Investigates adaptive waveform selection schemes where selection is based on overall target tracking system performance. Optimal receiver assumptions allow the inclusion of transmitted waveform specification parameters in the tracking subsystem defining equations. The authors give explicit expressions for two one-step ahead optimization problems for a single target in white Gaussian noise when the tracker is a conventional Kalman filter. These problems may be solved to yield the most improvement possible in tracking performance for each new transmitted pulse. In cases where target motion is restricted to one dimension, closed-form solutions to the local (one step ahead) waveform optimization problem have been obtained. The optimal waveform selection algorithms in the paper may be included with conventional Kalman filtering equations to form an enhanced Kalman tracker. Simulation examples are presented to illustrate the potential of the waveform selection schemes for the optimal utilization of the capabilities of modern digital waveform generators, including multiple waveform classes. The extension of the basic waveform optimization scheme to more complex tracking scenarios is also discussed. >

Optimal waveform selection for tracking systems

The effect of a beam traversing an optically active medium of turbulent fluctuations of the index of refraction in the medium is investigated theoretically. The general far-field case is solved by integration of the wave equation with both gain and fluctuations of the index refraction, simplified by neglecting the Laplacian of the fluctuating part of the electric field and one of the time derivatives. The far-field diffraction pattern is identical to that for a nonactive medium. For small to moderate extinction there is a loss of power from the coherent beam but only a slight change in the diffraction pattern, accompanied by wide-angle incoherent scattering. For high-gain cases it is possible to have the extinction coefficient less than the gain, but with complete extinction of the coherent beam and a large incoherent scattering pattern. The use of a more realistic turbulence spectrum than previously used has led to the discovery of a second branch in the resolution curve, in which the resolution far exceeds the previously given "limiting resolution" as the aperture size is increased beyond a certain limit, and approaches the diffraction pattern based on the aperture, although there may be considerable attenuation in the intensity in the diffraction pattern of the coherent beam. These results are applicable, for example, to the use of large aperture optics in turbulene wind tunnels.

Effect of turbulent fluctuations in an optically active fluid medium.

An adaptive, waveform selective probabilistic data association (WSPDA) algorithm for tracking a single target in clutter is presented. The assumption of an optimal receiver allows the inclusion of transmitted waveform specification parameters in the tracking subsystem equations, leading to a waveform selection scheme where the next transmitted waveform parameters are selected so as to minimize the average total mean-square tracking error at the next time step. Semiclosed form solutions are given to the local (one-step-ahead) adaptive waveform selection problem for the case of one-dimensional target motion. A simple simulation example is given to compare the performance of a tracking system using a WSFDA based tracking filter with that of a conventional system with a fixed waveform shape and probabilistic data association (PDA) tracking filter.

Waveform selective probabilistic data association

The distribution of the sum of log-normal variates is shown for most cases of interest to be very accurately represented by a log-normal distribution instead of a normal or Rayleigh distribution that might be expected from the central-limit theorem. As a result, observation of the log-normal distribution for the fluctuations of flux received after propagation through a random medium can be readily explained, regardless of the size of the receiving aperture.In every case, the log-normal distribution is a better representation than the normal for the distribution of the sum of log-normal variates. However, in some cases not even the log-normal distribution is very accurate. The questions of convergence and accuracy are examined in detail.

Permanence of the Log-Normal Distribution*

Waveform-agile sensing is fast becoming an important technique for improving sensor performance in applications such as radar, sonar, biomedicine, and communications. The paper provided an overview of research work on waveform-agile target tracking. From both control theoretic and information theoretic perspectives, waveforms can be selected to optimize a tracking performance criterion such as minimizing the tracking MSE or maximizing target information retrieval. The waveforms can be designed directly based on their estimation resolution properties, selected from a class of waveforms with varying parameter values over a feasible sampling grid in the time-frequency plane, or obtained from different waveform libraries.

Waveform-agile sensing for tracking

The amplitude and power of a large family of radio signals are observed to have log-normal probability density functions. Among these are signals propagated through random inhomogeneous media, a notable example being low frequency atmospheric radio noise. Of greater importance are certain radar targets that have been observed to have essentially log-normal density functions. Both ships and space vehicles may fall into this category. Curves of probability of detection vs. signal-to-noise ratio for the case of log-normal signals in Gaussian noise have been computed and are presented in this paper. The curves apply for square-law detection with varying degrees of postdetection linear integration. Both fully correlated and completely uncorrelated fluctuating signals are considered. It is shown that for log-normal signal distributions having large variances, the probability of detection differs significantly from that obtained using curves based on an assumed Rayleigh signal distribution.

Detection Probabilities for Log-Normally Distributed Signals

Although the properties of the linear FM signal have been studied previously in considerable detail, such studies have involved rather narrow aspects of the theory. This paper extends the work in several respects. By presenting three-dimensional projections of the conventional ambiguity function of the linear FM signal in more detail than was available before, we can study the sidelobe behavior off as well as on the axes, without weighting, with unilateral weighting in the receiver, and with bilateral weighting. These plots reveal interesting properties related to the signal symmetry in time and frequency. The matched-filter response is then extended to include Doppler distortions of the modulation function. The results show that Woodward's ambiguity function is valid only for signals with relatively modest sophistication, even though in most practical situations one is interested only in those undistorted parts of the matched-filter response in the vicinity of the delay axis. Plots of the response are presented for various degrees of distortion, for signals with and without weighting. Lastly, we consider the effects of a mismatch in range acceleration, again for the various cases of interest. The results convey a thorough insight into the properties of chirp radar under a broad range of operational conditions.

Matched-Filter Responses of the Linear FM Waveform

Radar for detecting and tracking short range airborne targets using a non-scanning beam to illuminate the entire search space, and processing the return signals from a plurality of spaced apart receive antennas. Target angle in one plane may be determined by coherent processing of the returns from the plurality of receive antennas. Spacing the receive antennas apart in three dimensions allows determining of two angles, such as azimuth and elevation. Processing of the returns may be coherent or noncoherent, or returns may be processed both coherently and noncoherently. Programmability of the processing algorithms and parameters provide flexibility in applications, as well as flexibility based on such things as the target type and its range. Exemplary applications are disclosed.

Non-scanning radar for detecting and tracking targets

A non-scanning radar for detecting and tracking multiple moving objects. The transmit antenna continuously illuminates the entire surveillance volume, which can even be omni-directional (hemispherical). Multiple receive antennas are employed, each covering part of the surveillance volume. Receivers are used in combination to measure angles of incidence via interferometry on objects that are resolved in range and Doppler. Very long processing times are used to compensate for the reduced antenna gain compared to any radar that scans. By continuously illuminating the surveillance volume, there is no hard limit to the number of objects that can be simultaneously tracked. The primary application for this technology is detection and tracking of such objects as bullets, artillery projectiles, mortar shells, and rockets, and determining the location of the weapon that fired them. Numerous other applications are also described.

Radar system for continuous tracking of multiple objects

The angular power pattern, or equivalently the image-plane intensity distribution, is obtained for a circular aperture receiving a plane wave perturbed in transmission through a turbulent atmosphere. The aperture is assumed always to be aligned so as to maximize the received power. This alignment is shown to be that effected by a conventional angle-tracking servomechanism. Homogeneous, isotropic turbulence is assumed, and the aperture is taken to be of moderate size so that the 5/3-power law phase-structure function predicted by use of the Obukhov–Kolmogorov theory of turbulence applies adequately over its full extent. It is assumed also that the departure of the incident wavefronts from planarity everywhere on the aperture is small relative to a wavelength. This condition is shown to be met for relatively large deviation of phase from that of an unperturbed plane wave. Power patterns for tracking and nontracking apertures are compared.

Richard L. Mitchell

Papers

Detection Probabilities for Log-Normally Distributed Signals

Matched-Filter Responses of the Linear FM Waveform

Non-scanning radar for detecting and tracking targets

Radar system for continuous tracking of multiple objects

Effect of a Turbulent Medium on the Power Pattern of a Wavefront-Tracking Circular Aperture*