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The autocorrelation function is a commonly used tool in statistical time series analysis. Under the assumption of Gaussianity, the sample autocorrelation function is the standard method used to estimate this function given a finite number of observations. Non-Gaussian, impulsive observation noise following probability density functions with thick tails, which often occurs in practice, can bias this estimator, rendering classical time series analysis methods ineffective. This work examines the robustness of two estimators of correlation based on memoryless nonlinear functions of observations, the Phase-Phase Correlator (PPC) and the Median- of-Ratios Estimator (MRE), which are applicable to complex-valued Gaussian random pro- cesses. These estimators are very fast and easy to implement in current processors. We show that these estimators are robust from a bias perspective when complex-valued Gaussian pro- cesses are contaminated with impulsive noise at the expense of statistical efficiency at the assumed Gaussian distribution. Additionally, iterative versions of these estimators named the IMRE and IPPC are developed, realizing an improved bias performance over their non- iterative counterparts and the well-known robust Schweppe-type Generalized M-estimator utilizing a Huber cost function (SHGM). An impulsive noise suppression technique is developed using basis pursuit and a priori atom weighting derived from the newly developed iterative estimators. This new technique is proposed as an alternative to the robust filter cleaner, a Kalman filter-like approach that relies on linear prediction residuals to identity and replace corrupted observations. It does not have the same initialization issues as the robust filter cleaner. Robust spectral estimation methods are developed using these new estimators and impulsive noise suppression techniques. Results are obtained for synthetic complex-valued Guassian processes and real-world digital television signals collected using a software defined radio. / Ph. D.

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Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N(^2). The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages, random sampling has been adopted in measurement Random sampling is a technique which overcomes the alias problem in regular sampling. The randomization, however, destroys the symmetry property of the transform kernel of the discrete Fourier transform. Hence, when transforming a randomly sampled sequence to its frequency spectrum, the Fast Fourier transform cannot be applied and the computational complexity is N"^. The objectives of this research project are (1) To devise sampling methods for random sampling such that computation may be reduced while the anti-alias property of random sampling is maintained : Two methods of inserting limited regularities into the randomized sampling grids are proposed. They are parallel additive random sampling and hybrid additive random sampling, both of which can save at least 75% , of the multiplications required. The algorithms also lend themselves to the implementation by a multiprocessor system, which will further enhance the speed of the evaluation. (2) To study the auto-correlation sequence of a randomly sampled sequence as an alternative means to confirm its anti-alias property : The anti-alias property of the two proposed methods can be confirmed by using convolution in the frequency domain. However, the same conclusion is also reached by analysing in the spatial domain the auto-correlation of such sample sequences. A technique to evaluate the auto-correlation sequence of a randomly sampled sequence with a regular step size is proposed. The technique may also serve as an algorithm to convert a randomly sampled sequence to a regularly spaced sequence having a desired Nyquist frequency. (3) To provide a rapid spectral estimation using a coarse kernel : The approximate method proposed by Mason in 1980, which trades the accuracy for the speed of the computation, is introduced for making random sampling more attractive. (4) To suggest possible applications for random and pseudo-random sampling : To fully exploit its advantages, random sampling has been adopted in measurement instruments where computing a spectrum is either minimal or not required. Such applications in instrumentation are easily found in the literature. In this thesis, two applications in digital signal processing are introduced. (5) To suggest an inverse transformation for random sampling so as to complete a two-way process and to broaden its scope of application. Apart from the above, a case study of realizing in a transputer network the prime factor algorithm with regular sampling is given in Chapter 2 and a rough estimation of the signal-to-noise ratio for a spectrum obtained from random sampling is found in Chapter 3. Although random sampling is alias-free, problems in computational complexity and noise prevent it from being adopted widely in engineering applications. In the conclusions, the criteria for adopting random sampling are put forward and the directions for its development are discussed.

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Industrial machines during their operation generate vibration due to dynamic forces acting on the machines. This vibration may create noise, abrasion in the machine parts, mechanical fatigue, degrade performance, transfer to other machines via floor or walls and may cause complete shutdown of the machine. To limit the vibration pre-installation, vibration isolation measures are usually employed in workshops and industrial units. However, such vibration isolation may not be sufficient due to varying operating and physical conditions, such as machine ageing, structural changes and new installations etc. Therefore, it is important to assess the quantity of vibration generated and transmitted during true operating conditions. The thesis work is aimed at the estimation of vibrational transmissibility or transfer from industrial machines to floor and to other adjacent installed machines. This study of transmissibility is based on the measurement and analysis of various spectral estimation tools such as Power Spectral Density (PSD), Frequency Response Function (FRF) and Coherence Function. The overall study is divided into three major steps. Firstly, the initial measurements are carried in BTH on simple Single Degree of Freedom (SDOF) systems to gain confidence in measurement and analysis. Then the measurements are performed on a Lathe machine “Quick Turn Nexus 300-II” in a laboratory at BTH. Finally, the measurements are taken on the machines of an Industrial workshop (KOSAB). The analysis results revealed that vibration measurements in industry are challenging and not easy as measurement in labs. Measurements are contaminated by noise from other machines, which degrade the coherence function. However, vibration transferred from one machine to the floor or other machines may be studied using FRF and PSD. Appropriate further isolations may be employed based on the spectral analysis.

In ultrasound contrast imaging, the discrimination between acoustic echoes from tissue and contrast microbubbles would have as a result the increase of the Contrast-to-Tissue-Ratio, improving therefore the quality of the imaging. The main idea is to differentiate the responses from those two kinds of signals based on their spectral content. The most important features of those sinusoidal signals are that they are very short in duration and than they are very likely to have many closely spaced frequency components. So, in order to achieve this target a novel Bayesian parametric spectral estimation technique has been originally designed by Yan Yan (PhD University of Edinburgh), that is supposed to have greater resolving capabilities than commonly used spectral estimation methods. The new technique uses a reversible jump Markov Chain Monte Carlo (rjMCMC) algorithm so as to identify the frequency components of a signal and it is called parametric because it assumes a model and then the problem of spectral estimation is reduced to that of estimating the parameters of the model. This new method has been initially tested with synthetic signals created in Matlab, so as to define on which parameters it depends and to extract mathematical equations that describe these dependences. And although some coarse comparisons with other techniques showed that the capabilities of this method were great, there was plenty room for improvements. Corrections in the Matlab code of this method, analysis of the code’s output in various ways so as to find which is superior, and the proposal of a new simpler model are just some of the changes that have evidently improved the method’s function. But the most important one is the completion of the amplitude estimation that was left unfinished in the past, as a complete spectral analysis implies both frequency and amplitude estimation. Now, signal reconstruction is possible and also, direct comparisons of the method’s resulting spectrum with the one of the Discrete Fourier Transform or of any other nonparametric (DFT-based) or parametric method can be made. The new version of the code has been applied apart from synthetic signals, to the real ones providing indeed information that was undisclosed in the past concerning the spectral content of those signals. However, further research is required, in order to take advantage of this information and in order to determine the exact performance and limitations of this method that remains still in experimental level. / Στην απεικόνιση με υπέρηχους όταν χρησιμοποιείται μέσο αντίθεσης, ο διαχωρισμός ανάμεσα στην ακουστική ηχώ που προέρχεται από τον ιστό και σε αυτή που προέρχεται από τo μέσο αντίθεσης όπως οι μικροφυσαλίδες, θα μπορούσε να έχει σαν αποτέλεσμα τη βελτίωση της ποιότητας της εικόνας. Η βασική ιδέα είναι να διαφοροποιηθούν οι αποκρίσεις από τα δύο διαφορετικά είδη σημάτων υπερηχοτομογραφίας με βάση το φασματικό τους περιεχόμενο. Τα κυριότερα χαρακτηριστικά αυτών των ημιτονοειδών σημάτων είναι ότι είναι πολύ μικρά σε χρονική διάρκεια και ότι είναι πολύ πιθανό να αποτελούνται από συχνοτικές συνιστώσες που βρίσκονται πολύ κοντά μεταξύ τους. Έτσι, για την επίτευξη αυτού του στόχου, μία καινούρια Μπαγιεσιανή παραμετρική μέθοδος για συχνοτική ανάλυση σχεδιάστηκε αρχικά από την Yan Yan (PhD Πανεπιστήμιο του Εδιμβούργου), η οποία θεωρητικά παρέχει μεγαλύτερη ακρίβεια από τις υπόλοιπες συμβατικές μεθόδους που ήδη χρησιμοποιούνται. Η τεχνική αυτή, κάνει χρήση ενός συγκεκριμένου αλγορίθμου (rjMCMC) έτσι ώστε να προσδιορίσει τις συχνοτικές συνιστώσες ενός σήματος και καλείται παραμετρική επειδή υποθέτει ένα αρχικό μοντέλο και στη συνέχεια «υποβιβάζει» το πρόβλημα της συχνοτικής ανάλυσης σε ένα πιο απλό, όπως αυτό του υπολογισμού των παραμέτρων του μοντέλου. Αρχικά, η μέθοδος δοκιμάστηκε σε συνθετικά σήματα, που δημιουργήθηκαν στο Matlab, ούτως ώστε να προσδιοριστούν οι παράμετροι από τις οποίες εξαρτάται και να εξαχθούν μαθηματικές εξισώσεις που να περιγράφουν τις εξαρτήσεις αυτές. Αν και από τις πρώτες συγκρίσεις με άλλες μεθόδους ήταν φανερό ότι η συγκεκριμένη έχει πολύ μεγάλες δυνατότητες, υπήρχαν πολλά περιθώρια βελτίωσης. Διορθώσεις στον κώδικα Matlab της μεθόδου, ανάλυση της εξόδου του με διάφορους τρόπους με σκοπό να προσδιοριστεί ο πιο αποτελεσματικός και η πρόταση ενός νέου πιο απλού μοντέλου είναι κάποιες από τις αλλαγές που αποδεδειγμένα βελτίωσαν τη λειτουργία της μεθόδου. Αλλά η πιο σημαντική αλλαγή είναι η ολοκλήρωση του κώδικα έτσι ώστε να περιλαμβάνει και προσδιορισμό των πλατών που αντιστοιχούν σε κάθε μια συχνότητα. Ο κώδικας για την εκτίμηση των πλατών αν και προϋπήρχε ως ένα βαθμό, δεν είχε ενσωματωθεί στον κυρίως κώδικα και έτσι η μέθοδος δεν μπορούσε να χαρακτηριστεί ολοκληρωμένη. Τώρα η ανακατασκευή σήματος είναι εφικτή καθώς και οι άμεσες συγκρίσεις του φάσματος της μεθόδου με το αντίστοιχο που προκύπτει από τον διακριτό μετασχηματισμό Fourier ή από άλλες μη παραμετρικές και παραμετρικές μεθόδους. Η καινούρια έκδοση του κώδικα εφαρμόστηκε, εκτός από τα συνθετικά σήματα, στα πραγματικά σήματα παρέχοντας πράγματι πληροφορίες που δεν είχαν φανερωθεί στο παρελθόν, σχετικά με το φασματικό περιεχόμενο των σημάτων αυτών. Ωστόσο, περαιτέρω έρευνα απαιτείται για να αξιοποιηθούν οι πληροφορίες αυτές αλλά και για να προσδιοριστούν οι ακριβείς περιορισμοί και οι επιδόσεις της μεθόδου που ακόμα και τώρα παραμένει σε πειραματικό στάδιο.

Digitized acoustical signals of Byzantine music performed by Iakovos Nafpliotis are used to extract the fundamental frequency of each note of the diatonic scale. These empirical results are then contrasted to the theoretical suggestions and previous empirical findings. Several parametric and non-parametric spectral parameter estimation methods are implemented. These include: (1) Phase vocoder method, (2) McAulay-Quatieri method, (3) Levinson-Durbin algorithm,(4) YIN, (5) Quinn & Fernandes Estimator, (6) Pisarenko Frequency Estimator, (7) MUltiple SIgnal Characterization (MUSIC) algorithm, (8) Periodogram method, (9) Quinn & Fernandes Filtered Periodogram, (10) Rife & Vincent Estimator, and (11) the Fourier transform. Algorithm performance was very precise. The psychophysical aspect of human pitch discrimination is explored. The results of eight (8) psychoacoustical experiments were used to determine the aural just noticeable difference (jnd) in pitch and deduce patterns utilized to customize acceptable performable pitch deviation to the application at hand. These customizations [Acceptable Performance Difference (a new measure of frequency differential acceptability), Perceptual Confidence Intervals (a new concept of confidence intervals based on psychophysical experiment rather than statistics of performance data), and one based purely on music-theoretical asymphony] are proposed, discussed, and used in interpretation of results. The results suggest that Nafpliotis' intervals are closer to just intonation than Byzantine theory (with minor exceptions), something not generally found in Thrasivoulos Stanitsas' data. Nafpliotis' perfect fifth is identical to the just intonation, even though he overstretches his octaveby fifteen (15)cents. His perfect fourth is also more just, as opposed to Stanitsas' fourth which is directionally opposite. Stanitsas' tendency to exaggerate the major third interval A4-F4 is still seen in Nafpliotis, but curbed. This is the only noteworthy departure from just intonation, with Nafpliotis being exactly Chrysanthian (the most exaggerated theoretical suggestion of all) and Stanitsas overstretching it even more than Nafpliotis and Chrysanth. Nafpliotis ascends in the second tetrachord more robustly diatonically than Stanitsas. The results are reported and interpreted within the framework of Acceptable Performance Differences.

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