Random Signals and Applications

(3-0-0-3)

CMPE Degree: This course is Selected Elective for the CMPE degree.

EE Degree: This course is Selected Elective for the EE degree.

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

Technical Interest Groups / Course Categories: Threads / ECE Electives

Course Coordinator: Xiaoli Ma

Prerequisites: ECE 2026 [min C] and (ECE 3077 [min D] or CEE 3770 [min D] or ISYE 3770 [min D] or MATH 3670 [min D] or MATH 3770 [min D])

Catalog Description

Introduction to random signals and processes with emphasis on applications in ECE. Includes basic estimation theory, linear prediction, and statistical modeling.

Textbook(s)

Probability, Statistics, and Random Processes for Engineers

Course Outcomes

Analyze random vectors,  their joint statistics, and functions of random vectors; 

Describe random vector properties using linear algebra

Optimally estimate random vectors given sets of observations

Decorrelate random vectors through transformations

Analyze random waveforms, both discrete and continuous observations, using multidimensional probability theory. 

Describe process characteristics in terms of ergodicity and various levels of stationarity (contiuous and discrete). 

Compute autocorrelation, autocovariance, and cross correlation functions for both non-stationary and stationary processes (continuous and discrete).

Analyze random signals put through linear filters both in time and frequency (contiuous and discrete). 

Model processes in terms of white noise applied to filters (continuous and discrete). 

Analyze random walks and Wiener processes.

Design linear predictors for autoregressive systems  (discrete-time only). 

Analyze short-time stationary processes along with time-dependent ACFs, CCFs, and PSDs (contiuous and discrete).

Design optimal causal and non-causal Wiener filters. 

Apply state-space descriptions to linear systems and random signals to employ Kalman filters  (discrete-time only).

Employ Markov chains to model evolving process properties.

Strategic Performance Indicators (SPIs)

N/A

Topic List

  1. Random Vectors
    1. joint distributions and transformation of random vectors
    2. mean vector and covariance matrix
    3. Gaussian random vectors
    4. estimating the mean vector and covariance matrix
    5. linear estimation and least-squares
    6. minimum mean-square error estimation
  2. Discrete-time random signals
    1. Bernoulli trials and random walks
    2. random sequences and discrete-time linear systems
    3. wide-sense stationary sequences and the power spectral density
    4. Markov processes
    5. hidden Markov models
  3. Introduction to statistical DSP
    1. discrete-time linear prediction
    2. the Wiener filter
    3. sequences of random vectors, state evolution and the Kalman filter
  4. Continuous-time random signals
    1. Poisson processes
    2. digital modulation
    3. Brownian motion
    4. Markov processes
    5. wide-sense stationary processes, the autocorrelation function, and the power spectral density
    6. continuous-time systems with random inputs
  5. Further topics
    1. graphical models
    2. Bayesian inference
    3. the expectation-maximization algorithm

Applications will be discussed alongside of general mathematical techniques.  Applications will include, but not be limited to, speech processing, tracking, modulation and detection for digital communications, radar, sigma-delta quantization, and financial modeling.