## Random Processes

(3-0-0-3)

**CMPE Degree:** This course is Not Applicable for the CMPE degree.

**EE Degree:** This course is Not Applicable for the EE degree.

**Lab Hours:** 0 supervised lab hours and 0 unsupervised lab hours.

**Technical Interest Group(s) / Course Type(s):** Telecommunications

**Course Coordinator:** Mary Ann Weitnauer

**Prerequisites:** ECE 3077

**Corequisites:** None.

### Catalog Description

To develop the theoretical framework for the processing of random signalsand data.

### Course Outcomes

Not Applicable

### Student Outcomes

In the parentheses for each Student Outcome:"P" for primary indicates the outcome is a major focus of the entire course.

“M” for moderate indicates the outcome is the focus of at least one component of the course, but not majority of course material.

“LN” for “little to none” indicates that the course does not contribute significantly to this outcome.

1. ( Not Applicable ) An ability to identify, formulate, and solve complex engineering problems by applying principles of engineering, science, and mathematics

2. ( Not Applicable ) An ability to apply engineering design to produce solutions that meet specified needs with consideration of public health, safety, and welfare, as well as global, cultural, social, environmental, and economic factors

3. ( Not Applicable ) An ability to communicate effectively with a range of audiences

4. ( Not Applicable ) An ability to recognize ethical and professional responsibilities in engineering situations and make informed judgments, which must consider the impact of engineering solutions in global, economic, environmental, and societal contexts

5. ( Not Applicable ) An ability to function effectively on a team whose members together provide leadership, create a collaborative and inclusive environment, establish goals, plan tasks, and meet objectives

6. ( Not Applicable ) An ability to develop and conduct appropriate experimentation, analyze and interpret data, and use engineering judgment to draw conclusions

7. ( Not Applicable ) An ability to acquire and apply new knowledge as needed, using appropriate learning strategies.

### Strategic Performance Indicators (SPIs)

Outcome 1 (Students will demonstrate expertise in a subfield of study chosen from the fields of electrical engineering or computer engineering):

1. Characterize a random process in terms of its autocorrelation function, distribution, and in the case of wide sense stationary processes, the power spectral density.

Outcome 2 (Students will demonstrate the ability to identify and formulate advanced problems and apply knowledge of mathematics and science to solve those problems):

1. Determine the dependence or independence of random variables, based on the properties of the joint probability density function.

2. Apply the properties of minimum mean squared error estimation.

3. Apply the properties of Markov Chains.

Outcome 3 (Students will demonstrate the ability to utilize current knowledge, technology, or techniques within their chosen subfield):

### Course Objectives

### Topical Outline

1. Review of Probability and Random Variables

a. Axioms and Properties of Probability

b. Conditional Probability, Independence

c. Random Variables, Density Functions, Expectation

d. Moments, Normal (Gaussian) Random Variables

2. Two Random Variables

a. Joint Density and Computation of Probability

b. Independence, Correlation

c. Linear Mean Square Estimation

3. Random Sequences

a. Conditional Densities, Chapman-Kolmogorov Equation

b. Normal Sequences, Sample Mean

i. Markov and Chebychev Inequalities

ii. Convergence of Sequences, Laws of Large Numbers, Central Limit

4. Random Processes

a. Definition, Mean, Autocorrelation, Autocovariance

b. Examples: Random Phase Sinusoid, Poisson Process, Telegraph Signal,

i. Random Walk, Wiener Process

5. Stationarity

a. Strict Sense, Wide Sense, Stationary Increments, Cyclostationarity

b. Properties of Auto- and Cross-correlation Functions

6. Power Spectral Density

a. Definition, Relation to Fourier Transform

b. Discrete-Time vs Continuous-Time

c. White Noise, Spectral Estimation

7. Response of Linear Systems to Random Inputs

a. Time Doman Analysis

b. Mean and Autocorrelation of Output, Crosscorrelation of Input with Output

c. Frequency Domain Analysis

d. Bandpass Signals and Filters

e. Shot Noise, ARMA Models

8. Ergodicity

a. Mean Ergodicity, Generally and for Wide Sense Stationary RP's

b. Correlation and Distribution Ergodicity

9. Expansions of Random Processes

a. Sampling

b. Karhunen-Loeve

10. Markov Processes

a. General Definition

b. Poisson Revisited

c. Queues

d. Discrete-Time, Discrete-State; Homogeneity, Reducibility, Recurrence

e. Continuous-Time, Discrete-State; Diffusion Equations

11. Simulation of Random Processes

12. Mean Square Estimation

a. Orthogonality Principle for N Observations, Whitening

b. Linear and Nonlinear Estimation

c. Continuous-Time Observations, Wiener Filter