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The objective of my thesis is to find optimal points and the quantization error for a probability measure defined on a Cantor set. The Cantor set, we have considered in this work, is generated by two self-similar contraction mappings on the real line with distinct similarity ratios. Then we have defined a nonhomogeneous probability measure, the support of which lies on the Cantor set. For such a probability measure first we have determined the n-optimal points and the nth quantization error for n = 2 and n = 3. Then by some other lemmas and propositions we have proved a theorem which gives all the n-optimal points and the nth quantization error for all positive integers n. In addition, we have given some properties of the optimal points and the quantization error for the probability measure. In the end, we have also given a list of n-optimal points and error for some positive integers n. The result in this thesis is a nonhomogeneous extension of a similar result of Graf and Luschgy in 1997. The techniques in my thesis could be extended to discretise any continuous random variable with another random variable with finite range.

No description available.

As Moore���s Law continues to give rise to ever shrinking channel lengths, circuits are becoming more digital and ever increasingly faster. Generating high frequency clocks in such scaled processes is becoming a tough challenge. Digital phase locked loops (DPLLs) are being explored as an alternative to conventional analog PLLs but suffer from issues such as low bandwidth and higher quantization noise. A digital multiplying delay locked loop (DMDLL) is proposed which aims at leveraging the benefit of high bandwidth of DLL while at the same time achieving the frequency multiplication property of PLL. It also offers the benefits of easier portability across process and occupies lesser area. The proposed DMDLL uses a simple flip-flop as 1-bit TDC (Time Digital Converter) for Phase Detector (PD). A digital accumulator acts as integrator for loop filter while a ��-�� DAC in combination with a VCO acts like a DCO. A carefully designed select logic in conjunction with a MUX achieves frequency multiplication. The proposed digital MDLL is taped out in 130nm process and tested to obtain 1.4GHz output frequency with 1.6ps RMS jitter, 17ps peak-to-peak jitter and -50dbC/Hz reference spurs. / Graduation date: 2012

We study the asymptotic behavior of the quantization error for general information functions and prove results for distributions P with regular support. We characterize the information functions for which the uniform distribution on the set of prototypes converges weakly to P. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

We study the empirical quantization error in case the number of prototypes increases with the size of the sample. We present a proof of the consistency of the empirical quantization error and of corresponding estimators of the quantization dimensions of distributions. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Une nouvelle approche du traitement de l'image, appelée SOM-QE, qui exploite quantization error (QE) des self-organizing maps (SOM) est proposée dans cette thèse. Les SOM produisent des représentations discrètes de faible dimension des données d'entrée de haute dimension. QE est déterminée à partir des résultats du processus d'apprentissage non supervisé du SOM et des données d'entrée. SOM-QE d'une série chronologique d'images peut être utilisé comme indicateur de changements dans la série chronologique. Pour configurer SOM, on détermine la taille de la carte, la distance du voisinage, le rythme d'apprentissage et le nombre d'itérations dans le processus d'apprentissage. La combinaison de ces paramètres, qui donne la valeur la plus faible de QE, est considérée comme le jeu de paramètres optimal et est utilisée pour transformer l'ensemble de données. C'est l'utilisation de l'assouplissement quantitatif. La nouveauté de la technique SOM-QE est quadruple : d'abord dans l'usage. SOM-QE utilise un SOM pour déterminer la QE de différentes images - typiquement, dans un ensemble de données de séries temporelles - contrairement à l'utilisation traditionnelle où différents SOMs sont appliqués sur un ensemble de données. Deuxièmement, la valeur SOM-QE est introduite pour mesurer l'uniformité de l'image. Troisièmement, la valeur SOM-QE devient une étiquette spéciale et unique pour l'image dans l'ensemble de données et quatrièmement, cette étiquette est utilisée pour suivre les changements qui se produisent dans les images suivantes de la même scène. Ainsi, SOM-QE fournit une mesure des variations à l'intérieur de l'image à une instance dans le temps, et lorsqu'il est comparé aux valeurs des images subséquentes de la même scène, il révèle une visualisation transitoire des changements dans la scène à l'étude. Dans cette recherche, l'approche a été appliquée à l'imagerie artificielle, médicale et géographique pour démontrer sa performance. Les scientifiques et les ingénieurs s'intéressent aux changements qui se produisent dans les scènes géographiques d'intérêt, comme la construction de nouveaux bâtiments dans une ville ou le recul des lésions dans les images médicales. La technique SOM-QE offre un nouveau moyen de détection automatique de la croissance dans les espaces urbains ou de la progression des maladies, fournissant des informations opportunes pour une planification ou un traitement approprié. Dans ce travail, il est démontré que SOM-QE peut capturer de très petits changements dans les images. Les résultats confirment également qu'il est rapide et moins coûteux de faire la distinction entre le contenu modifié et le contenu inchangé dans les grands ensembles de données d'images. La corrélation de Pearson a confirmé qu'il y avait des corrélations statistiquement significatives entre les valeurs SOM-QE et les données réelles de vérité de terrain. Sur le plan de l'évaluation, cette technique a donné de meilleurs résultats que les autres approches existantes. Ce travail est important car il introduit une nouvelle façon d'envisager la détection rapide et automatique des changements, même lorsqu'il s'agit de petits changements locaux dans les images. Il introduit également une nouvelle méthode de détermination de QE, et les données qu'il génère peuvent être utilisées pour prédire les changements dans un ensemble de données de séries chronologiques. / A new approach for image processing, dubbed SOM-QE, that exploits the quantization error (QE) from self-organizing maps (SOM) is proposed in this thesis. SOM produce low-dimensional discrete representations of high-dimensional input data. QE is determined from the results of the unsupervised learning process of SOM and the input data. SOM-QE from a time-series of images can be used as an indicator of changes in the time series. To set-up SOM, a map size, the neighbourhood distance, the learning rate and the number of iterations in the learning process are determined. The combination of these parameters that gives the lowest value of QE, is taken to be the optimal parameter set and it is used to transform the dataset. This has been the use of QE. The novelty in SOM-QE technique is fourfold: first, in the usage. SOM-QE employs a SOM to determine QE for different images - typically, in a time series dataset - unlike the traditional usage where different SOMs are applied on one dataset. Secondly, the SOM-QE value is introduced as a measure of uniformity within the image. Thirdly, the SOM-QE value becomes a special, unique label for the image within the dataset and fourthly, this label is used to track changes that occur in subsequent images of the same scene. Thus, SOM-QE provides a measure of variations within the image at an instance in time, and when compared with the values from subsequent images of the same scene, it reveals a transient visualization of changes in the scene of study. In this research the approach was applied to artificial, medical and geographic imagery to demonstrate its performance. Changes that occur in geographic scenes of interest, such as new buildings being put up in a city or lesions receding in medical images are of interest to scientists and engineers. The SOM-QE technique provides a new way for automatic detection of growth in urban spaces or the progressions of diseases, giving timely information for appropriate planning or treatment. In this work, it is demonstrated that SOM-QE can capture very small changes in images. Results also confirm it to be fast and less computationally expensive in discriminating between changed and unchanged contents in large image datasets. Pearson's correlation confirmed that there was statistically significant correlations between SOM-QE values and the actual ground truth data. On evaluation, this technique performed better compared to other existing approaches. This work is important as it introduces a new way of looking at fast, automatic change detection even when dealing with small local changes within images. It also introduces a new method of determining QE, and the data it generates can be used to predict changes in a time series dataset.

A large amount of today’s telecommunication consists of mobile and short distance wireless applications, where the effect of the channel is unknown and changing over time, and thus needs to be described statistically. Therefore the received signal can not be accurately predicted and has to be estimated. Since telecom systems are implemented in real-time, the hardware in the receiver for estimating the sent signal can for example be based on a DSP where the statistic calculations are performed. A fixed-point DSP with a limited number of bits and a fixed binary point causes larger quantization errors compared to floating point operations with higher accuracy. The focus on this thesis has been to build a library of functions for handling fixed-point data. A class that can handle the most common arithmetic operations and a least squares solver for fixed-point have been implemented in MATLAB code. The MATLAB Fixed-Point Toolbox could have been used to solve this task, but in order to have full control of the algorithms and the fixed-point handling an independent library was created. The conclusion of the simulation made in this thesis is that the least squares result are depending more on the number of integer bits then the number of fractional bits. / En stor del av dagens telekommunikation består av mobila trådlösa kortdistanstillämpningar där kanalens påverkan är okänd och förändras över tid. Signalen måste därför beskrivas statistiskt, vilket gör att den inte kan bestämmas exakt, utan måste estimeras. Eftersom telekomsystem arbetar i realtid består hårdvaran i mottagaren av t.ex. en DSP där de statistiska beräkningarna görs. En fixtals DSP har ett bestämt antal bitar och fast binärpunkt, vilket introducerar ett större kvantiseringsbrus jämfört med flyttalsoperationer som har en större noggrannhet. Tyngdpunkten på det här arbetet har varit att skapa ett bibliotek av funktioner för att hantera fixtal. En klass har skapats i MATLAB-kod som kan hantera de vanligaste aritmetiska operationerna och lösa minsta-kvadrat-problem. MATLAB:s Fixed-Point Toolbox skulle kunna användas för att lösa den här uppgiften men för att ha full kontroll över algoritmerna och fixtalshanteringen behövs ett eget bibliotek av funktioner som är oberoende av MATLAB:s Fixed-Point Toolbox. Slutsatsen av simuleringen gjord i detta examensarbete är att resultatet av minsta-kvadrat-metoden är mer beroende av antalet heltalsbitar än antalet binaler. / fixtal, telekommunikation, DSP, MATLAB, Fixed-Point Toolbox, minsta-kvadrat-lösning, flyttal, Householder QR faktorisering, saturering, kvantiseringsbrus

<p>A large amount of today’s telecommunication consists of mobile and short distance wireless applications, where the effect of the channel is unknown and changing over time, and thus needs to be described statistically. Therefore the received signal can not be accurately predicted and has to be estimated. Since telecom systems are implemented in real-time, the hardware in the receiver for estimating the sent signal can for example be based on a DSP where the statistic calculations are performed. A fixed-point DSP with a limited number of bits and a fixed binary point causes larger quantization errors compared to floating point operations with higher accuracy.</p><p>The focus on this thesis has been to build a library of functions for handling fixed-point data. A class that can handle the most common arithmetic operations and a least squares solver for fixed-point have been implemented in MATLAB code.</p><p>The MATLAB Fixed-Point Toolbox could have been used to solve this task, but in order to have full control of the algorithms and the fixed-point handling an independent library was created.</p><p>The conclusion of the simulation made in this thesis is that the least squares result are depending more on the number of integer bits then the number of fractional bits.</p> / <p>En stor del av dagens telekommunikation består av mobila trådlösa kortdistanstillämpningar där kanalens påverkan är okänd och förändras över tid. Signalen måste därför beskrivas statistiskt, vilket gör att den inte kan bestämmas exakt, utan måste estimeras. Eftersom telekomsystem arbetar i realtid består hårdvaran i mottagaren av t.ex. en DSP där de statistiska beräkningarna görs. En fixtals DSP har ett bestämt antal bitar och fast binärpunkt, vilket introducerar ett större kvantiseringsbrus jämfört med flyttalsoperationer som har en större noggrannhet.</p><p>Tyngdpunkten på det här arbetet har varit att skapa ett bibliotek av funktioner för att hantera fixtal. En klass har skapats i MATLAB-kod som kan hantera de vanligaste aritmetiska operationerna och lösa minsta-kvadrat-problem.</p><p>MATLAB:s Fixed-Point Toolbox skulle kunna användas för att lösa den här uppgiften men för att ha full kontroll över algoritmerna och fixtalshanteringen behövs ett eget bibliotek av funktioner som är oberoende av MATLAB:s Fixed-Point Toolbox.</p><p>Slutsatsen av simuleringen gjord i detta examensarbete är att resultatet av minsta-kvadrat-metoden är mer beroende av antalet heltalsbitar än antalet binaler.</p> / fixtal, telekommunikation, DSP, MATLAB, Fixed-Point Toolbox, minsta-kvadrat-lösning, flyttal, Householder QR faktorisering, saturering, kvantiseringsbrus

Telemetry system, Optimisation, Sensoric networks, Smart Grid, Internet of Things, Sensors, Information security, Cryptography, Cryptography algorithms, Cryptosystem, Confidentiality, Integrity, Authentication, Data freshness, Non-Repudiation.