Random Variables, Random vectors and Random processes; Second order characterization of stochastic processes, autocorrelation and covariance functions, special processes: Poisson process, Wiener process and White noise process. Stationary processes, types of stationarity: Strict sense stationary and Wide sense stationary processes, Gaussian processes, memory-less processes. Linear Systems with random inputs, Input-output Autocorrelation relations, Input-output Stationarity properties. Ergodicity and related results. Wide sense stationary processes, Autocorrelation function and power spectra. Spectral theory for linear systems. Rational spectra, Hilbert transforms, shot noise, thermal noise. Discrete time processes, Spectral factorization, Matched filters. Integral equations and series representation of stochastic processes: Karhunen Loeve (KL) expansion. Modulation, Band limited processes and sampling theory. Mean square estimation and the orthogonality principle. Linear Prediction and its Geometric interpretation. Levinson Recursion for one step predictors. Mean square estimation and normality. Mean square error for one and multi-step predictors. Smoothing and prediction as applications of orthogonality principle.
Prerequisite: EL 6303.