Uncorrelated Scattering Research Articles

Performance of coherent data transmission in frequency-selective Rayleigh fading channels

The error probability of digital transmission through a frequency-selective channel with random multipath fading due to uncorrelated scatterers is calculated. The error probability is averaged with respect to the randomly varying delays, with a given delay power spectrum. Two coherent receivers, suboptimum without any knowledge of the channel and optimum with perfectly known channel, are compared. The results are applied to continuous-phase-modulation (CPM) signals. >

A characterization of wave-front distortion in ultrasonic imaging

The influence of wave-front distortion in ultrasonic imaging is examined using a model that represents the signal in the imaging system aperture in terms of the aperture, the scattering source, and a distortion function. This model, along with assumptions that the beam pattern factors into the product of axial and lateral functions and that the distortion is represented by a time shift that is either a sinusoid or Gaussianly distributed, permits the development of expressions for the correlation function of the aperture and, thus, the intensity of the beam pattern. Although these assumptions are idealizations, the approach yields relatively simple expressions that show the way time delay couples into beamwidth to degrade resolution. Further simplifications follow from assuming that the distortion is small or that the axial and lateral beam functions have specific forms such as Gaussian shapes. The analysis indicates that compensation of distortion using time delays estimated from cross correlation of signals on adjacent elements of an aperture maximizes overall average intensity in an image of uncorrelated scatterers. The results provide representations of distortion in ultrasonic imaging systems and may be used to study the relationship between techniques that compensate for wave-front distortion.

Diffusion of Non‐Interacting Quantum Particles in a Disordered System. The Transport Length in the Classical Limit

AbstractWithin the model of uncorrelated scatterers and a long mean free path the recently proposed structure of the density matrix in a current‐carrying state is shown to be produced and reproduced by the superposition of all scattered waves. The corresponding transport length corresponds to its classical value, but the physical picture is a new one.

Communication over fading dispersive channels, optimal receivers and signals

We study the problem of designing the optimal receiver for a dispersive channel and the optimal signals to be transmitted over a given dispersive channel. The assumed transmission channel is WSSUS (Wide Sense Stationary Uncorrelated Scatterers), randomly time variant, so it is characterized by its scattering function. We study first the effective realization of the optimal receiver, given a scattering function, and we point out that the structure of the optimal receiver varies according to the discrete or continuous structure of the scattering function over the time frequency plane. We give some precise results in the case of multipath transmission and in the case of modulation-transmission. In a second part, we are interested in the signal design problem. Some new results are given which permit defining—in a way more direct than usual—the class of optimal signals for a given dispersive communication.

Integral representation of the incoherent field in disordered materials IV. Relation to multiple scattering theory and inclusion of pair correlations

AbstractThe theoretical scheme developed in Parts I to III is related to equations which exactly describe the multiple scattering problem. This procedure exhibits that the hitherto investigated simple variant is restricted to uncorrelated scatterers and that it rests on the same approximation idea as a previously developed medium approach to the coherent field. The neglect of fluctuations which is characteristic for a medium theory can be seen explicitly. The basic integral representation is generalized by the inclusion of pair correlations. The generalized representation is determined uniquely and forms the starting point for a generalized wave theory of diffusion.

Theory of ac and dc electric conductivity by noninteracting electrons in correlated arrays of fixed scatterers

It is shown that the procedure of averaging over positions of scatterers interacts synergetically with the application of the diagonal-projection-operator (..delta..,..delta..') decomposition to yield an exact expression for the ac and dc conductivity for an arbitrarily ordered array of fixed potentials. The formula is evaluated to lowest order in the scattering strength for arrays of scatterers with various degrees of order from uncorrelated to periodic; it is a function of density, frequency, and temperature. dc conductivity, for example, is a minimum for a perfectly random array, increasing to infinity for a periodic lattice at T = 0. Application to thermal-disorder resistivity in metals is mentioned. In appendices we discuss pervious attempts to solve this problem, simplify the quantum-mechanical Green-Kubo formula, and calculate the conductivity of classical electrons moving among dilute uncorrelated scatterers.

Diffraction-limited atmospheric imaging of extended objects*

Recent studies have shown that the use of predetection processing can greatly improve the quality of images formed through atmospheric turbulence. It is generally assumed that the object to be imaged lies within a single isoplanatic patch, and it is recognized that the angular subtense of the isoplanatic patch may be quite small, comparable to the seeing limit. In this paper the use of predetection processing for objects which comprise many isoplanatic elements is considered. It is shown that an atmosphere–lens system behaves macroscopically as though it consisted microscopically of a collection of uncorrelated scatterers with random scattering strengths which contribute image components with varying spatial and spatial-frequency displacements. In communication theory such systems are referred to as wide-sense stationary uncorrelated scatter (WSSUS) channels and are characterized by a macroscopic scattering function. The macroscopic scattering function for the atmosphere is derived and used to show that under typical turbulence-strength conditions the seeing limit can resolve the isoplanatic patch. Thus the fields from different isoplanatic patches of an extended object may be separated in the image plane of an objective lens and processed individually to obtain a diffraction-limited image.

Distributed-parameter state-variable model for time-variant channels

Signals transmitted over certain randomly time-varying linear channels undergo spreading both in time and in frequency. This paper considers a particular class of doubly spread channels: those with an impulse response that can be represented as the output of a distributed linear system, the dynamics of which are governed by partial differential state equations driven by a white noise process. The model is shown to include channels characterized by wide-sense stationary scattering (WSS), uncorrelated scattering (US), or both (WSSUS). When reception takes place in white Gaussian noise, the optimum, quadratic detector can be realized as an estimator-correlator. For the doubly spread channels encompassed by the model, the causal least-squares estimator of the channel response is obtained in the form of a linear filter with a distributed-parameter state-variable description. Certain error probability bounds and approximations can be evaluated upon solution of the receiver equations. Numerical results are given for an example: binary frequency-shift-keying (FSK) signaling over a particular WSSUS channel. The effects of the channel's frequency-time spread factor (BL) and the signaling interval on the optimum receiver performance, as measured by the Bhattacharyya distance, are investigated.

A study of the relationship between multipath distortion and wavenumber spectrum of refractive index in radio links

The study reported here is concerned with relating the multipath and frequency selective fading characteristics of radio links caused by scatter phenomena to the statistical spatial characteristics of the refractive index. Two specific antenna configurations are studied in detail: the LOS (line-of-sight) case where the two antennas are pointed concentrically and the OTH (over-the-horizon) case where the antennas point along a great circle path. For each of these antenna configurations two scatter situations are considered: isotropic scatter with the propagation medium controlling the focusing of energy and locally isotropic scatter with narrow-beam antennas. In all cases single scattering theory is used and narrow-band signal transmission is assumed. It is found that when multipath is of any importance in causing signal distortion for a LOS link, the scatter portion of the channel may be modeled as a continuum of uncorrelated scatterers. Also OTH links are found to be characterizable in the same way. Single integrals and specific analytic and numerical examples are given relating frequency correlation functions and delay power spectra to the wavenumber spectra of assumed refractive index fluctuations. It is shown that a measurement of the frequency correlation function or delay power spectrum can yield an estimate of the wavenumber spectrum of the refractive index fluctuations.

Characterization of Randomly Time-Variant Linear Channels

This paper is concerned with various aspects of the characterization of randomly time-variant linear channels. At the outset it is demonstrated that time-varying linear channels (or filters) may be characterized in an interesting symmetrical manner in time and frequency variables by arranging system functions in (timefrequency) dual pairs. Following this a statistical characterization of randomly time-variant linear channels is carried out in terms of correlation functions for the various system functions. These results are specialized by considering three classes of practically interesting channels. These are the wide-sense stationary (WSS) channel, the uncorrelated scattering (US) channel, and the wide-sense stationary uncorrelated scattering (WSSUS) channel. The WSS and US channels are shown to be (time-frequency) duals. Previous discussions of channel correlation functions and their relationships have dealt exclusively with the WSSUS channel. The point of view presented here of dealing with the dually related system functions and starting with the unrestricted linear channels is considerably more general and places in proper perspective previous results on the WSSUS channel. Some attention is given to the problem of characterizing radio channels. A model called the Quasi-WSSUS channel is presented to model the behavior of such channels. All real-life channels and signals have an essentially finite number of degrees of freedom due to restrictions on time duration and bandwidth. This fact may be used to derive useful canonical channel models with the aid of sampling theorems and power series expansions. Several new canonical channel models are derived in this paper, some of which are dual to those of Kailath.

Correlation functions in a tapped delay line model of the orbital dipole channel

This paper considers the properties of the tap gain functions in a tapped delay line model of the orbital dipole channel. Specifically, auto-correlation and cross-correlation functions are computed for these tap functions in terms of assumed statistics for the motion of illuminated dipoles. It is demonstrated that when a certain inequality involving the transmitted signal bandwidth is satisfied, these correlation functions become identical to those which would arise from the corresponding tap gain functions of a suitably defined channel consisting of a continuum of motionless scintillating uncorrelated scatterers. In effect, this inequality sets an upper bound to the bandwidth of the transmitted signal, beyond which the orbital dipole channel may not be modeled as such a continuum of uncorrelated scatterers. Simplified expressions for tap gain correlation functions are derived for the case of large signal bandwidths, including the subcase in which the above mentioned inequality is satisfied. In the latter case the tap gain functions are shown to become mutually uncorrelated for all time shifts.