Fractional order calculus (FOC) has wide applications in modeling natural behavior of systems related to different areas of engineering including bioengineering, viscoelasticity, electronics, robotics, control theory and signal processing. This thesis aims at modeling a lossy transmission line using fractional order calculus and identifying its parameters. A lossy transmission line is considered where its behavior is modeled by a fractional order transfer function. A semi-infinite lossy transmission line is presented with its distributed parameters R, L, C and ordinary AC circuit theory is applied to find the partial differential equations. Furthermore, applying boundary conditions and the Laplace transformation a generalized fractional order transfer function of the lossy transmission line is obtained. A finite length lossy transmission line terminated with arbitrary load is also considered and its fractional order transfer function has been derived. Next, the frequency responses of lossy transmission lines from their fractional order transfer functions are also derived. Simulation results are presented to validate the frequency responses. Based on the simulation results it can be concluded that the derived fractional order transmission line model is capable of capturing the phenomenon of a distributed parameter transmission line. The achievement of modeling a highly accurate transmission line requires that a realistic account needs to be taken of its parameters. Therefore, a parameter identification technique to identify the parameters of the fractional order lossy transmission line is introduced. Finally, a few open problems are listed as the future research directions. / Controls

Thesis (Ph. D.)--Virginia Polytechnic Institute, 1956. / Typewritten. Vita. Bibliography: p. 110-114. Also available via the Internet.

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In this paper, we derive the asymptotic distribution of the Augmented Dickey-Fuller t Test statistics, t_{ADF}, against a generalized fractional integrated process (for example: ARFIMA(p,1+d,q) ,|d|<1/2,and p, q be positive integer) by using the propositions of Lee and Shie (2003). Then we discuss why the power decreases with the increasing lags in the same and large enough sample size T when d is unequal to 0. We also get that the estimator of the disturbance's variance, S^2, has slightly increasing bias with increasing k. Finally, we support the conclusion by the Monte Carlo experiments.

An ISM-band frequency synthesizer is introduced in this thesis. The technique allows digital phase/frequency modulation to be achieved in a closed phase locked loop (PLL) without mixers and D/As. According to the simulation results using ADS, quantization noise will be filtered by the PLL bandwidth. But the data rate is also bounded by the PLL bandwidth. Two key components of this closed-loop architecture, Gaussian filter and delta-sigma modulator have been implemented by FPGA together with the Qualcomm Q3236 synthesizer IC.

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We study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. / text

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Some recently discussed stationary processes like fractionally integrated processes cannot be described by low order autoregressive or moving average (ARMA) models rendering the common algorithms for generation estimation and prediction partly very misleading. We offer an unified approach based on the Cholesky decomposition of the covariance matrix which makes these problems exactly solvable in an efficient way. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

In this thesis, we study a linear fractional Fokker-Planck equation that models non-local (`fractional') diffusion in the presence of a potential field. The non-locality is due to the appearance of the `fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular (Gaussian) diffusion. Motivated by the observation that, in contrast to the classical Fokker-Planck equation (describing regular diffusion in the presence of a potential field), there is no natural gradient flow formulation for its fractional counterpart, we prove existence of weak solutions to this fractional Fokker-Planck equation by combining a splitting technique together with a Wasserstein gradient flow formulation. An explicit iterative construction is given, which we prove weakly converges to a weak solution of this PDE. / Graduate