MVE135 Random Processes with Applications, 7.5 credit points, 2007/2008AimThe purpose of the course is to provide the students with the theoretical framework fundamental to the processing of signals with random variation. Starting from basic probability the course proceeds to a thorough study of models for stochastic processes which are relevant in processing of random signals, and gives techniques for manipulating and study of these signals. Practical methods for random signal analysis and filtering are also included. The level should be such that the student should be able to take an active part in designing and optimizing engineering systems involving random signals. Learning outcomes  define and explain fundamental probability tools used in the design and analysis of communication systems, with emphases on multidimensional joint distributions, the Gaussian one in particular, conditional expectation and conditional variance, convergence of random variables and limit theorems for sums of independent and identically distributed random variables describe the basic statistical principles involved in point and interval parameter estimation as well as in hypotheses testing  identify basic models of random processes and explain their use for the designing of components in communication systems and analysis of their effect on system performance. These models include the Poisson process, Markov processes, Gaussian processes, white noise, and stationary stochastic processes  use widesense stationary processes for modeling systems involving random signals and noise. In particular the students should have got a firm grasp on the important class of ARMA processes  estimate secondorder characteristics from data, including nonparametric estimation of the power spectral density, and understand the statistical properties of these estimates  explain the mathematical techniques for design of optimal
linear systems for signal processing, with emphasis on match filtering
and
the Wiener filter. Organization Course Book Course Content Basic Probability Theory: Review and Extension. Axioms of Probability. Conditional Probability. Independence of Events. Probability Distributions. Expectation and Variance. Random Variables. Functions of Random Variables.Multiple Random Variables. Conditional Distributions. Conditional Expectation and Conditional Variance. Multidimensional Gaussian Distribution. Convergence of Random Variables. Limit Theorems for Sums of Random Variables. Mathematical Statistics Parameter Estimation. Maximum Likelihood. Confidence Intervals. Hypothesis Testing. LikelihoodRatio tests. Random Processes with Application in Statistical Signal Processing Definition of a Random Process. Autocorrelation Functions. Some Special Random Processes: Random Walk Process, Poisson Process, Random Telegraph Signal, Wiener process, White Gaussian Noise. Continuity, Derivatives, and Integrals of Random Processes. WideSense Stationary Random Processes. Spectral Representation. Autoregressive Moving Average Processes. Analysis and Processing of Random Signals Through a Linear System. CrossCorrelation and CrossSpectrum. Statistical Signal Processing NonParametric Spectrum Estimation. Windowing and Frequency Resolution. Welch and BlackmanTukey Methods. Optimum Linear Systems. Prediction, Filtering and Smoothing. Wiener Filter. Introduction to Markov Processes Transition and State Probabilities. Characterization of Markov Chains. ContinuousTime Markov Chains.

MVE135 Random Processes with Applications, 7,5 credit points
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