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Über dieses Buch

This book is concerned with the processing of signals that have been sam­ pled and digitized. The fundamental theory behind Digital Signal Process­ ing has been in existence for decades and has extensive applications to the fields of speech and data communications, biomedical engineering, acous­ tics, sonar, radar, seismology, oil exploration, instrumentation and audio signal processing to name but a few [87]. The term "Digital Signal Processing", in its broadest sense, could apply to any operation carried out on a finite set of measurements for whatever purpose. A book on signal processing would usually contain detailed de­ scriptions of the standard mathematical machinery often used to describe signals. It would also motivate an approach to real world problems based on concepts and results developed in linear systems theory, that make use of some rather interesting properties of the time and frequency domain representations of signals. While this book assumes some familiarity with traditional methods the emphasis is altogether quite different. The aim is to describe general methods for carrying out optimal signal processing.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
This book is concerned with the processing of signals that have been sampled and digitized. The fundamental theory behind Digital Signal Processing has been in existence for decades and has extensive applications to the fields of speech and data communications, biomedical engineering, acoustics, sonar, radar, seismology, oil exploration, instrumentation and audio signal processing to name but a few [87].
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

2. Probabilistic Inference in Signal Processing

Abstract
In this chapter, the fundamental concepts and techniques underlying Bayesian inference are reviewed. We begin with a definition of the key problem of data analysis, which is to interpret data in the presence of noise. We advocate a Bayesian approach to this problem.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

3. Numerical Bayesian Inference

Abstract
The aim of this chapter is to provide an extensive survey of numerical Bayesian methods. The emphasis is placed primarily on the numerical integration of posterior densities to obtain approximations to marginal densities and Bayesian evidence.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

4. Markov Chain Monte Carlo Methods

Abstract
Sampling methods based on Markov chains were first developed for applications in statistical physics. Two branches of development originated in the 1950s. The classic paper by Metropolis et al [77] introduced what is now known as the Metropolis algorithm. This method was popularized for Bayesian applications, along with its variant the Gibbs sampler, by the influential papers of Geman and Geman [36], who applied it to image processing, and Gelfand and Smith [35], who demonstrated its application to Bayesian problems in general.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

5. Retrospective Changepoint Detection

Abstract
The problem of detecting and estimating the location of changepoints (or discontinuities) in data is fundamental to many areas of data analysis. Practical applications abound in diverse areas such as medicine (e.g. monitoring of drug levels in hospital patients), the detection of kickback in oil well pressure data [140] and edge detection in images [123]. In this chapter, optimal Bayesian techniques are developed for changepoint identification in one dimensional (time series) data. These use the probability density function (pdf) of the changepoint positions to estimate their positions in time series.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

6. Restoration of Missing Samples in Digital Audio Signals

Abstract
The aim of this chapter is to describe a novel method for interpolating autoregressive data. This is applied to the restoration of missing samples in digital audio signals. The section of audio signal in question is modelled as a stationary autoregressive process, and missing samples are imputed using the Gibbs sampler. The corresponding ML and EM algorithm solutions to the problem are developed and discussed, and the results are compared for both real and synthetic data.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

7. Integration in Bayesian Data Analysis

Abstract
The aim of this chapter is to demonstrate usage of the numerical methods developed in chapters 3 and 4 for the Bayesian analysis of data. The chapter begins with an example for which the Bayesian evidence (or rather the integrated likelihood) and marginal densities may be computed in closed form. These numerical techniques are then applied to a difficult problem in data analysis; namely, inferring the number of decaying exponentials and the values of the decays in real experimental data. The chapter concludes with examples of model selection by determining the appropriate noise model and signal model to use in a given data set.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

8. Conclusion

Abstract
This book was concerned with devising numerical methods for practical applications of Bayesian methods to signal processing. There were three components in this work. First, we considered optimisation for the location of the posterior mode. Second, we investigated different strategies for integrating the posterior density. Third, we attempted to simulate random samples from the posterior density.
Joseph J. K. Ó Ruanaidh, William J. Fitzgerald

Backmatter

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