Basic
Probability: 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
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 and density, 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, cross-correlation, covariance, Cauchy-Schwartz
inequality, conditional expectation, characteristic function, cental limit
theorem, transformations of single and multiple random variables, random
vectors, properties of Gaussian random vectors.
Random processes: Definition, stationarity, mean, correlation and
covariance, wide-sense stationary random processes, examples of random
processes, cross-correlation functions, joint wide-sense stationarity, time
averages and ergodicity, measuremen of mean and autocorrelation function,
transmission of random process through a linear filter - relationship between
input and output processes, power spectral density (PSD) - definition and
proporties, examples, relationship between input and output process PSD for a
linear filter, periodograms, cross spectral densities, Gaussian process -
properties, white noise, noise equivalent bandwidth, narrowband noise, bandpass
processes - representation, sampling.
Other topics (some of these will be covered depending on time available): Cyclostationary
random processes, PAM signals, Baseband shaping (raised cosine), optimum
transmitting and receiving filters for noise immunity, matched filtering,
sampling and expansion of random processes.
