EE 553
Stochastic Signals

Catalog Description

Random signals, correlation functions, power spectral densities, the Gaussian process, narrow band processes. Applications to communication systems

Credits: 3.0

Class Schedule: 3 lectures/week

Prerequisites: EE 410

Course Objectives 

1. Factual knowledge (such as terminology, definitions, conventions)
2. Conceptual knowledge (showing understanding of relationships, implications, and domain of validity)
3. Procedural knowledge (for deducing, calculating, or determining quantities specific to this subject)
4. Ability to apply the knowledge in engineering context (possibly involving modeling, approximation, or idealization)

 

Textbooks and References 

P.Z. Peebles, Probability, , Random Variables, and Random Signal Principles(McGraw Hill, 2001) 4th Edition.

Course Outline
Major Topics
1.Review of Probability theory

Definition of probability; axiomatic definition, probability space and measure Terms: random, experiment, trial, outcome, sample space, event, field of events,
Calculating event probabilities: counting ( combinatorics) method; identifying
Equally likely set of events; partition of space and disjoint events
Compound events and their probabilities
Union and intersection of events; independent events; joint events
Conditional probability, Bayes’ theorem
Repeated trials; Bernoulli trials; Binomial distribution and Multinomial distribution
Poisson distribution as a limit Binomial

2. Random Variables(RVs):

Definition; conditions for a function to be RV; Discrete and continuous RVs
Distribution and density functions; Probability mass func.(Impulse function)
Common discrete distribution; uniform, binomial, Poisson
Common continuous distribution; uniform, exponential, Gaussian, Rayleigh

3. Operations on One Random Variable

Expectation (mean, average) of RV; Expectation of a function of RV
Moments about origin and central; Variance; engineering interpretation
Transformation of  a continuous RV: monotonic and non-monotonic; discrete RV

4. Multiple RV (or Random Vector)

Joint and marginal distribution and density function and their properties
Conditional distribution and density functions; types of conditioning
Statistical independence; distribution and density functions for sums of RVs
Central Limit theorem

5. Operation on Multiple RVs ( or Random Vector)

Expected values of RVs and function of RVs; joint moments and central moments
Two and N jointly Gaussian RVs, their properties         
Transformation of random vectors; linear transformation of Gaussian RVs

6. Random Process (RPs)

Classifications of RPs; deterministic, stationary, ergodic, time and ensemble averages
Autocorrelation, cross-correlation, and covariance functions; their properties
Gaussian and Poisson RPs

7. RP Description in Spectral Domain

Power spectral density (PSD): definition, properties, bandwidth, determination
Power spectrum and autocorrelation function relationship
Cross-spectral density (CSD): definition, properties, relation to cross-correlation fn.

8. Linear System with Random input

Response of linear system to random excitation; its spectral characteristics
Bandwidth; bandpass, band-limited, and narrow-band process
Optimum linear systems: matched filter for max S/N; Wiener filter for min MSE

9. Markov Processes

Definition, classification, properties, finite and infinite state space
Discrete-time RP (Markov chains), state diagram, transition probability, irreducible
set of stat, recurrent and steady state behavior
Random walk models
Continuous-time Markov process, steady state properties

Prepared by: Dr. Madhu S. Gupta
Date of preparation: 10/23/2008