No / Bifractional displacement operators, are introduced by performing two fractional Fourier transforms on displacement operators. They are shown to be special cases of elements of the group G , that contains both displacements and squeezing transformations. Acting with them on the vacuum we get various classes of coherent states, which we call bifractional coherent states. They are special classes of squeezed states which can be used for interpolation between various quantities in phase space methods. Using them we introduce bifractional Wigner functions A(α,β;θα,θβ)A(α,β;θα,θβ), which are a two-dimensional continuum of functions, and reduce to Wigner and Weyl functions in special cases. We also introduce bifractional Q-functions, and bifractional P-functions. The physical meaning of these quantities is discussed.

Both the fast Fourier transform and discrete wavelet transform have been used extensively for signal denoising. Therefore, comparing the two for the purpose of denoising an electrocardiogram is of high interest. In this report, we outline the theory that both methods are built on, as well as develop MATLAB codes able to denoise an electrocardiogram using both methods. It was shown that the discrete wavelet transform performed significantly better in this context, which shows why it is the preferred method.

Thesis (M.S.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / We address the problem of efficiently computing, over narrow frequency bands, the short-time Fourier transform (STFT) and approximations to the STFT. This problem is important for the design of signal understanding systems that have to efficiently carry out STFT reprocessing of signals in order to examine detailed features of signal components that have already been located within narrow frequency bands. In the computation of the exact STFT we use an "overlap pruning" approach (Covell et al. 1992) for exploiting the commonality of computations between successive slices of the STFT with unity decimation interval. We have also extended this approach to the STFT with non-unity decimation intervals and combined it with a frequency pruning method (Sreenivas et al. 1980) to provide additional computational savings. In the computation of approximations to the STFT we use an algorithm (Khan et al. 1988) for efficiently computing Taylor series approximations over narrow frequency bands. Through examples involving real data we demonstrate the feasibility of using the approximated STFT to obtain more accurate estimates of the center frequency of spectral peaks, and to resolve multiple peaks that have been smeared due to the use of short window lengths. The efficiency of all the algorithms we have investigated is less than 0(N log N) multiplications per STFT slice and can be as small as O(N) multiplications per STFT slice in certain cases. Consequently, all the algorithms compare favourably with the standard FFT implementation of the STFT which requires O(N log N) multiplications per slice. All the algorithms considered in this thesis were implemented in software and tested on synthetic and real sound signals. / 2999-01-01

No description available.

Background: The efficient market hypothesis asserts that one cannot consistently achieve returns in excess of market returns by trading on publicly available information. Since there is no collective market return in the foreign exchange (FX) market, it has generally been perceived as impossible to consistently generate a profit. There is now empirical evidence which seriously call into question the efficiency of the FX market and opens up the possibility to turn a profit on the FX market by ways of analysis.Technical analysis is a method of analysis which by using historical price data tries to deduce future price changes. Technical analysis assumes that financial markets move in sine waves. There are stronger and weaker sine waves simultaneously. An accurate identification of the dominant sine wave gives the investor a good idea about future movement. Most technical trading tools approximate the length of the sine wave by default. This static approach does not consider the specific market or the recent lengths of the dominant sine wave. Spectral analysis will help to identify the dominant cycle, and thus determine the frequency of that cycle making the applied trading rules adaptive to the market. Purpose: The purpose is to investigate whether adding spectral analysis to existing technical analysis tools can create a higher and more stable return on investment on the FX market. Method: An experiment involving four different sets of trading rules was conducted to answer the purpose. In the first test, trades were performed based on a static approach commonly used by technical traders today. In the other three tests different transforms of spectral analysis were applied, thus making the input not static, but adaptive to the market. The four sets of trading rules where coded as an automatic trading algorithm and backtested on data collected for the currency-pair EURGBP during an 11-month period. All four tests were analysed in three different areas; performance, stability of return and crash risk. Results: The study shows that the application of spectral analysis to technical analysis methods on the FX market results in higher return on investment and better stability of returns. The win/lose ratio is significantly higher and the adaptive approach increases profit as well as decreases losses. / Bakgrund: Den effektiva marknadshypotesen stadgar att det inte är möjligt att stadigt generera högre avkastning än marknadens kollektiva avkastning genom att köpa och sälja baserat på tillgänglig information. Eftersom det inte finns någon kollektiv avkastning på valutamarknaden har det länge ansetts omöjligt att generera någon stabil vinst på denna marknad. Det finns numera empiriskt bevis som tydligt ifrågasätter valutamarknadens egentliga effektivitet och som också i sin tur öppnar upp för möjligheten att generera stabil avkastning på valutamarknaden genom analys.Teknisk analys är en analysmetod som genom avläsandet av historisk prisdata försöker utläsa framtida prisförändringar. Teknisk analys antar att finansiella marknader rör sig i sinuskurvor. Det finns starkare och svagare sinuskurvor. En exakt identifikation av den dominanta cykeln ger investeraren en god idé om framtida rörelser. De flesta tekniska analysverktygen uppskattar längden på cykeln statiskt och tar varken hänsyn till den specifika marknaden eller hur den dominanta cykeln har sett ut nyligen. Spektralanalys identifierar den dominanta cykeln varigenom frekvensen av densamma kan bestämmas och analysverktyget görs adaptivt till marknaden. Syfte: Syftet med uppsatsen är att ta reda på huruvida teknisk analys på valutamarknaden kan skapa en högre och mer stabil avkastning på investerat kapital genom användandet av spektralanalys för att mäta den dominanta cykeln. Metod: Ett experiment innehållande fyra olika uppsättningar av analysverktyg gjordes för att besvara syftet. Handel i det första testet baserades på en statisk ansats som normalt används av tekniska analytiker idag. På de andra tre testerna applicerades olika transformer av spektralanalys och gjordes därigenom adaptiva till marknaden. Analysverktygen kodades som en automatisk handelsalgoritm och testades retroaktivt på insamlad data för valutaparet EURGBP under elva månader. Samtliga fyra tester analyserades i tre olika områden; prestation, avkastningsstabilitet och risk att förlora hela kapitalet. Resultat: Studien visar att applikationen av spektralanalys på tekniska analysverktyg på valutamarknaden resulterar i högre avkastning på investerat kapital och högre avkastningsstabilitet. Vinst/förlust ration är väsentligt högre och den adaptiva ansatsen ökar avkastning samtidigt som den minskar förluster.

Speech has been the subject of interest for a very long time. Even with so much advancement in the processing techniques and in the understanding of the source of speech, it is, even today, rather difficult to generate speech in the laboratory in all its aspects. A simple aspect like how the speech can retain its intelligibility even if it is distorted or band passed is not really understood. This thesis deals with one small feature of speech viz., the intelligibility of speech is retained even when it is bandpassed with a minimum bandwidth of around 1 KHz located any where on the speech spectrum of 0-4 KHz. Several experiments have been conducted by the earlier workers by passing speech through various distortors like differentiators, integrators and infinite peak clippers and it is found that the intelligibility is retained to a very large extent in the distorted speech. The integrator and the differentiator remove essentially a certain portion of the spectrum. Therefore, it is thought that the intelligibility of the speech is spread over the entire speech spectrum and that, the intelligibility of speech may not be impaired even when it is bandpassed with a minimum bandwidth and the band may be located any where in the speech spectrum. To test this idea and establish this feature if it exists, preliminary experiments have been conducted by passing the speech through different filters and it is found that the conjecture seems to be on the right line. To carry out systematic experiments on this an experimental set up has been designed and fabricated which consists of a microprocessor controlled speech recording, storing and speech playback system. Also, a personal computer is coupled to the microprocessor system to enable the storage and processing of the data. Thirty persons drawn from different walks of life like teachers, mechanics and students have been involved for collecting the samples and for recognition of the information of the processed speech. Even though the sentences like 'This is devices lab' are used to ascertain the effect of bandwidth on the intelligibility, for the purpose of analysis, vowels are used as the speech samples. The experiments essentially consist of recording words and sentences spoken by the 30 participants and these recorded speech samples are passed through different filters with different bandwidths and central frequencies. The filtered output is played back to the various listeners and observations regarding the intelligibility of the speech are noted. The listeners do not have any prior information about the content of the speech. It has been found that in almost all (95%) cases, the messages or words are intelligible for most of the listeners when the band width of the filter is about 1 KHz and this is independent of the location of the pass band in the spectrum of 0-4 KHz. To understand how this feature of speech arises, spectrums of vowels spoken by 30 people have using FFT algorithms on the digitized samples of the speech. It is felt that there is a cyclic behavior of the spectrum in all the samples. To make sure that the periodicity is present and also to arrive at the periodicity, a moving average procedure is employed to smoothen the spectrum. The smoothened spectrums of all the vowels indeed show a periodicity of about 1 KHz. When the periodicities are analysed the average value of the periodicities has been found to be 1038 Hz with a standard deviation of 19 Hz. In view of this it is thought that the acoustic source responsible for speech must have generated this periodic spectrum, which might have been modified periodically to imprint the intelligibility. If this is true, one can perhaps easily understand this feature of the speech viz., the intelligibility is retained in a bandpassed speech of bandwidth 1 K H z . the pass band located any where in the speech spectrum of 0-4 KHz. This thesis describing the experiments and the analysis of the speech has been presented in 5 chapters. Chapter 1 deals with the basics of speech and the processing tools used to analyse the speech signal. Chapter 2 presents the literature survey from where the present problem is tracked down. Chapter 3 describes the details of the structure and the fabrication of the experimental setup that has been used. In chapter 4, the detailed account of the way in which the experiments are conducted and the way in which the speech is analysed is given. In conclusion in chapter 5, the work is summarised and the future work needed to establish the mechanism of speech responsible for the feature of speech described in this thesis is suggested.

<p><strong>Background: </strong></p><p>The efficient market hypothesis asserts that one cannot consistently achieve returns in excess of market returns by trading on publicly available information. Since there is no collective market return in the foreign exchange (FX) market, it has generally been perceived as impossible to consistently generate a profit. There is now empirical evidence which seriously call into question the efficiency of the FX market and opens up the possibility to turn a profit on the FX market by ways of analysis.Technical analysis is a method of analysis which by using historical price data tries to deduce future price changes. Technical analysis assumes that financial markets move in sine waves. There are stronger and weaker sine waves simultaneously. An accurate identification of the dominant sine wave gives the investor a good idea about future movement. Most technical trading tools approximate the length of the sine wave by default. This static approach does not consider the specific market or the recent lengths of the dominant sine wave. Spectral analysis will help to identify the dominant cycle, and thus determine the frequency of that cycle making the applied trading rules adaptive to the market.</p><p><strong>Purpose: </strong></p><p>The purpose is to investigate whether adding spectral analysis to existing technical analysis tools can create a higher and more stable return on investment on the FX market.</p><p><strong>Method: </strong></p><p>An experiment involving four different sets of trading rules was conducted to answer the purpose. In the first test, trades were performed based on a static approach commonly used by technical traders today. In the other three tests different transforms of spectral analysis were applied, thus making the input not static, but adaptive to the market. The four sets of trading rules where coded as an automatic trading algorithm and backtested on data collected for the currency-pair EURGBP during an 11-month period. All four tests were analysed in three different areas; performance, stability of return and crash risk.</p><p><strong>Results: </strong></p><p><strong></strong>The study shows that the application of spectral analysis to technical analysis methods on the FX market results in higher return on investment and better stability of returns. The win/lose ratio is significantly higher and the adaptive approach increases profit as well as decreases losses.</p> / <p><strong>Bakgrund: </strong></p><p>Den effektiva marknadshypotesen stadgar att det inte är möjligt att stadigt generera högre avkastning än marknadens kollektiva avkastning genom att köpa och sälja baserat på tillgänglig information. Eftersom det inte finns någon kollektiv avkastning på valutamarknaden har det länge ansetts omöjligt att generera någon stabil vinst på denna marknad. Det finns numera empiriskt bevis som tydligt ifrågasätter valutamarknadens egentliga effektivitet och som också i sin tur öppnar upp för möjligheten att generera stabil avkastning på valutamarknaden genom analys.Teknisk analys är en analysmetod som genom avläsandet av historisk prisdata försöker utläsa framtida prisförändringar. Teknisk analys antar att finansiella marknader rör sig i sinuskurvor. Det finns starkare och svagare sinuskurvor. En exakt identifikation av den dominanta cykeln ger investeraren en god idé om framtida rörelser. De flesta tekniska analysverktygen uppskattar längden på cykeln statiskt och tar varken hänsyn till den specifika marknaden eller hur den dominanta cykeln har sett ut nyligen. Spektralanalys identifierar den dominanta cykeln varigenom frekvensen av densamma kan bestämmas och analysverktyget görs adaptivt till marknaden.</p><p><strong>Syfte: </strong></p><p>Syftet med uppsatsen är att ta reda på huruvida teknisk analys på valutamarknaden kan skapa en högre och mer stabil avkastning på investerat kapital genom användandet av spektralanalys för att mäta den dominanta cykeln.</p><p><strong>Metod: </strong></p><p>Ett experiment innehållande fyra olika uppsättningar av analysverktyg gjordes för att besvara syftet. Handel i det första testet baserades på en statisk ansats som normalt används av tekniska analytiker idag. På de andra tre testerna applicerades olika transformer av spektralanalys och gjordes därigenom adaptiva till marknaden. Analysverktygen kodades som en automatisk handelsalgoritm och testades retroaktivt på insamlad data för valutaparet EURGBP under elva månader. Samtliga fyra tester analyserades i tre olika områden; prestation, avkastningsstabilitet och risk att förlora hela kapitalet.</p><p><strong>Resultat: </strong></p><p><strong></strong>Studien visar att applikationen av spektralanalys på tekniska analysverktyg på valutamarknaden resulterar i högre avkastning på investerat kapital och högre avkastningsstabilitet. Vinst/förlust ration är väsentligt högre och den adaptiva ansatsen ökar avkastning samtidigt som den minskar förluster.</p>

Context: We identify a material or thing that can be seen and touched in the world as having structures at both coarser and finer levels of scale. Scaling problem presents in a branch of science concerned with the description, prediction understanding of natural phenomena and visual arts. A moon, for instance, may appear as having a roughly round shape is much larger than stars when seen from the earth. In the closer look, the moon is much smaller than the stars. The fact that objects in the world appear in different ways depending upon the scale of observation has important implications when analyzing measured data, such as images, with automatic methods [1]. The type of information we are seeking from a one-dimensional signal or two-dimensional image is only possible when we have the right amount of scale for the structure of an image or signal data. In many modern applications, the right scale need not be obvious at all, and we all need a complete mathematical analysis on this scaling problem. This thesis is shown how a mathematical theory is formulated when data or signal is describing at different scales. Objectives: The subtle patterns deforming in data that can foretell of a scaling problem? The main objectives of this thesis are to address the dynamic scaling pattern problem in computers and study the different methods, described in the latest issue of Science, are designed to identify the patterns in data. Method: The research methodology used in this thesis is the Fractional Fourier Transform. To recognize the pattern for a different level of scale to one or many components, we take the position and size of the object and perform the transform operation in any transform angle and deform the component by changing to another angle which influences the frequency, phase, and magnitude.  Results: We show that manipulation of Fractional Fourier transform can be used as a pattern recognition system. The introduced model has the flexibility to encode patterns to both time and frequency domain. We present a detailed structure of a dynamic pattern scaling problem. Furthermore, we show successful recognition results even though one or many components deformed to different levels using one-dimensional and two-dimensional patterns. Conclusions: The proposed algorithm FrFT has shown some advantages over traditional FFT due to its competitive performance in studying the pattern changes. This research work investigated that simulating the dynamic pattern scaling problem using FrFT. The Fractional Fourier transform does not do the scaling. Manipulating the Fractional Fourier transform can be helpful in perceiving the pattern changes. We cannot control the deformation but changing the parameters allow us to see what is happening in time and frequency domain.

The fast Fourier transform (FFT), even though it has been widely applicable in Physics and Engineering, it has become attractive in Finance as well for it’s enhancement of computational speed. Carr and Madan succeeded in implementing the FFT for pricing of an option. This project, inspired by Carr and Madan’s paper, attempts to elaborate and connect the various mathematical and theoretical concepts that are helpful in understanding of the derivation. Further, we derive the characteristic function of the risk neutral probability for the logarithmic terminal stock price. The Black-Scholes-Merton (BSM) model is also revised including derivation of the partial deferential equation and the formula. Finally, comparison of the BSM numerical implementation with and without the FFT method is done using MATLAB.

This master’s thesis deals with signal evaluation using Fourier transform. In the theoretical section, different methods of signal analysis with an emphasis on a Fast Fourier Transform are described. Theoretical section also contains description of common machinery faults and their diagnosis using frequency spectrum analysis. In the practical section, an algorithm for signal evaluation with an emphasis on coherence condition was designed in the Matlab environment. This algorithm was later used to evaluate submitted signals.