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Improving Frequency Resolution of Discrete Spectra AGH University of Science and Technology, Kraków, Poland Faculty of Electrical Engineering, Automatics, Computer Science and Electronics Marek GĄSIOR Improving Frequency Resolution of Discrete Spectra A doctoral dissertation done under guidance of Professor Mariusz ZIÓŁKO Kraków 2006 - II - Marek Gąsior, Improving frequency resolution of discrete spectra Akademia Górniczo-Hutnicza w Krakowie Wydział Elektrotechniki, Automatyki, Informatyki i Elektroniki Marek GĄSIOR Poprawa rozdzielczości częstotliwościowej widm dyskretnych Rozprawa doktorska wykonana pod kierunkiem prof. dra hab. inż. Mariusza ZIÓŁKO Kraków 2006 Marek Gąsior, Improving frequency resolution of discrete spectra - III - - IV - Marek Gąsior, Improving frequency resolution of discrete spectra To my Parents Moim Rodzicom Acknowledgements I would like to thank all those who influenced this work. They were in particular the Promotor, Professor Mariusz Ziółko, who also triggered my interest in digital signal processing already during mastership studies; my CERN colleagues, especially Jeroen Belleman and José Luis González, from whose experience I profited so much. I am also indebted to Jean Pierre Potier, Uli Raich and Rhodri Jones for supporting these studies. Separately I would like to thank my wife Agata for her patience, understanding and sacrifice without which this work would never have been done, as well as my parents, Krystyna and Jan, for my education. To them I dedicate this doctoral dissertation. Podziękowania Pragnę podziękować wszystkim, którzy mieli wpływ na kształt niniejszej pracy. W szczególności byli to Promotor, Pan profesor Mariusz Ziółko, który także zaszczepił u mnie zainteresowanie cyfrowym przetwarzaniem sygnałów jeszcze w czasie studiów magisterskich; moi koledzy z CERNu, przede wszystkim Jeroen Belleman i José Luis González, z których doświadczenia skorzystałem tak wiele. Panowie Jean-Pierre Potier, Uli Raich oraz Rhodri Jones wspierali niniejszą pracę, za co jestem im bardzo wdzięczny. Osobne podziękowania składam mojej żonie Agacie – za cierpliwość, wyrozumiałość i poświęcenie, bez których ta praca nigdy by nie powstała, oraz moim Rodzicom, Krystynie i Janowi, za wykształcenie. Im dedykuję tę rozprawę doktorską. Marek Gąsior, Improving frequency resolution of discrete spectra - V - - VI - Marek Gąsior, Improving frequency resolution of discrete spectra Everything should be made as simple as possible, Wszystko powinno być robione najprościej jak można, but not simpler. ale nie prościej. Albert Einstein Abstract Discrete spectra can be used to measure frequencies of signal components. Such a measurement consists in digitizing the input signal, performing windowing of the signal samples and computing their discrete Fourier spectrum, usually by means of the Fast Fourier Transform algorithm. Frequencies of individual components can be evaluated from their locations in the discrete spectrum magnitude with resolution depending on the number of processed samples, which is usually limited by the time required for computing the spectrum. The subject of this dissertation is interpolation algorithms of discrete Fourier spectra allowing to increase the frequency resolution of such measurements by a few orders of magnitude, depending first of all the interpolation method, window used and noise present in the spectrum. The author focuses on three methods. All of them consist in fitting an interpolating curve through the three largest consecutive spectrum bins corresponding to the measured component of the input signal. The abscissa of the curve maximum determines the component frequency with improved resolution. The interpolation methods consist in fitting a parabola, a Gaussian curve and (so called in the dissertation) an exponential parabola and, for commonly used windows, allow resolution improvement, typically by one, two and four orders of magnitude, respectively. It is assumed that the measured frequency is constant during the signal sampling and that the corresponding spectral peak can be found in the discrete spectrum. The paper includes a description of the dependence of the algorithm efficiency on windowing used, noise present in the spectrum, interference from undesirable large components and exponential decay of the input signal. A direct application of the interpolation algorithms are systems measuring frequency of betatron oscillations of high energy particle beams. They are used presently in such systems in the European Organization for Particle Physics (CERN), Geneva, Switzerland. It seems that the algorithms presented in this dissertation may be also used in laboratory and industrial systems for frequency measurement of signals of limited periodicity, everywhere where real time, high resolution measurements, based on small sample sets are required. Streszczenie Widma dyskretne mogą służyć do pomiaru częstotliwości składowych sygnałów. Pomiar taki polega na zamianie sygnału wejściowego na postać cyfrową, poddaniu próbek sygnału procesowi okienkowania oraz wyznaczeniu ich fourierowskiego widma dyskretnego, najczęściej za pomocą algorytmu szybkiej transformacji Fouriera. Częstotliwości poszczególnych składowych sygnału mogą być określone z ich lokalizacji w module widma z rozdzielczością zależną od liczby próbek wziętych do jego wyznaczenia, która z reguły jest ograniczona czasem potrzebnym na obliczenie widma. Tematem niniejszej rozprawy są algorytmy interpolacji fourierowskich widm dyskretnych umożliwiające poprawę rozdzielczości częstotliwościowej takich pomiarów o kilka rzędów wielkości, zależnie przede wszystkim od użytej metody interpolacji, okienkowania oraz od szumu zawartego w interpolowanym widmie. Autor skupia się na trzech metodach. Wszystkie polegają na dopasowaniu krzywej interpolacyjnej na podstawie trzech węzłów wyznaczonych przez trzy kolejne największe prążki widmowe odpowiadające mierzonemu składnikowi sygnału wejściowego. Odcięta maksimum tej krzywej określa częstotliwość danej składowej sygnału z poprawioną rozdzielczością. Algorytmy interpolacyjne polegają na dopasowaniu paraboli, krzywej gaussowskiej oraz (tak zwanej w pracy) potęgowanej paraboli i dla powszechnie używanych okien umożliwiają poprawę rozdzielczości odpowiednio, o około rząd, dwa i cztery rzędy wielkości. Zakłada się, że mierzona częstotliwość jest stała podczas próbkowania sygnału oraz że odpowiadający jej prążek może być znaleziony w widmie dyskretnym. Praca zawiera opis zależności efektywności algorytmów od użytego okna, szumu obecnego w widmie, interferencji od niepożądanych silnych składowych sygnału wejściowego oraz wykładniczego tłumienia tego sygnału. Bezpośrednim zastosowaniem opisywanych algorytmów są systemy do pomiaru częstotliwości oscylacji betatronowych wiązek cząstek elementarnych wielkich energii. Obecnie są one używane w takich systemach w Europejskiej Organizacji Badań Jądrowych (CERN) w Genewie. Wydaje się, że opisane w rozprawie algorytmy mogą znaleźć także zastosowanie w systemach oraz technikach do laboratoryjnego i przemysłowego pomiaru częstotliwości sygnałów o ograniczonej okresowości, wszędzie tam gdzie wymaga się pomiarów w czasie rzeczywistym, o dużej rozdzielczości i na podstawie niewielkiej ilości próbek. Marek Gąsior, Improving frequency resolution of discrete spectra - VII - - VIII - Marek Gąsior, Improving frequency resolution of discrete spectra Table of contents List of most important symbols and abbreviations.......................................................................... XI 1. Introduction, theses, assumptions ............................................................................................... 1 2. From an analogue signal to its discrete spectrum.................................................................... 15 2.1. Analogue to digital conversion ............................................................................................ 15 2.2. Fourier and Discrete Fourier Transforms ............................................................................ 18 2.3. Sample windowing .............................................................................................................. 22 3. Three node interpolations of discrete spectra.......................................................................... 33 3.1. Main lobe shapes of window spectra................................................................................... 33 3.2. Parabolic interpolation (PI).................................................................................................. 36 3.3. Gaussian interpolation (GI) ................................................................................................. 40 3.4. Exponential parabolic interpolation (EPI) ........................................................................... 44 4. The interpolations on perturbed spectra.................................................................................. 49 4.1. Influence of noise on the interpolation methods.................................................................. 49 4.2. The interpolations on a peak distorted by a nearby interference ......................................... 66 4.3. The interpolations on spectra of exponentially decaying signals ........................................ 72 5. Application examples ................................................................................................................. 77 5.1. Measuring the betatron tune of an accelerator..................................................................... 77 5.2. A measurement on the
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