B. Drachman

Radar target discrimination by convolution of radar return with extinction-pulses and single-mode extraction signals

A new method of radar target discrimination and identification is presented. This new method is based on the natural frequencies of the target. It consists of synthesizing aspect-independent discriminant signals, called extinction-pulses (E-pulses) and single-mode extraction signals which, when convolved numerically with the late-time transient response of an expected target, lead to zero or single-mode responses. When the synthesized, discriminant signals for an expected target are convolved with the radar return from a different target, the resulting signal will be significantly different from the expected zero or single-mode responses, thus, the differing targets can be discriminated. Theoretical synthesis of discriminant signals from known target natural frequencies and experimental synthesis of them for a complex target from its measured pulse response are presented. The scheme has been tested with measured responses of various targets in the laboratory.

Frequency domain E-pulse synthesis and target discrimination

A frequency domain approach to the E -pulse radar target discrimination scheme is introduced. This approach is shown to allow easier interpretation of E -pulse convolutions via the E -pulse spectrum, and leads to a simplified calculation of pulse basis function amplitudes in the E -pulse expansion. Experimental evidence obtained using aircraft models verifies the single-mode discrimination scheme, as well as the aspect-independent nature of the E -pulse technique. This leads to an integrated technique for target discrimination combining the E -pulse with single mode extraction waveforms.

A continuation method for identification of the natural frequencies of an object using a measured response

The identification of the natural frequencies of an object using measured data is an ill-conditioned problem. A method and algorithm to solve the problem based on regularization by a continuation method is presented. The algorithm is applied to the measured response of a model aircraft, and the superiority of this method to Pronys method in the presence of noise is demonstrated.

Computation of normal forms

Abstract In this paper a method of computing a normal form for a system of ordinary differential equations is given. A program using the symbolic manipulator MACSYMA is used on several examples.

Two methods to deconvolve: L 1 -method using simplex algorithm and L 2 -method using least squares and parameter

If r(t) is the linear scattering response of an object to an excitation waveform e(t) , then r(t) = (e \ast h) (t) . One would like to deconvolve and solve for h(t) , the impulse response. It is well-known that this is often an ill-conditioned problem. Two methods are discussed. The first method replaces the discretized matrix form E \cdot H = R by the following problem. Minimize \|h_{1}\|+ \ldots + \|h_{n}\| subject to R - \lambda \leq E \cdot H \leq R + \lambda where \lambda is a column vector chosen sufficiently small to yield acceptable residuals, yet large enough to make the problem well-conditioned. This problem is converted to a linear programming problem so that the simplex algorithm can be used. The second method is to minimize \parallel E \cdot H - R \parallel^{2} +\lambda \parallel H \parallel^{2} where again \lambda is chosen small enough to yield acceptable residuals and large enough to make the problem well-conditioned. The method will be demonstrated with a Hilbert matrix inversion problem, and also by the deconvolution of the impulse response of a simple target from measured data.

Accurate evaluation of Sommerfeld integrals using the fast Fourier transform

It is shown that the fast Fourier transform (FFT) combines naturally with Simpsons rule for Sommerfeld-type integral computation. The principal advantage of using the FFT is that a single subroutine call yields a set of sample values of an integral (i.e. the integral for various values of an integrand parameter). Such samples could be useful in themselves. In other applications Sommerfeld integrals represent Greens functions nested within other spatial integrals, so samples from the FFT might be useful in approximating the outernested integral. Several examples are provided to illustrate the process. >

Approximation of certain functions given by integrals with highly oscillatory integrands

There is often a need to approximate integrals of highly oscillatory functions when studying scattering and diffraction of electromagnetic waves. This paper presents a method of estimating certain types of these integrals by evaluating one interpolating function and performing one or two relatively easy numerical integrations. The method is demonstrated for the case of a Fresnel integral. >

A Brief Introduction to Interval Analysis

Some major advances in mathematics have occurred through the extension of existing number systems. The natural numbers were extended to the real numbers, the real numbers to the complex numbers, and so on.

Basic Review and Elementary Facts

Inequalities lie at the heart of mathematical analysis. They appear in the definitions of continuity and limit (and hence in the definitions of the integral and the derivative). They play crucial roles in generalizing the notions of distance and vector magnitude. But many problems of physical interest also rely on simple inequality concepts for their solution. In engineering, it is not always best to think in terms of equality. Let us illustrate this statement with a few examples.

Methods from the Calculus

In this chapter we revisit some facts from mathematical analysis and show how these may be used to establish important inequalities. We begin by reviewing convergence of real number sequences and continuity of real functions of a single variable.