Pulsed-Beam Wavelets are exact, causal solutions of the inhomogeneous wave equation or Maxwell's equations whose `wavelet parameters' specify physically relevant attributes of the associated pulsed beams. Their point of emission and launch time, as well as the pulse width, collimation, direction of propagation, and duration. Their time-domain radiation patterns have no sidelobes and can be made arbitrarily well-focused. We compute the source distribution necessary to synthesize such beams. It is a generalized function supported on the disk aperture, consisting of a circular line source concentrated on the rim plus a single and double layer distributed over the disk interior. We speculate that such pulsed beams may be generated physically by realizing their source distributions. If so, they could have important applications in radar, sonar, and secure communications.
Physical Wavelets are exact, caused pulsed-beam solutions of the inhomogeneous wave- or Maxwell equations whose `wavelet parameters' specify the point of emission, launch time, radius of the emitting aperture, direction of propagation, duration and width of the pulse. We describe their time- domain radiation patterns, which show that the beams can be made arbitrarily well focused by choosing the wavelet parameters accordingly. We also find the source distribution as a generalized function supported on the disk aperture determined by the wavelet parameters. As the radius of the disk shrinks to zero, the distribution reduces to the usual point source represented by the retarded Green function. It is suggested that such `physical wavelets' may be synthesized in practice by realizing their source distributions.
Pulsed-Beam Wavelets are exact, causal solutions of the inhomogeneous wave equation or Maxwell's equations whose `wavelet parameters,' instead of giving a time and scale, specify the point of emission, launch time, the radius of the emitting aperture, direction of propagation, duration, and width of the pulse. We derive their far fields and time- domain radiation patterns and confirm that the beams can be made arbitrarily well focused by choosing the wavelet parameters accordingly. We also find the source distribution as a generalized function supported on the compact source region determined by the wavelet parameters. As the radius of the disk shrinks to zero, the distribution reduces to the usual point source represented by the causal Green function. It is suggested that such `physical wavelets' may be synthesized in practice.
We study tropospheric aerothermal probe data by using the orthogonal Haar wavelet averages of the dissipation to segment the data. Segmenting the data in this way allows us to isolate regions of distinct mean dissipation. We then use the Haar wavelet transform to derive spectra and structure functions for the segmented regions, thereby recovering Kolmogorov statistics. We also comment on wavelet derived structure functions and point out data anomalies only visible in the wavelet domain.
For a given basic wavelet (psi) (t), two distinct correspondences (called C1 and C2) are established between frequency filters, defined in the frequency domain through multiplication by a transfer function W(f), and scale filters, defined in the wavelet domain through multiplication by a scale transfer function w((sigma) ). W(f) is obtained by performing a scaling convolution of w((sigma) ) with (psi) (f)* (for C1) or its spectral energy density (psi) (f) 2 (for C2). For a large class of transfer functions W(f), this relation can be solved for w((sigma) ) by applying the Mellin transform. We call such frequency filters and their associated time-domain convolution operators C1- or C2-admissible with respect to (psi) . In particular, the identity operator (W(f) equalsV 1) is C2-admissible if and only if the wavelet (psi) is admissible in the conventional sense. The implementation of the correspondence C1 is computationally simpler than C2, but C2 can be generalized to time-dependent filters. Applications are proposed to the analysis of atmospheric turbulence data and wideband Doppler filtering.
It is shown that any convolution operator in the time domain can be represented exactly as a multiplication operator in the time-scale (wavelet) domain. The Mellin transform establishes a one-to-one correspondence between frequency filters (system or transfer functions) and scale filters, which are defined as multiplication operators in the scale domain, subject to the convergence of the defining integrals. Applications to the denoising of random signals are proposed. We argue that the present method is more suitable for removing the effects of atmospheric turbulence than the conventional procedures based on Fourier analysis because it is ideally suited for resolving spectral power laws.
The electromagnetic and acoustic wavelets recently introduced by the author are here redefined in a much simplified and generalized form. An application to remote sensing is proposed, resulting in a formulation of radar and sonar as problems in the calculus of variations which reduce to the usual wideband ambiguity function formalisms when the reflectors are assumed to move with a uniform velocity. The new wavelets also lend themselves naturally to multiresolution analysis of electromagnetic and acoustic signals, which could have applications for communication.
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