Stochastic signal processing plays a central role in telecommunication and information processing systems, and has a wide range of applications in speech technology, audio signal processing, channel equalisation, radar signal processing, pattern analysis, data forecasting, decision making systems etc. The theory and application of signal processing is concerned with the identification, modelling, and utilisation of patterns and structures in a signal process. The observation signals are often distorted, incomplete and noisy. Hence, noise reduction and the removal of channel distortions is an important part of a signal processing system. The aim of this book is to provide a coherent and structured presentation of the theory and applications of stochastic signal processing and noise reduction methods. This book is organised in fourteen chapters. Chapter 1 begins with an introduction to signal processing, and provides a brief review of the signal processing methodologies and applications. The basic operations of sampling and quantisation are reviewed in this chapter. Chapter 2 provides an introduction to the theory and applications of stochastic signal processing. The chapter begins with an introduction to random signals, stochastic processes, probabilistic models and statistical measures. The concepts of stationary, non-stationary and ergodic processes are introduced in this chapter, and some important classes of random processes such as Gaussian, mixture Gaussian, Markov chains, and Poisson processes are considered. The effects of transformation of a signal on its distribution are considered.

Table of Contents:
1 Introduction.- 1.1 Signals and Information.- 1.2 Signal Processing Methods.- 1.3 Applications of Digital Signal Processing.- 1.4 Sampling and Analog to Digital Conversion.- 2 Stochastic Processes.- 2.1 Random Signals and Stochastic Processes.- 2.2 Probabilistic Models of a Random Process.- 2.3 Stationary and Nonstationary Random Processes.- 2.4 Expected Values of a Stochastic Process.- 2.5 Some Useful Classes of Random Processes.- 2.6 Transformation of a Random Process.- Summary.- 3 Bayesian Estimation and Classification.- 3.1 Estimation Theory: Basic Definitions.- 3.2 Bayesian Estimation.- 3.3 Estimate-Maximise (EM) Method.- 3.4 Cramer-Rao Bound on the Minimum Estimator Variance.- 3.5 Bayesian Classification.- 3.6 Modelling the Space of a Random Signal.- Summary.- 4 Hidden Markov Models.- 4.1 Statistical Models for Nonstationary Processes.- 4.2 Hidden Markov Models.- 4.3 Training Hidden Markov Models.- 4.4 Decoding of Signals Using Hidden Markov Models.- 4.5 HMM-based Estimation of Signals in Noise.- Summary.- 5 Wiener Filters.- 5.1 Wiener Filters: Least Squared Error Estimation.- 5.2 Block-data Formulation of the Wiener Filter.- 5.3 Vector Space Interpretation of Wiener Filters.- 5.4 Analysis of the Least Mean Squared Error Signal.- 5.5 Formulation of Wiener Filter in Frequency Domain.- 5.6 Some Applications of Wiener Filters.- Summary.- 6 Kalman and Adaptive Least Squared Error Filters.- 6.1 State-space Kalman Filters.- 6.2 Sample Adaptive Filters.- 6.3 Recursive Least Squares (RLS) Adaptive Filters.- 6.4 The Steepest Descent Method.- 6.5 The LMS Adaptation Method.- Summary.- 7 Linear Prediction Models.- 7.1 Linear Prediction Coding.- 7.2 Forward, Backward and Lattice Predictors.- 7.3 Short-term and Long-term Predictors.- 7.4 MAP Estimation of Predictor Coefficients.- 7.5 Signal Restoration Using Linear Prediction Models.- Summary.- 8 Power Spectrum Estimation.- 8.1 Fourier Transform, Power Spectrum and Correlation.- 8.2 Non-parametric Power Spectrum Estimation.- 8.3 Model-based Power Spectrum Estimation.- 8.4 High Resolution Spectral Estimation Based on Subspace Eigen Analysis.- Summary.- 9 Spectral Subtraction.- 9.1 Spectral Subtraction.- 9.2 Processing Distortions.- 9.3 Non-linear Spectral Subtraction.- 9.4 Implementation of Spectral Subtraction.- Summary.- 10 Interpolation.- 10.1 Introduction.- 10.2 Polynomial Interpolation.- 10.3 Statistical Interpolation.- Summary.- 11 Impulsive Noise.- 11.1 Impulsive Noise.- 11.2 Stochastic Models for Impulsive Noise.- 11.3 Median Filters.- 11.4 Impulsive Noise Removal Using Linear Prediction Models.- 11.5 Robust Parameter Estimation.- 11.6 Restoration of Archived Gramophone Records.- Summary.- 12 Transient Noise.- 12.1 Transient Noise Waveforms.- 12.2 Transient Noise Pulse Models.- 12.3 Detection of Noise Pulses.- 12.4 Removal of Noise Pulse Distortions.- Summary.- 13 Echo Cancellation.- 13.1 Telephone Line Echoes.- 13.2 Adaptive Echo Cancellation.- 13.3 Acoustic Feedback Coupling.- 13.4 Sub-band Acoustic Echo Cancellation.- Summary.- 14 Blind Deconvolution and Channel Equalisation.- 14.1 Introduction.- 14.2 Blind Equalisation Using Channel Input Power Spectrum.- 14.3 Equalisation Based on Linear Prediction Models.- 14.4 Bayesian Blind Deconvolution and Equalisation.- 14.5 Blind Equalisation for Digital Communication Channels.- 14.6 Equalisation Based on Higher-Order Statistics.- Summary.- Frequently used Symbols and Abbreviations.