170 research outputs found
Synthesis methods for linear-phase FIR filters with a piecewise-polynomial impulse response
his thesis concentrates on synthesis methods for linear-phase ïŹnite-impulse response ïŹlters with a piecewise-polynomial impulse response. One of the objectives has been to ïŹnd integer-valued coeïŹcients to efficiently implement ïŹlters of the piecewise-polynomial impulse response approach introduced by Saramšaki and Mitra. In this method, the impulse response is divided into blocks of equal length and each block is created by a polynomial of a given degree. The arithmetic complexity of these ïŹlters depends on the polynomial degree and the number of blocks. By using integer-valued coefficients it is possible to make the implementation of the subïŹlters, which generates the polynomials, multiplication-free. The main focus has been on ïŹnding computationally-efficient synthesis methods by using a piecewise-polynomial and a piecewise-polynomial-sinusoidal impulse responses to make it possible to implement high-speed, low-power, highly integrated digital signal processing systems. The earlier method by Chu and Burrus has been studied. The overall impulse response of the approach proposed in this thesis consists of the sum of several polynomial-form responses. The arithmetic complexity depends on the polynomial degree and the number of polynomial-form responses. The piecewise-polynomial-sinusoidal approach is a modiïŹcation of the piecewise-polynomial approach. The subresponses are multiplied by a sinusoidal function and an arbitrary number of separate center coefficients is added. Thereby, the arithmetic complexity depends also on the number of complex multipliers and separately generated center coefficients. The ïŹlters proposed in this thesis are optimized by using linear programming methods
Design Of Polynomial-based Filters For Continuously Variable Sample Rate Conversion With Applications In Synthetic Instrumentati
In this work, the design and application of Polynomial-Based Filters (PBF) for continuously variable Sample Rate Conversion (SRC) is studied. The major contributions of this work are summarized as follows. First, an explicit formula for the Fourier Transform of both a symmetrical and nonsymmetrical PBF impulse response with variable basis function coefficients is derived. In the literature only one explicit formula is given, and that for a symmetrical even length filter with fixed basis function coefficients. The frequency domain optimization of PBFs via linear programming has been proposed in the literature, however, the algorithm was not detailed nor were explicit formulas derived. In this contribution, a minimax optimization procedure is derived for the frequency domain optimization of a PBF with time-domain constraints. Explicit formulas are given for direct input to a linear programming routine. Additionally, accompanying Matlab code implementing this optimization in terms of the derived formulas is given in the appendix. In the literature, it has been pointed out that the frequency response of the Continuous-Time (CT) filter decays as frequency goes to infinity. It has also been observed that when implemented in SRC, the CT filter is sampled resulting in CT frequency response aliasing. Thus, for example, the stopband sidelobes of the Discrete-Time (DT) implementation rise above the CT designed level. Building on these observations, it is shown how the rolloff rate of the frequency response of a PBF can be adjusted by adding continuous derivatives to the impulse response. This is of great advantage, especially when the PBF is used for decimation as the aliasing band attenuation can be made to increase with frequency. It is shown how this technique can be used to dramatically reduce the effect of alias build up in the passband. In addition, it is shown that as the number of continuous derivatives of the PBF increases the resulting DT implementation more closely matches the Continuous-Time (CT) design. When implemented for SRC, samples from a PBF impulse response are computed by evaluating the polynomials using a so-called fractional interval, ”. In the literature, the effect of quantizing ” on the frequency response of the PBF has been studied. Formulas have been derived to determine the number of bits required to keep frequency response distortion below prescribed bounds. Elsewhere, a formula has been given to compute the number of bits required to represent ” to obtain a given SRC accuracy for rational factor SRC. In this contribution, it is shown how these two apparently competing requirements are quite independent. In fact, it is shown that the wordlength required for SRC accuracy need only be kept in the ” generator which is a single accumulator. The output of the ” generator may then be truncated prior to polynomial evaluation. This results in significant computational savings, as polynomial evaluation can require several multiplications and additions. Under the heading of applications, a new Wideband Digital Downconverter (WDDC) for Synthetic Instruments (SI) is introduced. DDCs first tune to a signal\u27s center frequency using a numerically controlled oscillator and mixer, and then zoom-in to the bandwidth of interest using SRC. The SRC is required to produce continuously variable output sample rates from a fixed input sample rate over a large range. Current implementations accomplish this using a pre-filter, an arbitrary factor resampler, and integer decimation filters. In this contribution, the SRC of the WDDC is simplified reducing the computational requirements to a factor of three or more. In addition to this, it is shown how this system can be used to develop a novel computationally efficient FFT-based spectrum analyzer with continuously variable frequency spans. Finally, after giving the theoretical foundation, a real Field Programmable Gate Array (FPGA) implementation of a novel Arbitrary Waveform Generator (AWG) is presented. The new approach uses a fixed Digital-to-Analog Converter (DAC) sample clock in combination with an arbitrary factor interpolator. Waveforms created at any sample rate are interpolated to the fixed DAC sample rate in real-time. As a result, the additional lower performance analog hardware required in current approaches, namely, multiple reconstruction filters and/or additional sample clocks, is avoided. Measured results are given confirming the performance of the system predicted by the theoretical design and simulation
Linear Matrix Inequality Formulation of Spectral Mask Constraints With Applications to FIR Filter Design
Abstract-The design of a finite impulse response (FIR) filter often involves a spectral "mask" that the magnitude spectrum must satisfy. The mask specifies upper and lower bounds at each frequency and, hence, yields an infinite number of constraints. In current practice, spectral masks are often approximated by discretization, but in this paper, we will derive a result that allows us to precisely enforce piecewise constant and piecewise trigonometric polynomial masks in a finite and convex manner via linear matrix inequalities. While this result is theoretically satisfying in that it allows us to avoid the heuristic approximations involved in discretization techniques, it is also of practical interest because it generates competitive design algorithms (based on interior point methods) for a diverse class of FIR filtering and narrowband beamforming problems. The examples we provide include the design of standard linear and nonlinear phase FIR filters, robust "chip" waveforms for wireless communications, and narrowband beamformers for linear antenna arrays. Our main result also provides a contribution to system theory, as it is an extension of the wellknown Positive-Real and Bounded-Real Lemmas
Biorthogonal partners and applications
Two digital filters H(z) and F(z) are said to be biorthogonal partners of each other if their cascade H(z)F(z) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. We first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above-mentioned applications
Digital Filters
The new technology advances provide that a great number of system signals can be easily measured with a low cost. The main problem is that usually only a fraction of the signal is useful for different purposes, for example maintenance, DVD-recorders, computers, electric/electronic circuits, econometric, optimization, etc. Digital filters are the most versatile, practical and effective methods for extracting the information necessary from the signal. They can be dynamic, so they can be automatically or manually adjusted to the external and internal conditions. Presented in this book are the most advanced digital filters including different case studies and the most relevant literature
Polynomial Procrustes problem : paraunitary approximation of matrices of analytic functions
In the narrowband case, the best least squares approximation of a matrix by a unitary one is given by the Procrustes problem. In this paper, we expand this idea to matrices of analytic functions, and characterise a broadband equivalent to the narrowband case: the polynomial Procrustes problem. Its solution is based on an analytic singular value decomposition, and for the case of spectrally majorised, distinct singular values, we demonstrate the application of a suitable algorithm to three problems â time delay estimation, paraunitary matrix completion, and general paraunitary approximations â in simulations
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
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