Course Content | Various definitions of probability, axioms of probability, basic properties derived from the axioms, conditional probability, total probability, Bayes’ rule, Independence of events, combined experiments and independence, binary communication channel example (MAP and ML decoding).Random variables: Definition, cumulative distribution function (cdf), continuous, discrete and mixed random variables, probability density function (pdf), examples of random variables, physical interpretation of pdf’s (histograms), multiple random variables, joint distribution – definition and properties, joint density – definition and properties, marginal distribution anddensity, conditional distribution and density, independence of random variables, expectations, moments, central moments, properties of expectation operator, mean, variance, Markov inequality, Chebyshev inequality, Chernoff bound, effect of linear transformations on mean and variance, autocorrelation, crosscorrelation, covariance, Cauchy-Schwartzinequality, conditional expectation, characteristic function, central limittheorem, transformations of single and multiple random variables, random vectors, properties of Gaussian random vectors.Random processes: Definition, stationarity, mean, correlation and covariance, wide-sensestationary random processes, examples of random processes, cross-correlation functions, joint wide-sense stationarity, time averages and ergodicity, measurement of mean and autocorrelation function, transmission of random process through a linear filterrelationship between input and output processes, power spectral density (PSD) – definition and properties, examples, relationship between input and output processes PSD for a linearfilter, periodograms, cross spectral densities, Gaussian process – properties, white noise, noise equivalent bandwidth, narrowband noise, bandpass processes – representation, sampling. |