Time-varying Medium Research Articles

Image restoration under random time-varying blur

We present a method for restoring a fixed scene, which is imaged through a random time-varying medium, from a number of observation frames contaminated by additive detection noise. The method is based on frame-pixel separability. First, frames are deblurred individually (ignoring additive noise). Then a pixel is treated individually by finding an estimate of its value based on its values in all deblurred frames. In some conditions the final estimate turns out to depend on only two statistics: the frame-averaged image and the frame-averaged spatial correlation function. The latter is the basis of speckle interferometry.

Electromagnetic propagation and scattering in time-dependent moving media.

A new formalism is introduced for discussing propagation of electromagnetic waves in space- and time-varying media. The approach taken here is to supplement the velocity-independent solution by a correction factor. This factor involves a four-dimensional line integral and is therefore a WKB-type solution. The formalism is relativistically exact to the first order in v/c. Special cases are discussed, demonstrating how time-harmonic velocity fields modulate the propagating electromagnetic fields. Scattering problems involving time-varying surfaces have been considered in the past. The scatterers were situated in free space (vacuum), or if material media were involved, the mechanical interaction of the moving surfaces with the medium was usually ignored. The new theory presented here facilitates the analysis of electromagnetic scattering problems in the presence of combined surface and medium motion. Consequently, mass continuity at the moving surfaces can be preserved. This makes the modeling of such problems much more realistic. Simplified canonical problems involving interaction of electromagnetic and mechanical waves are discussed to highlight the effects resulting from the new boundary considerations. Surprisingly, for bounded surfaces the time-varying medium effects vanish in the far field. This means that previous results which simply ignored the medium motion are still valid as good approximations.

Sunspot Number Series Envelope and Phase

The sunspot number series R( t) from 1700 to date is found to be representable by R( t) = I Jf' {Re( E( t) exp[i {wo t + ( t) I]) + U( t) 1 I, where Wo is the angular frequency corresponding to a period of 22 years, E(t) is the instantaneous envelope amplitude, (t) is the instantaneous phase of a complex time-varying analytic function, U(t) is an undulation of low amplitude and period about 30 (22-year) cycles and jy is a nonlinear operator whose main effect is to introduce a small amount of third harmonic (period about 7 years). The justification for the 22-year period is the known fact that the observable sunspot magnetic fields reverse polarity every 11 years or so at the time of sunspot minimum; the undulation has been demonstrated, and its period determined, in fossil records discovered by Williams; and the third harmonic is an expected consequence of minor nonlinearity in the dependence of the arbitrarily defined R( t) on the physical cause of sunspots. The algebraic representation is established by the Hilbert transform method of forming a complex analytic function as proposed by Gabor. The method reveals three obscuring features that may be alleviated as follows: use the alternating series R� (t) in which alternate II-year cycles take opposite signs, remove the third harmonic, and subtract the undulation. These justifiable steps remove artificial components, such as sum and difference frequencies, that are gratuitously and nonlinearly introduced by conventional Fourier analysis as applied to the rectified, or absolute, value of the 22-year oscillation. Then a complex envelope E( t) exp {i ( t)j can be discerned whose intrinsic behaviour can be studied to reveal statistics that bear on the physical origin of the solar cycle. The results favour a deep monochromatic oscillator whose influence is propagated to the observable surface via a time-varying medium. The r.m.s. value of the component of E(t) is 0�4 of the mean and the characteristic time is a century. Frequency analysis of the envelope does not support a 78-year period in the modulation noticed by Wolf. Both the statistical frequency distribution of the amplitude E( t) and its spectrum are subject to refinement by analysis of fossil solar records. The results do not favour the theory that the 22-year period is set by the natural frequency of a resonator with characteristic damping subject to random turbulent excitation. Also disfavoured is the theory of energy release at intervals determined by a relaxation process. Correlation has been found between the phase departure ~(t) from linear and envelope amplitude and attributed to propagation of the magnetic .cycles through a time-varying, such as a convecting, medium. A correlation not depending on Hilbert transform analysis is predicted between the reciprocal cycle length and envelope amplitude and found to� exist. Variability of the sunspot cycle length can be viewed as a Doppler shift due to propagation in a time-varying medium and the Wolf modulation then represents the concomitant intensity change. Agreement has been found between E(t) and '(t) but not explained. If the explanation is dispersion in the propagation of the assumed magnetic flux waves then there is a mode of oscillation. that has the characteristics required for the undulation U( t). Extra buoyancy possessed by the magnetic field of strong cycles accounts for the fast rise time of strong cycles.

Green functions for transport problems in time-varying media

The Green functions are formulated for boundary-value problems associated with the phenomenological parabolic transport equation with time-dependent coefficients.

Electromagnetic Wave Energy and Momentum Equations in Transparent, Dispersive, Space- and Time-Varying Media

It is shown that the transport equations for the electromagnetic wave energy density W k and momentum density P k in transparent, dispersive, space- and time-varying media are given by d W k /d t =ω k -1 ∂ω k /∂ t W k +2γ k W k and by d P k /d t = - k -1 ·∂ω k /∂ rP k + 2γ k P k , respectively, where d/d t denotes the total time derivative along the ray trajectory and γ k is the growth rate. The terms ω k -1 ∂ω k /∂ t W k and - k -1 ·∂ω k /∂ rP k result from the fact that the wave energy and momentum density are not adiabatic invariants in space- and time-varying media. It is assumed that the geometric optics approximation and the nonlocal lincar response theory are valid.

A numerical method for transient analysis of coupled antennas

A numerical method for finding the transient response induced on coupled thin cylindrical antennas excited by an electromagnetic pulse (EMP) is introduced. Numerical examples are given for two coupled thin antenna that are broadside incident by an idealized EMP. Applications of the method are made to calculate the induced currents on corner reflector antennas, and numerical examples are given to illustrate how the transient responses are modified by coupling effects. Examples of applying the method to include the effects of the time-varying medium on the responses are also shown with an assumed time-varying air conductivity.

Reflection of electromagnetic wave from a time-varying medium

For the purpose of radiowave propagation in the ionosphere, in a plasma and in other media, the reflection and transmission of an electromagnetic wave from a suddenly changed conductivity, when a plane wave is normally incident upon an interface separating two conducting media, are studied. The field expressions are exactly obtained through the use of the Laplace transform. The discussion is primarily on the time behaviour of the reflected wave.

Space-time considerations for doppler effects in inhomogeneous time-varying media

When signals are propagating along ray paths in timevarying inhomogeneous media, there are cumulative effects, leading to Doppler phenomena. By representing the ray path as a two-parameter function, which takes into account its space-time evolution, now these effects are analysed by using special-relativistic concepts. This facilitates the discussion of high-velocity signals, as encountered in the electromagnetic theory, and serves as a check on previous results.

The group doppler effect

The Doppler effect for the modulation of a wave signal is investigated. This group Doppler effect is considered for rays in inhomogeneous, dispersive, absorbing, anisotropic and time-varying media. In addition, the receiver or scatterer is allowed to move with respect to the medium. It is shown that the new effects depend on the group velocity and its direction relative to the normal.

Generalized doppler effect in time-varying media

Doppler effects for time-varying inhomogeneous media are considered. The ray path is represented in terms of a two-parameter function, this takes into account its time evolution. Time dilation along the path is discussed. This provides, in a systematic way, an alternative derivation to a heuristic expression previously given by Gill. The phase function is integrated along the ray path and yields the various Doppler effects in a straightforward way. It is shown that the generalized Doppler effect is produced by the relative motion of source and receiver with respect to the medium, and the cumulative rate of change of frequency and propagation vector along the path.

Numerical method for computing multipath interference in a time-varying medium

Ray-model simulations of multipath interference of CW signals require precise computation of travel times along the paths between source and receiver. The travel times for each ray path must be known with sufficient precision to resolve the phase of the arrival. The effects of time-varying sound-speed profiles can then be simulated by allowing the state of the medium to change slightly for each multipath summation. This letter is concerned with a numerical method for making such computations.

Compression of the Wave Power Density by Means of a Time-Varying Medium

Propagation of waves in a medium homogeneous in space but with time-varying properties is considered. It is shown that this can result in compression of a signal and a gain in power. The signal becomes also frequency modulated. For nondispersive waves, such as acoustic waves, there is an increase in frequency, whereas for transverse electromagnetic waves the frequency is lowered for the particular scheme of propagation considered.

Invariant imbedding in time-varying homogeneous nondispersive media

The invariant imbedding approach is applied to electromagnetic wave propagation in homogeneous temporally varying media. As a convenient example a plane wave is considered. The result is a coupled pair of complex first-order linear differential equations for the amplitude of the forward and backward propagating waves which arise when the index of refraction of the medium is changed in an arbitrary way from one steady-state value to another. This approach provides an alternative point of view in the analysis of time-varying medium devices such as electrooptic phase modulators.

Transmission of electromagnetic waves into time-varying media

Expressions are obtained for the electromagnetic fields transmitted into the time-varying medium when a plane wave is incident upon either a dielectric or dispersive half-space. Solutions are obtained for the case when the medium is changed in a stepwise fashion, and also for the case when the medium varies slowly and continuously.

Wave propagation in time-varying media

This paper examines the propagation of waves in media which are spatially homogeneous but whose properties vary with time abruptly or continuously. Emphasis is placed on the excitation problem and on source-dependent phenomena which are not evident when attention is given only to the source-free case. After a presentation and interpretation of various exact closed-form solutions for simple nondispersive and dispersive media undergoing sudden or gradual temporal changes, attention is given to integral representations required under more general conditions. In the ensuing far-zone asymptotic analysis, dispersion surfaces and space-time rays are used extensively for identification of wave packets and other wave constituents descriptive of the field, and a comparison is made between the asymptotic results and the exact solutions noted above. Asymptotic field solutions are then derived by a direct ray procedure, without intervention of integral representations. The examples considered exhibit a variety of phenomena, among which the most interesting is the focusing of reflected waves when a dispersive medium undergoes a sudden change.

Diversity of modulation—a new concept in communication technology

A new concept called ‘diversity of modulation’ has been presented. Its purpose is to reduce the fading range of the signals received through a time-varying multipath transmission medium. A digital-communication system with N-order diversity utilising this concept has also been described, and its performance is compared with the conventional diversity systems.

Electromagnetic pulse scattering in time-varying inhomogeneous media

A finite difference solution technique is used to solve Maxwell's equations directly in the treatment of electromagnetic pulse scattering in a time-varying inhomogeneous medium. In particular the scattering from a cylindrical rod inside a cylindrical waveguide is considered where axial symmetry is obtained.

Correlation Measurements on an HF Transmission Link

In data transmission over long-haul HF circuits, the signal is subjected to the vagaries of a turbulent ionospheric medium. To permit accurate performance predictions it is necessary to obtain a comprehensive channel description. This paper describes a program designed to achieve two goals. 1) To determine the feasibility of performing an experiment with sufficient accuracy to determine the time-variant medium transfer characteristic. 2) To obtain sample descriptions of these characteristics. The statistical medium descriptions obtained included first-order probability distribution functions of phase and amplitude and the secondorder time-frequency correlation functions. Records of the above medium descriptors obtained over a more extensive testing period will aid in the determination of optimum signaling techniques and in reliable data performance predictions.

Signal processing in a nonperiodically time-varying magneto-elastic medium

Signal processing in a nonperiodically time-varying magneto-elastic medium

Charge density response in a lossy time-varying medium

Williams' work on charge density response in a lossy yet constant medium has been extended te a lossy time-varying medium.