**Instructor**

B. Srikrishna

Office: ESB 337A

Phone: 2257 4439

**Course Content**

**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 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 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.

**Evaluation**

Quiz I: September 14, 2007

Quiz II: October 19, 2007

Final exam: November 28, 2007

**References**

[1] A. Papoulis, S. U. Pillai, "Probability, Random Variables and Stochastic Processes", 4th edition, McGraw-Hill, 2002.

[2] R. M. Gray, L. D. Davisson, "An Introduction to Statistical Signal
Processing," Cambridge University Press, 2004. http://ee.stanford.edu/~gray/sp.html

[3] H. Stark and J. W. Woods, "Probability and Random Processes with Applications to Signal Processing", 3rd edition, Pearson Education, 2002.

[4] Y. Viniotis, "Probability and Random Processes for Electrical Engineers", McGraw-Hill, 1998.

[5] S. Haykin, "Communication Systems", 4th edition, John Wiley & Sons, 2001.

[6] J. G. Proakis and M. Salehi, "Communication Systems Engineering", 2nd
edition, Pearson Education, 2002.

[7] G. Grimmett, D. Stirzaker, "Probability and Random Processes," 3rd edition, Oxford Univ. Press, 2001.

**Problem sets**

Problem Set 1
Solution to Problem Set 1

Problem Set 2
Solution to Problem Set 2

Problem Set 3
Solution to Problem Set 3

Problem Set 4
Solution to Problem Set 4

Problem Set 5
Solution to Problem Set 5

Problem Set 6
Solution to Problem Set 6

Problem Set 7
Solution to Problem Set 7

Problem Set 8
Solution to Problem Set 8