Wavelets and Fractals
Wavelets and Fractals

Mathematical Constructions of Wavelets
Wavelet bases are a powerful mathematical tool. They allow multiscale representations of functions and operators. This feature, combined with the possibility of sparse representations obtained by simple thresholding of wavelet coefficients, allows to design lowcomplexity algorithms for the numerical simulation of relevant EMC problems, like transmission line structures. However, the intrinsic nature of such structures requires wavelet bases to be natively constructed on bounded domains, instead of the usual constructions defined on translationinvariant domains. Significant work has been devoted to the mathematical construction of biorthogonal wavelet systems on bounded domains to be used as basis functions for advanced numerical modeling algorithms. As an example, in the picture you can see a 2D wavelet function in the Bspline(2,2) system.

Interconnects Modeling using Wavelets
This activity is focused on highly efficient algorithms for transient analysis of nonuniform transmission lines characterized by the Telegraphers Equations with spacedependent parameters. Wavelet bases are used to represent the spatial variation of voltages and currents. The intrinsic multiresolution properties of the adopted waveletbased representation is exploited by an ad hoc timeadaptive algorithm that selects automatically the few relevant wavelet basis functions that are strictly necessary to describe voltage and current with a given accuracy. When time progresses during a simulation, the wavelet expansion is automatically adapted to the particular waveforms being computed. As an example, the picture on the right shows the localization of the basis functions in the spacetime plane during propagation of a single trapezoidal pulse along a transmission line. This activity on 1D propagation modeling has been complemented by a theoretical analysis of numerical dispersion for 3D waveletbased algorithms for fullwave transient simulations.

Adaptive Signal Processing and Applications
Wavelets allow the decomposition of signals into their elementary timefrequency components. This implies the capability of detecting waves and wave packets with a precise definition both of their frequency and their time localization. This feature allows the design of automatic wave detection algorithms making use of such localization properties. Some past research activities in cooperation with NASA/Goddard Space Flight Center have been focused on the design of such algorithms finalised to the detection of gravity waves from atmospheric pressure time series. In this activity, several concurrent data sources from a network of sensors were processed in order to detect the wave packets, extract their signature, and study their propagation characteristic. As an example, the picture shows the timefrequency signature of one of highenergy gravity waves that were detected by our algorithm.

Modeling of Random Wire Bundles
The Transmission Lines equations provide an excellent physical model for uniform or almostuniform interconnects. However, accurate modeling is only possible when the perunitlength parameters are known. This implies that for nonuniform structures like wire bundles (see picture), which are characterized by random twists and wire displacements, a deterministic approach is not realistic. The propagation characteristics and the crosstalk in a wire bundle are quite dependent on the variations of the parameters along the propagation direction. A statistical approach is the only one really feasible. Our efforts have been concentrated on two main aspects. On one hand, the geometrical modeling of wires and their displacements. This led to the use of fractal geometry for the generation of realistic random bundle models. On the other hand, sophisticated statistical techniques like the Kriging method have been applied for crosstalk prediction.

Noise Characterization
This research activity focused on the relations between time and frequency specifications of noise. Specifically, we concentrated on the socalled “1/f” noise and we investigated possible representations making use of fractal geometry. In fact, looking at typical realizations of such a noise process (see picture), one can recognize the signatures of fractal processes. Analytical relations between the fractal dimension of the process and its spectral characterizations were derived. In addition, the interaction of fractal noise processes with linear systems specified via their transfer functions were thoroughly investigated.

Modeling of Lightning Channel and Radiation
This less recent activity focused on several aspects of lightning modeling. On one hand, pure geometrical approaches using fractal theory were applied for the derivation of realistic approximation of the typical shape of the lightning channel (see the picture). These fractal models were used to investigate the radiated fields and the coupling properties to sensitive structures like transmission lines. On the other hand, a number of studies were conducted on propagation models for current along the channel, using transmission line methods. These models included also the effects of branching and of secondary channels, like those you can see in the picture.