The Gaussian distribution is typically used to model the additive noise affecting communication systems. However, in many cases the noise cannot be modeled by a Gaussian distribution. In this thesis, we investigate the performance of different communication systems perturbed by non-Gaussian noise. Three families of noise are considered in this work, namely the generalized Gaussian noise, the Laplace noise/interference, and the impulsive noise that is modeled by an α-stable distribution. More specifically, in the first part of this thesis, the impact of an additive generalized Gaussian noise is studied by computing the average symbol error rate (SER) of one dimensional and two dimensional constellations in fading environment. We begin by the simple case of two symbols, i.e. binary phase shift keying (BPSK) constellation. From the results of this constellation, we extended the work to the average SER of an M pulse amplitude modulation (PAM). The first 2 − D constellation is the M quadrature amplitude modulation (QAM) (studied for two geometric shapes, namely square and rectangular), which is the combination of two orthogonal PAM signals (in-phase and quadrature phase PAM). In the second part, the system performance of a circular constellation, namely M phase shift keying (MPSK) is studied in conjunction with a Laplace noise with independent noise components. A closed form and an asymptotic expansion of the SER are derived for two detectors, maximum likelihood and minimum distance detectors. Next, we look at the intra cell interference of a full duplex cellular network which is shown to follow a Laplacian distribution with dependent, but uncorrelated, complex components. The densities of that interference are expressed in a closed form in order to obtain the SER of several communication systems (BPSK, PAM, QAM, and MPSK). Finally, we study the statistics of the α-stable distribution. Those statistics are expressed in closed form in terms of the Fox H function and used to get the SER of BPSK, PAM, and QAM constellations. An approximation and an asymptotic expansion for high signal to noise ratio are presented also and their efficiency is proved using Monte Carlo simulations. It is worth mentioning that all the error rates presented in this work are averaged over a generalized flat fading, namely the extended generalized K, which has the ability to capture most of the known fading distribution. Many special cases are treated and simpler closed form expressions of the probability of error are derived and compared to some previous reported results.

No / Abstract: We develop a novel approach to estimate the n unknown constituent frequencies of a sinusoidal signal that comprises of unknown number, n, of sinusoids of unknown phases and unknown amplitudes. The approach has been applied to multiple sinusoidal signals in the presence of white Gaussian noise with varying signal to noise ratio (SNR). The approach is based on eigenspace analysis of Hankel matrix formed with the samples from averaged frequency spectrum of the signal obtained through multiple measurements. The eigenspace analysis is based on the newly developed 3M relationship which reflects and exploits the relationship between the consecutive sets of Maximum, Middle and Minimum eigenvalues of square symmetric matrix of the Hankel matrix. The 3M relationship exhibits a pattern in line with the order of the Hankel matrix and leads to parametric estimation of the constituent sinusoids. This paper also presents the relationship equation between the size of 3M relationship pattern and the dimensions of the Hankel matrix. The performance of the developed approach has been tested to correctly estimate multiple constituent frequencies within a noisy signal.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

Στην παρούσα διπλωματική εργασία επιχειρείται η ανάλυση και η εκτίμηση της συχνότητας απλών ημιτονοειδών σημάτων υπό την παρουσία λευκού Γκαουσιανού θορύβου (AWGN).Η εκτίμηση παραμέτρων απλών ημιτονοειδών σημάτων υπό την παρουσία προσθετικού Γκαουσιανού θορύβου αποτελεί ένα κλασσικό πρόβλημα και σημαντικό αντικείμενο μελέτης εξαιτίας της πληθώρας των εφαρμογών που έχει στην θεωρία ελέγχου, στην επεξεργασία σημάτων, στις ψηφιακές επικοινωνίες, στην βιοϊατρική τεχνολογία κ.α.Η εκτίμηση της συχνότητας είναι συνήθως το θέμα ‘ζωτικής σημασίας’ του προβλήματος για δύο σημαντικούς λόγους. Αφ’ενός οι συχνότητες πρέπει να εκτιμηθούν διότι αποτελούν μη-γραμμικές συναρτήσεις στην ληφθείσα ακολουθία δεδομένων και αφ’ ετέρου έχοντας καθοριστεί οι συχνότητες, οι υπόλοιπες παράμετροι του σήματος όπως είναι το πλάτος και η φάση του, μπορούν να υπολογιστούν άμεσα. Αρχικά γίνεται μία σύντομη εισαγωγή στις βασικές έννοιες πάνω στις οποίες δομείται η εκτίμηση παραμέτρων ενός ημιτονοειδούς σήματος και έπειτα παρουσιάζονται μερικοί αλγόριθμοι εκτίμησης. Πιο συγκεκριμένα παρουσιάζεται η διαδικασία κατασκευής τους και αναλύονται οι επιδόσεις τους. Τέλος παραθέτουμε και προσομοιώσεις μέσω υπολογιστή για κάθε αλγόριθμο ξεχωριστά και συγκρίνουμε την επίδοση του καθενός με τους υπόλοιπους. Από την σύγκριση αυτή γίνεται εξαγωγή χρήσιμων συμπερασμάτων σχετικά με τον προσδιορισμό των παραμέτρων κάθε αλγόριθμου αλλά και με την καταλληλότητα κάθε αλγόριθμου για συγκεκριμένες συνθήκες θορύβου. / In this thesis attempts to analyze and estimate the frequency of single sinusoid signals in Additive White Gaussian Noise (AWGN). Parameter estimation of sinusoids has been a classical problem and it is still an important research topic because of its numerous applications in multiple disciplines such as control theory, signal processing, digital communications, biomedical engineering etc. Estimation of the frequencies is often the crucial step in the problem for two principally reasons. Firstly, frequencies should be estimated because they are nonlinear functions in the received data sequence and secondly, once frequencies have been determined, the remaining parameters, such as amplitude and phase, can then be computed straightforwardly. Primarily we introduce some basic concepts on parameters estimation of sinusoid signals and then several estimation algorithms. More specifically shows the fabrication process of these algorithms and analyze their performance. Finally, we quote computer simulations for each algorithm separately and compare their performance. From these comparisons we can draw conclusions on the determination of parameters for each algorithm and the appropriateness of algorithms for specific noise conditions.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

This thesis focuses on comparing methods of denoising by deep learning and their implementation. In the last few years, it has become clear that it is not necessary to have paired data, as for noisy and clean pictures, to train convolution neural networks but it is sufficient to have only noisy pictures for denoising in particular cases. By using methods described in this thesis it is possible to effectively remove i.e. additive Gaussian noise and what more, it is possible to achieve better results than by using statistic methods, which are being used for denoising these days.

This master´s work is focused on the comparison of subjective assessment of the quality of video sequences. In this study, data are obtained by hardware and sofware techniques and they are compared. In the introduction, methods of video compressions are described. The main part of this work deals wtih the exploring of different methods of subjective assessment of the quality of video sequences. Finally, obtained results from different methods, are evaluated and discussed.

Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possible definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive understanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. 1

Title: Long range dependence in time series Author: Alexander Till Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Michaela Prokešová, Ph.D. Abstract: The diploma thesis demonstrates the necessity of a study of long range dependence, introduces fractional Gaussian noise and discusses possi- ble definitions of long memory. It is done by notions of ergodic theory and by second moment characteristics and spectral density. These definitions are confronted with the model of fractional Gaussian noise and with intuitive un- derstanding of long range memory. Relations and connections between these criteria are studied as well. The work is restricted to the study of discrete time processes. Method for Hurst index estimation for fractional Gaussian noise and it's application on logarithmic returns of shares of selected produ- cers of beer are included in this work. 1