Scattering from a 2D sea fractal surface: fractal analysis of the scattered signal

For first part of the work see ibid., vol.50, no.4, p.426-34 (2002). We analytically demonstrate that the electromagnetic (EM) sea scattered signal retains certain fractal properties of the sea surface. Starting from the expression of the scattering coefficient derived in the first part of the work, we show theoretically that the fractal dimension of the in phase and quadrature components of the EM signal scattered from a sea fractal surface is linearly related to the fractal dimension of the sea surface. This result holds under the hypothesis of Kirchhoff approximation. Bearing in mind that the sea fractal dimension gives a measure of the degree of surface irregularity, we can exploit this result to make a remote estimation of the surface roughness through the fractal analysis of the scattered signal. This is a very interesting result and it could be successfully used in remote sensing applications for classification, target detection, sea surface traffic monitoring, sea current analysis, oil slick and ship wake recognition, pollution analysis etc.     

Scattering from natural soils modeled by dielectric fractal profiles: the forward-backward approach

The forward-backward (FB) method is an efficient technique for numerical evaluation of electromagnetic scattering from rough surfaces. In its usual formulation, this technique can be only applied to perfectly or highly conducting surfaces. In addition, up to now FB has been employed to compute scattering from surfaces modeled by Gaussian stochastic processes with Gaussian or Pierson-Moscowitz spectra. Accordingly, this technique can be fruitfully used for numerical simulations of scattering from sea surfaces. However, in order to properly deal with natural soil surfaces, extension to the dielectric interface case and to fractal surface models is needed. Extension of the FB method to the dielectric interface case has been recently presented, whereas application to fractal surface models is presented here. Original contribution of the present paper is twofold. First of all, the FB method for dielectric profiles is framed within the theory of iterative methods for the solution of linear systems. In addition, application of the FB method to dielectric band-limited fractional Brownian motion fractal one-dimensional surfaces is explored. Numerical experiments show that, for most of realistic values of dielectric constant and fractal parameters actually encountered for natural soil profiles, the FB method is very rapidly convergent, and its results are in perfect agreement with "exact" ones (i.e. with results of method of moments solved via a direct method).     

The Luneburg-Kline expansion determined from bandlimited scattering data

A method for determining the coefficients of an asymptotic1/omega-expansion, known as Luneburg-Kline expansion, from bandlimited scattering data is presented. The coefficients are derived from the output signals of an orthogonal filter. The method is applicable to calculate geometric quantities as principal radii or principal directions of curvature of simple scatterers. Furthermore early parts of transient backscattered signals can be analyzed, e.g., to reconstruct the early ramp or impulse response.     

Moving average restoration of bandlimited signals from noisyobservations

The purpose of this paper is to describe the extension of the Whittaker-Shannon sampling theorem to reconstruction of bandlimited functions in the presence of zero mean, uncorrelated noise. It is shown that the classical Whittaker-Shannon sampling scheme is not consistent in the case of noisy measurements, and new reconstruction algorithms based on the moving average smoothing are proposed. The weak and strong consistency of the algorithms is established, and the rate of convergence is investigated. The theory is verified in the computer simulations     

Beyond bandlimited sampling

Digital applications have developed rapidly over the last few decades. Since many sources of information are of analog or continuous-time nature, discrete-time signal processing (DSP) inherently relies on sampling a continuous-time signal to obtain a discrete-time representation. Consequently, sampling theories lie at the heart of signal processing devices and communication systems. Examples include sampling rate conversion for software radio and between audio formats, biomedical imaging, lens distortion correction and the formation of image mosaics, and super-resolution of image sequences.     

Filterbank reconstruction of bandlimited signals from nonuniformand generalized samples

This paper introduces a filterbank interpretation of various sampling strategies, which leads to efficient interpolation and reconstruction methods. An identity, which is referred to as the interpolation identity, is developed and is used to obtain particularly efficient discrete-time systems for interpolation of generalized samples as well as a class of nonuniform samples, to uniform Nyquist samples, either for further processing in that form or for conversion to continuous time. The interpolation identity also leads to new sampling strategies including an extension of Papoulis' (1977) generalized sampling expansion     

Limits on bandlimited signals

The problem of estimating the errors arising in a variety of numerical operations involving bandlimited functions is considered. The errors are viewed as responses of suitably created systems, and the analysis is based on the evaluation of the maximum response of these systems in terms of the energy or power of their input. The investigation includes deterministic and random signals, and it is extended to two-dimensional functions and Hankel transforms. Finally, the results are related to the uncertainty principle in one and two variables.     

Reconstruction of multidimensional bandlimited signals from nonuniform and generalized samples

This paper addresses the problem of multidimensional signal reconstruction from nonuniform or generalized samples. Typical solutions in the literature for this problem utilize continuous filtering. The key result of the current paper is a multidimensional "interpolation" identity, which establishes the equivalence of two multidimensional processing operations. One of these uses continuous domain filters, whereas the other uses discrete processing. This result has obvious benefits in the context of the afore mentioned problem. The results here expand and generalize earlier work by other authors on the one-dimensional (1-D) case. Potential applications include two-dimensional (2-D) images and video signals.     

The optimization of bandlimited systems

Bandlimited functions play important roles in the design of signal transmission systems; antennas; digital, analog, and optical filters; tapered transmismon lines, etc. The basic problem arising in all the above-mentioned applications consists in approximating a given nonbandlimited response with a bandlimited system function. This paper presents 1) a unified treat-merit of the mathematical properties of bandlimited system functions; 2) analytical and numerical methods for the optimization of bandlimited systems; and 3) the application of the theoretical results to practical engineering problems.     

Uniform linear prediction of bandlimited processes from past samples (Corresp.)

Forx(t)either a deterministic or stochastic signal band-limited to the normalized frequency intervalmidomegamid leq pi, explicit coefficients{ a_{kn} }are exhibited that have the property that begin{equation} lim_{n rightarrow infty} parallel x(t) - sum_{1}^n a_{kn} x(t - kT) parallel = 0 end{equation} in an appropriate norm and for any constant intersample spacingTsatisfying0 < T < fac{1}{2}; that is,x(t)may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation ofx(t)is uniform in the sense that the coefficients{ a_{kn} }do not depend on the detailed structure ofx(t)but are absolute constants for any choice ofT. The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required.     

Nonuniform sampling of bandlimited modulated signals

In this paper it is shown that an unequally spaced sampling can be implemented on the base of sinc-composition function usually used in the uniform sampling expansion. The nonuniform sampler is based on a level crossing detector (LCD) which produces a sample of carrier whenever the integral of modulating function crosses a threshold level. The information about the modulated signal is contained in the carrier samples taken at the instants which are defined by modulating function. The Riemann-Stieltjes uniform integral sum representation of the Fourier transform of signal is used as a mathematical base for the derivation of the principal results. In the particular case of the representation of single-valued real positive function, the suggested method leads to the signal representation by the equally weighted sampling at the nonuniformly distributed instants. The error estimation and numerical example illustrating the methods are given.     

Optimal reconstruction of bandlimited bounded signals

A formula is presented for restoring bounded, bandlimited signals from a finite number of samples. The restoration is linear in the samples used, and the error bound of the method is the theoretical minimum.     

Adaptive trellis-coded modulation for bandlimited meteor burstchannels

The throughput performances of three adaptive information rate techniques on the bandlimited meteor burst channel are investigated. Closed-form expressions for throughput are derived based on the channel model commonly used in the literature. The throughput performance is compared to the conventional fixed information rate modem and upper bounds on throughput improvement over the fixed rate modem are derived. It is shown that an adaptive technique that uses trellis-coded modulation (TCM) with three phase-shift keyed (PSK) signal sets can increase throughput over the conventional fixed rate modem by more than a factor of 3. Data from the US Air Force High Latitude Meteor-Scatter Test Bed confirm the superiority of the adaptive TCM technique. A practical implementation is suggested that uses a single rate 1/2 convolutional code for all three PSK signal sets. The use of this single code, versus the three best Ungerboeck codes, results in a throughput loss of less than 2%. An expression for the theoretical information capacity of the bandlimited meteor burst channel is derived     

The predictability of continuous-time, bandlimited processes

In communications and signal processing, we can find examples of applications that could benefit from the prediction of a bandlimited random process. We consider a continuous-time linear predictor applied to a bandlimited process. We show that if the past values of the process are known over an interval of arbitrary positive length, the mean squared prediction error may be made arbitrarily small, regardless of how far in the future we wish to make the prediction. We also show that this is no longer true when a certain energy constraint is applied to the predictor. Furthermore, we discuss what this means for the case in which the prediction is based on past values that are corrupted by estimation errors     

Fitting a bandlimited signal to given points

One of the fundamental problems in communications is the transmission of a signal through a bandlimited channel. It should, therefore, be of great interest to find a general method of transmitting an arbitrarily close approximation to any finite signal through a channel which is arbitrarily bandlimited. The present paper develops such a method and evaluates the cost in time and energy to accomplish this feat. The problem of fitting an arbitrarily bandlimited signal to a finite number of arbitrary points is solved, and the minimum energy signal, fitting the points and having a spectrum confined to the given pass band, is found. The behavior of this signal, with shrinking bandwidth, is investigated.     

Scattering from three-dimensional cracks

Scattering from three-dimensional cracks is analyzed and measured. The crack geometry is modeled as a rectangular groove in a perfectly conducting surface. The groove forming the crack may be terminated with an open aperture creating a slit in the conducting surface or with an impedance boundary creating a trough. The scattered fields from a crack are analyzed with two types of scattering mechanisms: a component directly related to the scattered fields from a two-dimensional crack, and a traveling-wave component     

Scattering from wedge-tapered absorbers

The scattering of a plane electromagnetic wave, arbitrarily polarized and normally impinging on a wedge-tapered absorbing structure, is theoretically and experimentally studied. The theoretical solution is found in two asymptotic cases: spacing between consecutive edges, small and large with respect to the incident wavelength. In the former case, a transmission line analogy is set up; in the latter, quasi-optical techniques are applied. Numerical and experimental results are presented; the latter are in good agreement with the theory. The possibility of using the studied structure as an absorbing termination is discussed in some detail, and optimizing criteria, as well as the possibility of obtaining polarization-independent performance, are indicated.     

Scattering from non-spherical hydrometeors

A new theoretical formulation for scattering from a wide class of non-spherical hydrometeors has been developed recently using Waterman’s extended integral equation technique. The transition or T-matrix formulation is an exact solution to the scattering problem and computer programs have been developed to handle both homogeneous and imbedded bodies (e.g., a dielectric imbedded within another dielectric body). This theory will be briefly reviewed followed by sample computations of backscattering calculations of (a)oblate spheroidal icestones of varying sizes and eccentricities, (b)ice-stones with surface perturbations to model roughness, and (c)rough ice-stones covered with a thin coating of water. The method is realistically applicable for sizes up to D ≈3 λand for a wide range of dielectric constants. Comparisons with measurements of backscattering from a non-concentric dielectric-clad spherical perfect conductor is included.     

Scattering from loaded grooves

Electromagnetic scattering from a two-dimensional groove recessed in an arbitrarily thick conducting screen is studied. The groove may be empty or loaded with a lossy material which may or may not completely fill the cavity. For the partially loaded groove, the filling material is assumed electrically dense so that the standard impedance boundary condition is applicable at the top surface of the material. Employing a full-wave analysis, integral equations are derived for the tangential components of the electric field over the aperture. It is shown that the equations are identical for both partially loaded and completely loaded (or empty) cases provided that the aperture admittance of the groove is treated as the equivalent admittance of the internal medium looking into the aperture, thus simplifying the integral equations. When the groove is completely filled by a dense material, the formulation reduces to that corresponding to a direct application of the impedance boundary condition over the aperture.