Amplitude Distribution Function Research Articles

Structural stabilization and quenching by dither in nonlinear systems

A dither is a high frequency signal introduced into a nonlinear system for the purpose of augmenting stability or quenching undesirable jump-phenomena. In an earlier paper [2], it had been shown that the effects of a dither depend on its amplitude distribution function. The stability of a dithered system was related to that of an equivalent smoothed system, whose nonlinear element is the convolution of the dither distribution and the original nonlinearity. In this paper, which is a continuation of [2], the ability of dithers to quench jump-phenomena (i.e., to induce continuity) is demonstrated. A notion of structural-stability for feedback equations is introduced, and it is shown that dithers can structurally stabilize large classes of nonlinear systems subject to a second-order Lipschitz condition. The quenching properties of dithers are explained in terms of an effective narrowing of the nonlinear incremental sector.

A model for energy absorption in a laser-produced plasma

Energy absorption is assumed to take place through the generation of large amplitude plasma waves propagating outwards from the critical surface, and the effect of these waves on the particle distribution is described by a quasi-linear equation with non-linear broadening of the resonance. Instead of solving for the wave amplitude and particle distribution function simultaneously, a form for the wave amplitude is chosen from physical considerations, with a damping coefficient chosen to give conservation of energy. With this simplification results are obtained over a range of parameters, without a great deal of computational effort.

Dither in nonlinear systems

A dither is a high-frequency signal introduced into a nonlinear system with the object of augmenting stability. In this paper,[1] it is shown that the effects of dither depend on its amplitude distribution function. The stability of a dithered system is related to that of an equivalent smoothed system, whose nonlinear element is the convolution of the dither distribution and the original nonlinearity. The ability of dithers to stabilize large classes of nonlinear systems is explained in terms of an effective narrowing of the nonlinear sector. A feature of the approach taken here is that a deterministic (i.e., strong) concept of stability is established under probabilistic (i.e., weak) assumptions on the dither.

Statistical properties of magnetization discontinuities in technical steels

The application of statistical methods to the study of technical steels is described and a statistical amplitude distribution function for the magnetization discontinuities is presented, making use of the correlation between the amplitudes observed and the structure of the material. It is shown by experimental measurements that the optically measured grain size distribution of fine-grain low-carbon steel correlates closely over a wide range with the distribution of the pulse heights measured. The results of previously performed measurements have been investigated statistically using a multichannel analyser, and these pulses are shown to obey Weibull's distribution. The physical characteristics of the distribution function of the magnetic discontinuities and its computed parameters are discussed.

Fourier Analysis of Spherical Aberrations

Aberration in imaging systems due to the presence of spherical surfaces is usually treated by the expansion of the aberration function in terms of a complete set of polynomials that are orthogonal over the interior of a circle of unit radius. This method involves a set of complicated programs, and the interpretation of the components in the expansion in terms of physical quantities is not possible. To avoid such complications, Fourier transform techniques are used. This method is applied to determine the diffraction pattern of a circular acoustical reflector. It is shown that the amplitude distribution function using this technique and the language of J. W. Goodman's Introduction to Fourier Optics is consistent with experimental results obtained in the laboratory. In particular, the effects of varying the radius of the spherical surface, index of refraction, and wavelength in the reduction of spherical aberration are obvious in the resulting equation. [This work supported by ARPA and monitored by ONR.]

Distribution of Click Amplitudes

The amplitude of a click is defined as the output of an ideal FM demodulator at the time of occurrence of the click. It is shown that the amplitude of a click (when the carrier is unmodulated) is a function of two random variables whose join distribution function is derived under the condition that a positive click has occurred. The distribution of positive click amplitudes is derived with the carrier-to-noise power ratio (CNR) at the output of the intermediate frequency (IF) filter as a parameter The results are extended to include negative clicks. The probability density and distribution functions of click amplitudes are plotted for the special case of a rectangular IF and CNRs of 4 and 9 The effect of modulation on the distribution of click amplitudes is determined. Results are given for the special case of a sinusoidal modulating signal fully deviating the carrier and a rectangular IF

Complete model for the statistical composition of the end-plate potential

A statistical model for the composition of end-plate potentials has been designed and used for detailed study of end-plate potential's amplitude distribution function. The model is based on the present knowledge of synaptic function and takes into account the miniature end-plate potential's amplitude distribution, pulse shape and latency fluctuation as well as the residual potential difference across the membrane, non-linear summation of unit quanta and the mean quantum content. The building up of end-plate potentials is a stochastic process in which all of the above factors have significant influence. The computer model for statistical building of end-plate potentials is designed on the basis of the Monte Carlo technique. The end-plate potentials' distribution is not calculated, but is built up in the same way as in a living neuromuscular junction through the addition of miniature end-plate potentials which have random amplitudes and random times of arrival. Large numbers of experiments have been performed with the model with the mean quantum content in the range from 2 to 200, mean amplitudes of miniature end-plate potentials in the range of 0·2 to 0·8 mv, coefficient of variation of miniature end-plate potentials in the range of 0·1 to 0·2, number of depolarizations per experiment up to 25,000. Latency fluctuation and pulse shape have been taken according to the measurements by Katz & Miledi (1965). Very good agreement between results produced by the theoretical model and real experiments was obtained. Coefficients of variation of end-plate potentials produced by the model are in good agreement with experimental results for all values of mean quantum content. This is due partly to the fact that not only non-linear summation but also the latency fluctuation diminishes the coefficient of variation.

Amplitude distribution of an electromagnetic pulse propagated through the atmosphere

Peak signal strength is one critical parameter in analyzing the vulnerability of electronic systems to the electromagnetic pulse (EMP) generated by a nuclear burst. Two probability distribution functions of EMP signal amplitudes are derived using the central limit theorem. These distribution functions express peak amplitude as a percentage of total incident signal energy.

The limiting behaviour of the turbulent transverse velocity component close to a wall

Electrochemical techniques are used to measure the circumferential component of the velocity gradient sz at the wall of a pipe through which a turbulent fluid is flowing. The ratio of the root-mean-squared value of sz to the time-averaged velocity gradient is found to be 0·09 or 0·1, depending on whether corrections are made for frequency response. The frequency spectrum is similar to that for the component of the wall velocity gradient in the direction of mean flow. The amplitude distribution function for sz is very roughly approximated by a Gaussian distribution.

Radiometric Measurement of Attenuation and Emission by the Earth's Atmosphere at Wavelengths From 4 cm to 8 mm

Measurement instrumentation for the investigation of atmospheric noise fluctuations at 4 cm to 8 mm wavelengths is described. A brief review of the published literature on this subject is presented in support of the methods used in the measurement. Observations using the sun as a background source to obtain a measure of atmospheric opacity are described as providing an average value over an observing period frequently longer than the time during which a significant change in atmospheric opacity is detectable at these short wavelengths. Preliminary observational data are presented which indicate the potential of the measurement instrument to obtain the amplitude distribution function of sky noise fluctuations for various weather cases.

On the Application of Some Digital Sequences to Communication

The effects of filtering a random binary sequence with certain finite memory linear and nonlinear filters are considered. Of interest are the statistical properties of the filtered sequence such as the power spectral density and amplitude distribution function, as well as methods for recovering the original sequence. Applications of these techniques to communication systems are discussed.

<tex>Z</tex>-transform theory for linear array analysis

The author wishes to amplify Christiansen's recent communication [P.L. Christiansen, On the closed form of the array factor for linear arrays, IEEE Trans. on Antennas and Prop. (Commun.), vol. AP-11, p. 198, March, 1963.] on the use of Z-transform theory for obtaining a closed form for the array polynomial of linear antenna arrays. He is convinced that his proposed approach provides a concise and convenient way for linear array analysis, and his new results will be useful. In his communication Christiansen stressed that the Z-transform approach does not require that the envelope function of the current distribution be periodic, nor is the approach limited to envelope functions with a linear phase relation. That a unit gate function could be used in cases where the amplitude envelope is not representable by a periodic function was pointed out previously by the present author [1960]. Christiansen gave the array factor of a very complicated amplitude distribution function without showing the intermediate steps. In an effort to clarify the situation, two examples are provided that may prove to be helpful to interested readers.

Comparison of Acoustical and Telephonic Speech-Signal Instantaneous Amplitude-Density Functions

Recordings of speech material, read in an anechoic chamber and picked up simultaneously both by a telephone handset and by a high-quality condenser microphone, were made with a system which preserves waveform. Both signals were kept polarized. The signal from the condenser microphone is a good replica of the acoustical pressure wave, at least for the lower frequencies where the greater part of the speech energy lies; and, consequently, the statistics of the telephone signal can be compared with those of the high-quality signal to indicate the statistical transformation a given speech signal undergoes when transduced by a standard telephone transmitter. An analog-to-digital converter and a computer were used to get fine-grained histograms from which density and distribution functions of the instantaneous amplitudes of the 2 signals could be calculated, and these have been compared as to peak factor and asymmetry for a sufficiently large population, so that the statistics for English-speaking males begin to stabilize.

Radio-Frequency Noise of an Immersed Langmuir Probe

The rf noise power spectrum at a Langmuir probe in a cold-cathode neon discharge is observed over a 50-kc to 10-Mc range. The statistical noise amplitude distribution function is found to be Gaussian and the probe noise can be analyzed in terms of shot noise of the probe electron and ion currents using a substantially constant probe sheath impedance at different probe potentials. The noise current spectral density increases at low frequencies in agreement with hot-cathode discharge probe observations, has a minimum at about the shot noise current level, and has a high-frequency increase corresponding to that observed by Shimada [K. Shimada, Proc. I.R.E. 49, 632 (1961)]. It is believed that freedom from wall effects can be attained, since relative insensitivity of the noise to discharge pressure, current, and purity is observed. The introduction of stationary striations does not influence the power spectrum other than to superimpose very sharp oscillatory peaks which are easily altered by changing discharge current or pressure. Low-frequency fluctuations in the plasma can be largely eliminated by using large diameter discharge tubes.

Noise performance of transistors

The accuracy with which the Beatie lumped model represents transistor noise performance is experimentally verified. The noise model consists of three statistically independent shot-noise generators connected across the lumped elements representing the generation-recombination and diffusion mechanisms of the transistor. In addition a thermal-noise generator is used to represent the noise contributed by the base spreading resistance. By comparing measurements with calculations based on the model, not only is good over-all agreement obtained, but it becomes evident that certain bias and source conditions allow separate confirmation of the internally postulated noise generators. The statistical properties of transistor noise are studied using a laboratory-constructed amplitude-probability density analyzer. The results show that the first-order amplitude distribution function of transistors is indeed gaussian.

One-Dimensional Plasma Model

A one-dimensional plasma model consisting of a large number of identical charge sheets embedded in a uniform fixed neutralizing background is investigated by following the sheet motions on a high-speed computer. The thermalizing properties and ergodic behavior of the system are examined and found to be in agreement with the assumption that one is equally likely to find the system in equal volumes of the available phase space. The velocity distribution, Debye shielding, drag on fast and slow sheets, diffusion in velocity space, the Landau damping of the Fourier modes, the amplitude distribution function for the Fourier modes, and the distribution of electric fields felt by the sheets were obtained for the plasma in thermal equilibrium and compared with theoretically predicted values. In every case, except one, the drag on a slow sheet, the numerical results agreed with theory to within the statistical accuracy of the results. The numerical results for the drag on a slow sheet were about a factor of 2 lower than the theory predicated indicating that the approximations made in the theory are not entirely valid. An understanding of the cause of the discrepancy might lead to a better understanding of collisional processes in plasmas.

The Discrepancy from the Normal Distribution of the Probability of the Response of Nonlinear Control Systems Subjected to a Gaussian Random Input

In this paper, the discrepancy from the normal distribution of the response of nonlinear control systems excited by a stationary Gaussian signal is quantitatively discussed by evaluating the skewness, β1, and the kurtosis, β2, for the probability distribution of response. The approach is carried out by introducing the theory of a Markov process and solving the Fokker-Planck equation. Nonlinear characteristic considered here is a symmetric one. The discussion is, therefore, directed to the evaluation of the kurtosis, β2, as function of the various values of nonlinearity. The remainder of this paper is devoted to justify the previous consideration as already presented by the authors that the response has the same amplitude distribution function as the system input. For this purpose, a new quantity, the confidence, is defined. As a result, the dependency of nonlin-earity in the system on the probability distribution of response is obtained and the confidence that the probability distribution function can be regarded as the normal one is also evaluated by numerical computations.

Random noise with bias signals in nonlinear devices

A number of investigations have been made in recent years about the transmission of Gaussian noise through nonlinear devices. In many cases, simplification or approximations were needed to make analytical solutions possible, and only zero-average Gaussian input signals were used when the results were applied to feedback control systems. This paper presents a different approach to the problem of noise transmission through non-linear single-valued elements. Basically, amplitudes removed by nonlinear saturation or deadzones are replaced by impulses in the amplitude distribution functions of the output signals, and the resulting first and second moments of the output distribution are computed to yield the average and rms value of the output signal. The solution may be found by graphical or mathematical integration, a visual representation of the phenomenon is obtained, and input signals with any distributions having non-zero average values may be considered. It is shown that there is an equivalent transmission function or describing function for the average value of the noise, another for the rms value, and that one is a function of the other. Examples of the functions are given and the simpler functions with zero-average values are compared to the results obtained by other methods. Finally, the application of the noise describing functions to feedback control systems is discussed. Theoretical results are compared with those obtained from analog simulations.

Effect of arbitrary phase errors on the gain and beamwidth characteristics of radiation pattern

Simple expressions have been obtained for predicting the maximum loss in antenna gain when the peak value of the aperture phase deviation is known. It is not necessary to know the exact amplitude or phase distribution function as long as the phase errors are relatively small; and the same expressions may be used for both rectangular and circular aperture cases. Relations have also been established such that the maximum change in the main-lobe beamwidth can be predicted from the knowledge of the amplitude distribution function and the peak phase deviation.

Experimental Study of the Spectrum of Isotropic Turbulence, I

In order to clarify the detailed structure of turbulence, measurements were made on the fluctuation of spectral component which is the output of band-pass filter. Using a voltage integrator and a counter-chronograph, two quantities were observed in the so-called isotropic turbulence produced by a grid in a wind-tunnel. One is the distribution function of amplitude averaged in a certain time interval. Dispersions of distribution under various experimental conditions are computed and compared. The other is the time correlation function of fluctuating output. The magnitude of correlation is not so small even for a time interval of 0.5 second and curves seem to be classified into two groups according to the central frequency.