This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: given complex numbers $z_1,z_2,...,z_N$ of modulus at most $1$ and $w_1,w_2,...,w_N$ additional complex numbers, what is a necessary and sufficiency condition for the existence of an analytic function $f: \mathbb{D} \rightarrow \mathbb$ satisfying $f(z_i) = w_i$ for $1 \leq i \leq N$ and $\vert f(z) \vert \leq 1$ for each $z \in \mathbb{D}$? The key idea is to realize bounded, analytic functions (the algebra $H^\infty$) as the \emph of the Hardy class of analytic functions, and apply dilation theory to this algebra. This operator theoretic approach may then be applied to a wider class of interpolation problems, as well as their matrix-valued equivalents. This also yields a fundamental distance formula for $H^\infty$, which provides motivation for the study of completely isometric representations of certain quotient algebras. Our attention is then turned to a related interpolation problem. Here we require the interpolating function $f$ to satisfy the additional property $f'(0) = 0$. When $z_i =0$ for some $i$, we arrive at a special case of a problem class studied previously. However, when $0$ is not in the interpolating set, a significant degree of complexity is inherited. The dilation theoretic approach employed previously is not effective in this case. A more function theoretic viewpoint is required, with the proof of the main interpolation theorem following from a factorization lemma for the Hardy class of analytic functions. We then apply the theory of completely isometric maps to show that matrix interpolation fails when one imposes this constraint.

In this thesis, we propose a modified MT-CDMA system, which can improve channel estimation accuracy by using transmit diversity of pilot signals. We not only expound the principles and structures of the system we proposed, but analyze its performance in slow Rayleigh fading channel environment. According to different ways to assign data symbols of transmitted signals, we have two different bit error rate results. At the same transmit power, the simulation results show that when we combine comb-type pilot signals of two parallel channels to estimate channel gains, we can recover the drawbacks of comb-type pilot arrangement, which can not perform well in frequency selective fading channel. Finally, the numerical results will be also shown.

Thèse de doctorat : Informatique : Lyon 2 : 2000. / Titre provenant de l'écran-titre. Bibliogr. Index.

Kassel, Universiẗat, Diss., 2008.

Zugl.: Kaiserslautern, Techn. Univ., Diss., 2009

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010. / Committee Chair: Gangbo, Wilfrid; Committee Member: Chow, Shui-Nee; Committee Member: Harrell, Evans; Committee Member: Swiech, Andrzej; Committee Member: Yezzi, Anthony Joseph. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Includes bibliographical references (p. 55-57).

High-speed and low resolution analog-to-digital converters (ADC) are key elements in the read channel of optical and magnetic data storage systems. The required resolution is about 6-7 bits while the sampling rate and effective resolution bandwidth requirements increase with each generation of storage system. Folding is a technique to reduce the number of comparators used in the flash architecture. By means of an analog preprocessing circuit in folding A/D converters the number of comparators can be reduced significantly. Folding architectures exhibit low power and low latency as well as the ability to run at high sampling rates. Folding ADCs employing interpolation schemes to generate extra folding waveforms are called "Folding and Interpolating ADC" (F&I ADC). The aim of this research is to increase the input bandwidth of high speed conversion, and low latency F&I ADC. Behavioral models are developed to analyze the bandwidth limitation at the architecture level. A front-end sample-and-hold unit is employed to tackle the frequency multiplication problem, which is intrinsic for all F&I ADCs. Current-mode signal processing is adopted to increase the bandwidth of the folding amplifiers and interpolators, which are the bottleneck of the whole system. An operational transconductance amplifier (OTA) based folding amplifier, current mirror-based interpolator, very low impedance fast current comparator are proposed and designed to carry out the current-mode signal processing. A new bit synchronization scheme is proposed to correct the error caused by the delay difference between the coarse and fine channels. A prototype chip was designed and fabricated in 0.35μm CMOS process to verify the ideas. The S/H and F&I ADC prototype is realized in 0.35μm double-poly CMOS process (only one poly is used). Integral nonlinearity (INL) is 1.0 LSB and Differential nonlinearity (DNL) is 0.6 LSB at 110 KHz. The ADC occupies 1.2mm2 active area and dissipates 200mW (excluding 70mW of S/H) from 3.3V supply. At 300MSPS sampling rate, the ADC achieves no less than 6 ENOB with input signal lower than 60MHz. It has the highest input bandwidth of 60MHz reported in the literature for this type of CMOS ADC with similar resolution and sample rate.

In this paper, we turn the interpolation problem of coarsely-sampled data into a denoising problem. From this point of view, we illustrate the benefit of random sampling at sub-Nyquist rate over regular sampling at the same rate. We show that, using nonlinear sparsity promoting optimization, coarse random sampling may actually lead to significantly better wavefield reconstruction than equivalent regularly sampled data.

This thesis introduces fractal interpolation functions, exposes advantages of fractal interpolation of real world objects and presents some newly developed procedures, associated with fractal interpolation process. The work briefly presents the context needed for introduction of fractal approach and relevant definitions. Also, the detailed description of fractal generating algorithms (deterministic, random iteration, “escape time”) as well as fractal classifications is presented. Since the research object is theoretical and practical aspects of fractal interpolation function analysis, special attention is paid to geometric fractals, obtained using systems of iterated functions (IFS). The notion of a fractal interpolation function is introduced in the work. The author shows that it is possible to generate fractal interpolation functions for various types of data. The generated functions are “close” (in the sense of Housdorf dimension) to the data under processing, i.e., it is possible to ensure that the fractal interpolation graph dimension were equal to the fractal dimension of experimental data (graph). The random iteration algorithm is used for the analysis of fractal interpolation functions, since it is relatively simple and fast enough. The author makes an attempt to analyze and solve the problem of choosing interpolation points (general case). A few approaches are proposed, namely the uniform distribution of interpolation points (for the interactive use) and collage. On... [to full text]