the entrance pupil of a spectrometer is set in the interference pattern of a two-beam device illuminated in white light, the coloured spectrum is crossed by dark bands that are closer and closer as the difference between the optical paths increases. Along a time frequency (or wave-number) scale, the dark bands are sinusoidal. In this manner, one can say that the time spectrum is the Fourier transform of the couple of the wave trains. The time spectrum of white light being broad, a beam of white light should be considered as juxtaposition of very short wave groups. So, one can consider two individual wave groups passing along each arm of the interferometer. The same reasoning holds as one deals with several well-arranged wave trains, issued for instance from a grating or a multiple beam interferometer, or even further, with a sequence of wave trains in complete disorder. The situation is summarized in the diagram (fig. i): an incident plane wave is diffracted when transmitted through various devices. The spectrometer slit is placed in the region where the parallel beams combine with each other, at infinity or not. One should note the Fourier relationships of the spectral modulation curves with each sequnce of wave trains drawn along a time axis. These considerations immediately open out into new possibilities of conveying and collecting information. While a familiar way of transmission is to let the signal modulate the carrier which is in optics a parallel beam of monochromatic light, here the carrier consists of a parallel beam of polychromatic light, the modulation law being of temporal type. In turn this law is determined by the configuration of the optical system. Therefore any given function can be transmitted to any receiver station having a spectrometer at disposal - the role of this spectrometer being the spectral analysis of the message (or of its Fourier Transform). As an example suppose the tansmission of a (sinX/X)2 function is desired. It is the intensity distribution in the far-field diffraction pattern of a rectangular aperture. Therefore one has to perform the diffraction of a beam of white light by a slit of suitable width, a. In a first approximation (fig. ii), in a direction θ, the sequence of wave trains is limited by a rectangle function of width a cosθ and the power spectrum is expected to be given by the square modulus of a sinc-function. Likewise another test signal for studying the transmission mechanism would be a pair of thin slits that yields a cos2 function, or else a pupil of gaussian transmittance would be nessary to build up a message described by the reciprocal gaussian law. The previous process suffers of limitations. It applies to real and positive function whose Fourier transforms are real and positive. For a wider class of functions, namely complex ones, one has to resort to an equivalent of holography. The transposition comes to send a reference wave-group before or after the information proper. The phase terms are thus kept, same as in conventional holography. Such a simple experiment was carried out by means of a narrow slit set in the same plane as that of the diffracting pupil. A temporal hologram was then recorded. It appears as an image of the time spectrum striped with thin dark fringes of sinusoidal profile, carrying the amplitude and phase temrs. The usual reconstruction process in monochromatic light applies to that temporal Fourier hologram. Information concerning the previous rectangular pupil has been transmitted in the form of a hologram recorded at the output of a spectrometer 10 meters away from the source. Incidentally a much more interesting and promising technique enables real-time operating. All one has to do is get the temporal autocorrelation function of the whole of the message with its reference waves group. The pair of antisymmetrical side band distribution represents the image and its conjugate. Practically such phenomena are observable as the amplitude and phase modulation of the visibility function of the fringes displayed at the output of a Michelson interferometer. An important requirement is the use of a spatially coherent source of high luminance, with a broad spectrum (its transform - the reference - would be narrow then). The input function of the Michelson interferometer is the sequence of wave trains. Therefore this represents the basic arrangement in Fourier transform spectroscopy, with a slight difference: instead of gathering a signal and making an interferogram whose spectrum would describe the source, one gets here the autocorrelation function of the information under test, that is of the time holographic signal coming from the pupil. As a matter of fact the time-hologram itself is never formed; no intermediate recording is needed as one directly reaches three terms: the central one being the superposition of the autocorrelation of the signal and the reference respectively, the other two the symetrical side-bands displaying, as said before, the cross correlations of the signal by the reference which was assumed as a narrow impulse along a time axis. A more rigourous treatment has proved necessary to give a full account of the mechanisms involved in the transmission of a time message. In particular a time impulse response and a time transfer function can be defined for any pupil. Looking at figure iii, one notes that the scale of the Fraunhofer spatial diffraction pattern of a rectangular aperture varies as the reciprocal of the time frequency ν. As ν of the incident plane wave is tuned continuously from 0 to ∞, an observer set in a fixed position at infinity will see the « breathing » of the pattern according to the law sin Kν/Kν To complete the determination of the amplitude H(ν) at the observation point, one must add an important fact: usually, in the calculation of the amplitude diffracted by any pupil by means of the Fresnel-Kirchhoff integral, one omits the multiplicative factor, - j/λ. Therefore: H(ν) = (j2πν sin Kν/Kν. The reciprocal of ν in the time domain is t. The time response of the system to a unit impulse h(u, t) in the u-direction is then the Fourier Transform of H(ν). After a well-known derivative theorem, the transform of H(ν) is the derivative of a rectangle function. The same procedure would show that if one considers any pupil, whatever its contour and amplitude distribution, the time impulse response is the derivative of the pupil function. A similar result is obtained with phase pupils. Generally, the time impulse response of a complex pupil in a given direction is described by the derivative of the complex pupil function projected on this direction. This is illustrated by various drawings of functions corresponding to one or several slit apertures. Applications are envisaged, namely in metrology - for instance in the assessment of the quality of surfaces and the measurement of thicknesses.

Spectral analysis of speech by linear prediction

Abstract: The autocorrelation method of linear prediction is formulated in the time, autocorrelation, and spectral domains. The analysis is shown to be that of approximating the short-time signal power spectrum by an all-pole spectrum. The method is compared with other methods of spectral analysis such as analysis-by-synthesis and cepstral smoothing. It is shown that this method can be regarded as another method of analysis-by-synthesis where a number of poles is specified, with the advantages of noniterative computation and an error measure which leads to a better spectral envelope fit for an all-pole spectrum. Compared to spectral analysis by cepstral smoothing in conjunction with the chirp z transform (CZT), this method is expected to give a better spectral envelope fit (for an all-pole spectrum) and to be less sensitive to the effects of high pitch on the spectrum. The normalized minimum error is defined and its possible usefulness as a voicing detector is discussed.

Predictive filtering and smoothing of short records by using maximum entropy

Abstract: This paper presents a method of filtering and smoothing short records which depends on the prediction of the data by means of J. P. Burg's algorithm for estimating the maximum entropy power spectrum. A method of determining the prediction filter for prediction distance m from the unit distance prediction filter is developed. Examples of notch filtering and of optimum smoothing of short records are considered.

A Nonstationary Analysis of the Electroencephalogram

Abstract: A statistical technique is described which allows description of the statistical characteristics of nonstationary electroencephalograms (EEG's) The EEG is investigated in terms of its nonstationary power spectrum The instantaneous power spectra of certain transition processes in the EEG (the evolution and blocking of the alpha wave) are calculated and described on the time-frequency plane

Response of the impact damper to stationary random excitation

Abstract: The response of an impact damper system to an excitation with approximately white‐power spectral density and Gaussian probability distribution is determined, using two independent methods: digital computer and electronic‐analog techniques. Results are given for mean‐squared level, power spectral density, probability density, probability distribution, and amplitude probability density of the response. The impact damper is found to be a practical and efficient device for reducing the response amplitude of systems subjected to random excitation.

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.

Membrane shot-noise in electrically depolarized nodes of Ranvier.

Abstract: The power spectra of the spontaneous voltage fluctuations (membrane noise) of the node of Ranvier were measured in the frequency range from 0.3 to 1500 cycles per second at different levels of the membrane potential (−90 to +30 mV, inside negative). Up from about −30 mV the power spectrum shows a\(\frac{1}{{1 + \left( {2\pi f\tau } \right)^2 }}\) component, which increases with depolarization. This shot like noise component is independent of and occurs in addition to the 1/f component. The source of this shot like noise component is probably given by fluctuations in the conductance for potassium ions. With the use of a minimum parameter model which consists of channels that switch randomly in time from the closed to the open state and vice versa, independent of each other, the number of active channels per μm2 appears to be of the order of 1000. The elementary unit of the potassium system conductance is then of the order of 10−11S per channel1 and the mean frequency of switches per second per channels is about 160.

The estimation of signal spectra and related quantities by means of the multiple coherence function

Abstract: A seismic trace recorded with suitable gain control can be treated as a stationary time series. Each trace, χ j ( t ), from a set of traces, can be broken down into two stationary components: a signal sequence, α j ( t ) * s ( t —τ j ), which correlates from trace to trace, and an incoherent noise sequence, n j ( t ), which does not correlate from trace to trace. The model for a seismic trace used in this paper is thus χ j ( t ) =α j ( t ) * s( t —τ j ) n j ( t ) where the signal wavelet α j ( t ), the lag (moveout) of the signal τ j , and the noise sequence n j ( t ) can vary in any manner from trace to trace. Given this model, a method for estimating the power spectra of the signal and incoherent noise components on each trace is presented. The method requires the calculation of the multiple coherence function γ j ( f ) of each trace. γ j ( f ) is the fraction of the power on traced at frequency f that can be predicted in a least-square error sense from all other traces. It is related to the signal-to-noise power ratio ρ j ( f ) by where K j ( f ) can be computed and is in general close to 1.0. The theory leading to this relation is given in an Appendix. Particular attention is paid to the statistical distributions of all estimated quantities. The statistical behaviour of cross-spectral and coherence estimates is complicated by the presence of bias as well as random deviations. Straightforward methods for removing this bias and setting up confidence limits, based on the principle of maximum likelihood and the Goodman distribution for the sample multiple coherence, are described. Actual field records differ from the assumed model mainly in having more than one correctable component, components other than the required sequence of reflections being lumped together as correlated noise. When more than one correlatable component is present, the estimate for the signal power spectrum obtained by the multiple coherence method is approximately the sum of the power spectra of the correlatable components. A further practical drawback to estimating spectra from seismic data is the limited number of degrees of freedom available. Usually at least one second of stationary data on each trace is needed to estimate the signal spectrum with an accuracy of about 10%. Examples using synthetic data are presented to illustrate the method.

Analysis of the 5 min Oscillatory Photospheric Motion. I: A Problem in Waveform Classification

Abstract: Four Mt. Wilson measurements (T>4 h) of the photospheric motion at one point on the Sun are shown to have the characteristics of a narrow-band random process. The motion is shown to have a characteristic correlation time of 23 min and a mean power spectrum that is a smooth, single-peaked function centered at 3.4 mHz. In order to make this classification we use the analytic signal to estimate the amplitude, phase, and frequency as functions of time. The power spectrum analysis differs from the common approaches in that it uses the theoretical expression for the mean spectrum for a sequence of random pulses. Because of the random nature of the motion, we doubt the existence of more than one eigenfrequency characteristic of the photosphere as a whole. Likewise, any description of the observed motion in terms of simple deterministic functions will be inadequate for the data used here.

Power spectrum of density irregularities in the solar wind plasma

Abstract: Observations of interplanetary scintillations are compared with predictions based on a spatial electron density spectrum extrapolated from the power law shape at wave numbers 10−6–10−5 rad/km measured from spacecraft. The influences of the power law exponent, a high wave number cutoff, and the Fresnel filter are examined with respect to scintillation index and pattern scale. The comparison suggests a model in which the high wave number density spectrum lies below the extrapolated power law but has a relatively flat part before the cutoff.

Spectra of short-term fluctuations of line-of-sight signals: Electromagnetic and acoustic

Abstract: Theoretical expressions for the spectral density of amplitude, amplitude-difference, and phase-difference fluctuations of waves propagating over a line-of-sight path in a weakly scattering turbulent medium are derived. Experimental observations made at radio (35 GHz) and acoustic (3 kHz) frequencies are in good agreement with the theoretical predictions under a variety of meteorological conditions. Comparison of experimental and theoretical spectra yields a measure of the average across-the-path wind velocity and the average refractive-index structure constant Cn. In general, wind speeds and refractive-index structure constants inferred from simultaneous meteorological measurements agree with those obtained from propagation data.

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.

On Cyclic Autocorrelation and the Walsh-Hadamard Transform

Abstract: It is shown that the complete set of circular shift invariants called the Q-spectrum, of the Walsh-Hadamard transform (WHT) of a periodic sequence is related to the cyclic autocorrelation of the given sequence through the Hadamard matrices. It is also shown that the modified WHT (MWHT) of the cyclic autocorrelation yields the Q-spectrum within some scale factors. This is analogous to the discrete Fourier transform (DFT) case, i.e., the DFT of the autocorrelation of a sequence yields the shift invariant power spectrum. The Q-spectrum can be computed efficiently using the MWHT rather than the WHT. A physical interpretation for the Q-spectrum is also provided. The motivation for this study is to show that while the WHT is inherently associated with the notion of dyadic time shifts, it does have analogous properties with respect to cyclic time shifts.

Continuity of Gaussian Processes and Random Fourier Series

Abstract: This paper is mainly a survey of results on the problem of finding necessary and sufficient conditions for a Gaussian process to be continuous. The relationship between this problem and the same one for random Fourier series is explored. Some new results are presented that give continuity conditions for stationary Gaussian processes in terms of the spectrum of the process. Let $X(t)$ be a real-valued stationary Gaussian process; $EX(t) = 0, EX^2(t) = 1$. Define $F$ by the equation $EX(t + h)X(t) = \int^\infty \cos \lambda h dF(\lambda)$. Assume that $F(\lambda)$ is concave for $\lambda \geqq\lambda_0 > 0$ then $X(t)$ is continuous a.s. if and only if $$\int^\infty \frac{(1 - F(x))^{\frac{1}{2}}}{x(\log x)^{\frac{1}{2}}} dx < \infty.$$ A similar result holds for Fourier series with normal coefficients.

A Power Spectral Estimate which is Insensitive to Transients

Abstract: We indicate an estimate of the power spectrum, of a stationary time series of interest, that is insensitive to brief transient signals imposed on the series. In the case that no transient is imposed, all that one loses by computing the estimate is two degrees of freedom, asymptotically. The estimate may be computed using existing cross-spectral programs.

Electrical noise from PVC-membranes.

Abstract: Voltage fluctuations were measured across a polyvinylchloride (PVC)-membrane, acting as a constraint between two identical aqueous salt solutions. Membranes with de-resistances between about 2 and 10 megohms. cm2 were studied in solutions of KCl between 0.01 and 0.10 moles per liter. Positive and negative currents between 10 and 100 nanoamps per square centimeter of membrane were passed. The noise was measured in the frequency range between 1.5 and 150 hertz. The spectral density of the noise power, in excess of that measured at equilibrium, was found to be proportional to the mean power being dissipated in the membrane, and inversely proportional to frequency.

A direct electronic Fourier transform device for imaging

Abstract: A novel device employing elastophotoconductivity in CdS to create electronic signals representing the spatial Fourier transform of an image is described.

Synthetic Aperture Radar System Design for Random Field Classification

Abstract: The optimum design of synthetic aperture radar (SAR) systems intended to classify randomly reflecting areas, such as agricultural fields, characterized by a reflectivity density spectral density is studied. Assuming areas of known shape and location, the binary case, and a certain Gaussian signal field property, and ignoring interfield interference, the problem solution is given. The optimum processor includes conventional matched filter processing, but is nonlinear; a coherent optical system realization is outlined. The performance is approximated using a x2 assumption and bounded by the Cernov bound. A fundamental design problem involves the system bandwidth analogously, in a special case, as in diversity communication systems; a solution is given based on the Cernov bound. A set of summary design curves is given and exemplified by a satellite SAR system design. Also discussed is the measurement of reflectivity spectral density amplitude with imaging sidelooking (synthetic or ?brute-force?) radars and the maximum likelihood estimator's accuracy and realization with a coherent optical system. It is also shown that a CW modulation is useable if the random reflectivity is, effectively, isotropic. Finally, the reflectivity density spectral density amplitude, when constant over the spatial bandpass of the measuring system, is related to the scattering cross-section density commonly measured.

Folded-path weighting function for a high-frequency spherical wave

Abstract: Analysis of signal characteristics observed in the receiving plane of an optical beam has yielded information about the turbulence characteristics and cross winds over the propagation path. The measurements are generally made over relatively short paths, near the ground, with a laser at one end and one or more detectors situated at the opposite end of the propagation path. In some cases it may be desirable to have the laser and detectors at the same location. This can be accomplished by use of a reflector at the far end of the path to direct the transmitted energy back to the detectors. In a previous publication, the characteristics were developed for the optical filter function appropriate to a high-frequency plane wave propagating over a folded path. The purpose of this communication is to extend the previous analysis to the case of a spherical wave propagating over a folded path. In particular, an expression is developed showing the dependence of the observed amplitude covariance on the spectral density of the turbulence and the path geometry.

Application of on-line digital noise analysis to reactor diagnosis in JMTR

Abstract: The reactor noise analysis technique is particularly useful in reactor diagnosis for on-line monitoring if the raw noise signals can be processed in almost real time. An on-line reactor noise analysis system has been developed with use made of the mini-computer HITAC-10. This system utilizes functions for calculating the power spectral density in almost real time, plots the output by digital incremental plotter, and displays the results by means of color graphic display equipment, in order to detect anomalous reactor conditions with the statistical technique. Using this system, reactor noise signals have been measured and analyzed under various operational conditions in the JMTR. The variance of the power spectral density is found to fit a logarithmic probability density function. This function is independent of the frequency, but is dependent on the number of sampling functions. A logical procedure for anomaly detection based on statistical characteristics has been developed. It is applied to a case wher...

S = 1 2 , XY Model on Cubic Lattices. I. Susceptibility and Fluctuation near Critical Temperature

Abstract: For the $S=\frac{1}{2}$, $\mathrm{XY}$ model of a quantum lattice fluid or a ferromagnet the conventional order parameter does not commute with the Hamiltonian. As a result, the mean-square fluctuation of the order parameter and the isothermal susceptibility are not related in the usual way the general fluctuation theorem. For the above model, arguments are here presented to support the idea that as $T\ensuremath{\rightarrow}{T}_{c}$ the quantum effect due to the noncommutation becomes masked and the two quantities have the same critical behavior. This work is consistent with the exact results of Falk and Bruch who defined a certain moment of the spectral density and used inequalities to establish that if the moment \ensuremath{\rightarrow} 0 as $T\ensuremath{\rightarrow}{T}_{c}$, then the susceptibility-fluctuation ratio becomes unity thus ensuring coinciding critical behavior. The latter result applies to a large class of models including the one considered here.

On the Atmospheric Kinetic Energy Spectrum and Its Estimation at Some Selected Stations

Abstract: The motion of the atmosphere, with its climatological trend and periodic components excluded, is considered to be a stationary random process, under the control of macroscopic factors such as the solar constant, the rate of earth's rotation, the distribution and physical natures of land and sea, etc. In an effort to develop understanding of the statistical features of this process, the Eulerian frequency energy spectra of the large-scale atmospheric motions were constructed at 12 North American and nearby island stations from lengthy wind data. Examination of these spectra reveals that: 1) they are red-noise in character, 2) there is no systematic relationship between the shape of the spectrum and the local rate of earth's rotation, 3) there is a cut-down of spectral energy at low frequencies at 950 mb at some stations, 4) the shape of the spectrum is very much the same at levels in the mid and high troposphere, and 5) a spectral peak at some stations may correspond to the frequency of cyclone oc...