172 research outputs found
Pade interpolation by F-polynomials and transfinite diameter
We define -polynomials as linear combinations of dilations by some
frequencies of an entire function . In this paper we use Pade interpolation
of holomorphic functions in the unit disk by -polynomials to obtain
explicitly approximating -polynomials with sharp estimates on their
coefficients. We show that when frequencies lie in a compact set
then optimal choices for the frequencies of interpolating
polynomials are similar to Fekete points. Moreover, the minimal norms of the
interpolating operators form a sequence whose rate of growth is determined by
the transfinite diameter of .
In case of the Laplace transforms of measures on , we show that the
coefficients of interpolating polynomials stay bounded provided that the
frequencies are Fekete points. Finally, we give a sufficient condition for
measures on the unit circle which ensures that the sums of the absolute values
of the coefficients of interpolating polynomials stay bounded.Comment: 16 page
Fast, area-efficient 32-bit LNS for computer arithmetic operations
PhD ThesisThe logarithmic number system has been proposed as an alternative to floating-point.
Multiplication, division and square-root operations are accomplished with fixedpoint
arithmetic, but addition and subtraction are considerably more challenging.
Recent work has demonstrated that these operations too can be done with similar
speed and accuracy to their floating-point equivalents, but the necessary circuitry is
complex. In particular, it is dominated by the need for large lookup tables for the
storage of a non-linear function.
This thesis describes the architectures required to implement a newly design
approach for producing fast and area-efficient 32-bit LNS arithmetic unit. The
designs are structured based on two different algorithms. At first, a new cotransformation
procedure is introduced in the singularity region whilst performing
subtractions in which the technique capable to generate less total storage than the cotransformation
method in the previous LNS architecture. Secondly, improvement to
an existing interpolation process is proposed, that also reduce the total tables to an
extent that allows their easy synthesis in logic. Consequently, the total delays in the
system can be significantly reduced.
According to the comparison analysis with previous best LNS design and
floating-point units, it is shown that the new LNS architecture capable to offer
significantly better in speed while sustaining its accuracy within floating-point limit.
In addition, its implementation is more economical than previous best LNS system
and almost equivalent with existing floating-point arithmetic unit.University Malaysia Perlis:
Ministry of Higher Education, Malaysia
Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2
On arbitrary polygonal domains , we construct hierarchical Riesz bases for Sobolev spaces . In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from to . Since the latter range includes , with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned
Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers
We propose a solution strategy for a multimaterial minimum compliance
topology optimization problem, which consists in finding the optimal allocation
of a finite number of candidate (possibly anisotropic) materials inside a
reference domain, with the aim of maximizing the stiffness of the body. As a
relevant and novel application we consider the optimization of self-assembled
structures obtained by means of diblock copolymers. Such polymers are a class
of self-assembling materials which spontaneously synthesize periodic
microstructures at the nanoscale, whose anisotropic features can be exploited
to build structures with optimal elastic response, resembling biological
tissues exhibiting microstructures, such as bones and wood. For this purpose we
present a new generalization of the classical Optimality Criteria algorithm to
encompass a wider class of problems, where multiple candidate materials are
considered, the orientation of the anisotropic materials is optimized, and the
elastic properties of the materials are assumed to depend on a scalar
parameter, which is optimized simultaneously to the material allocation and
orientation. Well-posedness of the optimization problem and well-definition of
the presented algorithm are narrowly treated and proved. The capabilities of
the proposed method are assessed through several numerical tests
Efficient algorithms for arbitrary sample rate conversion with application to wave field synthesis
Arbitrary sample rate conversion (ASRC) is used in many fields of digital signal processing to alter the sampling rate of discrete-time signals by arbitrary, potentially time-varying ratios.
This thesis investigates efficient algorithms for ASRC and proposes several improvements. First, closed-form descriptions for the modified Farrow structure and Lagrange interpolators are derived that are directly applicable to algorithm design and analysis. Second, efficient implementation structures for ASRC algorithms are investigated. Third, this thesis considers coefficient design methods that are optimal for a selectable error norm and optional design constraints.
Finally, the performance of different algorithms is compared for several performance metrics. This enables the selection of ASRC algorithms that meet the requirements of an application with minimal complexity.
Wave field synthesis (WFS), a high-quality spatial sound reproduction technique, is the main application considered in this work. For WFS, sophisticated ASRC algorithms improve the quality of moving sound sources. However, the improvements proposed in this thesis are not limited to WFS, but applicable to general-purpose ASRC problems.ï»żVerfahren zur unbeschrĂ€nkten Abtastratenwandlung (arbitrary sample rate
conversion,ASRC) ermöglichen die Ănderung der Abtastrate zeitdiskreter
Signale um beliebige, zeitvarianteVerhÀltnisse. ASRC wird in vielen
Anwendungen digitaler Signalverarbeitung eingesetzt.In dieser Arbeit wird
die Verwendung von ASRC-Verfahren in der Wellenfeldsynthese(WFS), einem
Verfahren zur hochqualitativen, rÀumlich korrekten Audio-Wiedergabe,
untersucht.Durch ASRC-Algorithmen kann die WiedergabequalitÀt bewegter
Schallquellenin WFS deutlich verbessert werden. Durch die hohe Zahl der in
einem WFS-Wiedergabesystembenötigten simultanen ASRC-Operationen ist eine
direkte Anwendung hochwertigerAlgorithmen jedoch meist nicht möglich.Zur
Lösung dieses Problems werden verschiedene BeitrÀge vorgestellt. Die
KomplexitÀtder WFS-Signalverarbeitung wird durch eine geeignete
Partitionierung der ASRC-Algorithmensignifikant reduziert, welche eine
effiziente Wiederverwendung von Zwischenergebnissenermöglicht. Dies
erlaubt den Einsatz hochqualitativer Algorithmen zur Abtastratenwandlungmit
einer KomplexitÀt, die mit der Anwendung einfacher konventioneller
ASRCAlgorithmenvergleichbar ist. Dieses Partitionierungsschema stellt
jedoch auch zusÀtzlicheAnforderungen an ASRC-Algorithmen und erfordert
AbwĂ€gungen zwischen Performance-MaĂen wie der algorithmischen
KomplexitÀt, Speicherbedarf oder -bandbreite.Zur Verbesserung von
Algorithmen und Implementierungsstrukturen fĂŒr ASRC werdenverschiedene
MaĂnahmen vorgeschlagen. Zum Einen werden geschlossene,
analytischeBeschreibungen fĂŒr den kontinuierlichen Frequenzgang
verschiedener Klassen von ASRCStruktureneingefĂŒhrt. Insbesondere fĂŒr
Lagrange-Interpolatoren, die modifizierte Farrow-Struktur sowie
Kombinationen aus Ăberabtastung und zeitkontinuierlichen
Resampling-Funktionen werden kompakte Darstellungen hergeleitet, die sowohl
Aufschluss ĂŒber dasVerhalten dieser Filter geben als auch eine direkte
Verwendung in Design-Methoden ermöglichen.Einen zweiten Schwerpunkt bildet
das Koeffizientendesign fĂŒr diese Strukturen, insbesonderezum optimalen
Entwurf bezĂŒglich einer gewĂ€hlten Fehlernorm und optionaler
Entwurfsbedingungenund -restriktionen. Im Gegensatz zu bisherigen AnsÀtzen
werden solcheoptimalen Entwurfsmethoden auch fĂŒr mehrstufige
ASRC-Strukturen, welche ganzzahligeĂberabtastung mit zeitkontinuierlichen
Resampling-Funktionen verbinden, vorgestellt.FĂŒr diese Klasse von
Strukturen wird eine Reihe angepasster Resampling-Funktionen
vorgeschlagen,welche in Verbindung mit den entwickelten optimalen
Entwurfsmethoden signifikanteQualitÀtssteigerungen ermöglichen.Die
Vielzahl von ASRC-Strukturen sowie deren Design-Parameter bildet eine
Hauptschwierigkeitbei der Auswahl eines fĂŒr eine gegebene Anwendung
geeigneten Verfahrens.Evaluation und Performance-Vergleiche bilden daher
einen dritten Schwerpunkt. Dazu wirdzum Einen der Einfluss verschiedener
Entwurfsparameter auf die erzielbare QualitÀt vonASRC-Algorithmen
untersucht. Zum Anderen wird der benötigte Aufwand bezĂŒglich
verschiedenerPerformance-Metriken in AbhÀngigkeit von Design-QualitÀt
dargestellt.Auf diese Weise sind die Ergebnisse dieser Arbeit nicht auf WFS
beschrÀnkt, sondernsind in einer Vielzahl von Anwendungen unbeschrÀnkter
Abtastratenwandlung nutzbar
- âŠ