3 research outputs found
Regular and irregular signal resampling
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2002.Includes bibliographical references (p. 107-110).In this thesis, we consider three main resampling problems. The first is the sampling rate conversion problem in which the input and output grids are both regularly spaced. It is known that the output signal is obtained by applying a time-varying filter to the input signal. The existing methods for finding the coefficients of this filter inherently tradeoff computational and memory requirements. Instead, we present a recursive scheme for which the computational and memory requirements are both low. In the second problem which we consider, we are given the instantaneous samples of a continuous-time (CT) signal taken on an irregular grid from which we wish to obtain samples on a regular grid. This is referred to as the nonuniform sampling problem. We present a noniterative algorithm for solving this problem, which, in contrast to the known iterative algorithms, can easily be implemented in real time. We show that each output point may be calculated by using only a finite number of input points, with an error which falls exponentially in the number of points used. Finally we look at the nonuniform lowpass reconstruction problem. In this case, we are given regular samples of a CT signal from which we wish to obtain amplitudes for a sequence of irregularly spaced impulses. These amplitudes are chosen so that the original CT signal may be recovered by lowpass filtering this sequence of impulses. We present a general solution which exhibits the same exponential localization obtained for the nonuniform sampling problem. We also consider a special case in which the irregular grid is obtained by deleting a single point from an otherwise regular grid. We refer(cont.) to this as the missing pixel problem, since it may be used to model cases in which a single defective element is present in a regularly spaced array such as the pixel arrays used in flat-panel video displays. We present an optimal solution which minimizes the energy of the reconstruction error, subject to the constraint that only a given number of pixels may be adjusted.by Andrew I. Russell.Ph.D
Regular and Irregular Signal Resampling
Thesis Supervisor: Alan V. Oppenheim
Title: Ford Professor of EngineeringIn this thesis,w e consider three main resampling problems. The first is the sampling rate conversion problem in which the input and output grids are both regularly spaced.
It is known that the output signal is obtained by applying a time-varying filter to the input signal. The existing methods for finding the coefficients of this filter inherently tradeoff computational and memory requirements. Instead,w e present a recursive scheme for which the computational and memory requirements are both low. In the second problem which we consider,w e are given the instantaneous samples of a continuous-time (CT) signal taken on an irregular grid from which we wish to obtain samples on a regular grid. This is referred to as the nonuniform sampling problem.
We present a noniterative algorithm for solving this problem,whic h,in contrast to
the known iterative algorithms,can easily be implemented in real time. We show that
each output point may be calculated by using only a finite number of input points,
with an error which falls exponentially in the number of points used. Finally we look
at the nonuniform lowpass reconstruction problem. In this case,w e are given regular
samples of a CT signal from which we wish to obtain amplitudes for a sequence of
irregularly spaced impulses. These amplitudes are chosen so that the original CT signal may be recovered by lowpass filtering this sequence of impulses. We present a general solution which exhibits the same exponential localization obtained for the nonuniform sampling problem. We also consider a special case in which the irregular grid is obtained by deleting a single point from an otherwise regular grid. We refer to this as the missing pixel problem,since it may be used to model cases in which a single defective element is present in a regularly spaced array such as the pixel arrays used in flat-panel video displays. We present an optimal solution which minimizes the energy of the reconstruction error,sub ject to the constraint that only a given number of pixels may be adjusted
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