**No:** EE341

**Title:** DISCRETE-TIME LINEAR SYSTEMS

**Credits:** 5

**Coordinator:** Eve Riskin, Professor, Electrical Engineering

**Goals:** To provide students with the fundamental concepts of
digital signal processing. To study discrete-time signal and system
analysis using time-domain, Fourier and Z-transform techniques. To
build proficiency in signal analysis with Matlab.

**Learning Objectives:**

At the end of this course, students will be able to:

*Describe*discrete-time signals in different domains (time, frequency, and Z) and*map*characteristics in one domain to those in another (e.g. distinguish between high and low frequency components of time signals).*Understand*the implications of different system properties and how to test for them.*Perform*convolutions for arbitrary and closed-form discrete-time signals.*Analyze*LTI systems given different system representations (including input-output equations, impulse response, frequency response and transfer function), and*translate*between these different representations.*Solve*linear difference equations associated with linear digital filters by classical techniques and the Z-Transform method.*Use and understand*standard EE terminology associated with digital filtering and LTI systems (e.g. LPF, HPF, impulse response, step response, etc.)*Implement*filters,*analyze*frequency content of signals and responses of systems,*design*filters and*synthesize*signals using Matlab tools.

**Textbook:**
A. Oppenheim, A. Willsky with S. Hamid, *Signals and Systems (2nd Edition),* Prentice Hall, 1997.

**Reference Texts:** None

**Prerequisites by Topic:**

- Continuous-time linear systems (EE235)
- Complex numbers and signals

**Topics:**

- Chapter 1: Signals and Systems
- Introduction, notation and description of discrete time signals and systems, intro to transformations (Ch 1.0-1.2)
- time axis transformations (1.2, 1.5)
- more on time axis transformations, even and odd signals
- even and odd signals, periodicity (1.2.2, 1.2.3)
- common discrete time signals (1.3)
- unit step and unit impulse (1.4)
- system properties: memory, invertibility, causality, stability, linearity, time-invariance (1.5,1.6)

- Chapter 2: Linear Time-Invariant Systems
- Intro and the convolution Sum (2.0,2.1)
- convolution properties and visualization of algorithm, mathematical convolution, examples (2.1)
- LTI system properties (2.3)
- LTI systems described bydDifference equations (2.4)

- Chapter 3: Fourier Series Representation of Periodic Signals
- Intro to Discrete Time Frequency Analysis
- Motivation of the DFT by sampling theory, begin understanding of the DFS (3.6)
- DFS, start defining DFT (3.6 and Supplemental reading)
- DFT and IDFT Examples, Transform pairs (3.6, 3.7, and Supplemental reading)
- Circular Convolution (Supplemental reading)
- FFT (Supplemental reading)

- Chapter 5: The Discrete-Time Fourier Transform
- Intro to DTFT, Convergence, IDTFT (5.1)
- Periodic Signals (5.2)
- Properties of the DTFT (5.3-5.6)
- Inverse systems, difference equations (5.8)

- Chapter 6: Time and Frequency Characterization of Signals and Systems
- Block Diagrams, Magnitude Response, Phase Response, dB spectrum 6.0-6.2)

- Chapter 7: Sampling
- Introduction, Sampled continuous time signal (pulse train) (7.0-7.1)
- Reconstruction, Aliasing (7.2,7.3)

- Chapter 10: The Z-Transform
- Introduction, Definition of Transform and ROC (10.0-10.2)
- Z transform inverse Problems (10.3)
- Poles and Zeros, ROC Properties (10.5,10.6)
- Difference equations, Z Transform Properties (10.7)

**Course Structure:** The class meets 4 times a week for a 50
minute lecture and each student participates in one laboratory session
that meets for 1 hour a week. There is weekly homework and several
laboratory exercises that must be done in Matlab. There is a midterm
and final exam and possibly additional quizzes depending upon the
instructor.

**Computer Resources:** The course uses MATLAB for the
laboratory exercises and also for checking homework problems. The
recommended platforms are PC workstations in the EE Computer Labs,
preferably the SCC Lab in Sieg 232, but general purpose labs can
also be used (EE1
351, 365, 371). Students can do some of the labs on home computers,
but the student edition of Matlab may not support all functions used
in the lab and array size limits can be problematic. The students
complete an average of 1.5 hours of computer work per week.

**Laboratory Resources:** (see Computer Resources)

**Outcome Coverage:**

(a) *An ability to apply knowledge of mathematics, science, and enigneering.*
The majority of the lectures and homework deal with the derivations and
application of linear mathematics theory to solve difference equations,
perform convolutions and transform signals. (H)

(b) *An ability to design and conduct experiments, as well as to analyze and interpret data. *
The labs include assignments where students experiment with different
signals and systems to learn fundamental DSP concepts, including: i)
exploring different system configurations in the Z-domain to learn about
the associated time-domain and frequency-domain behavior, and ii)
exploring different configurations of FFT parameters. (M)

(c) *An ability to design a system, component or process to meet desired needs within realistic constraints such as economic, environmental, social, political, ethical, health and safety, manufacturability and sustainability. *
The labs include open-ended assignments in signal synthesis and in digital
filter design, with constraints on performance and computation. Social
constraints are posed in a music synthesis lab. (L)

(e) *An ability to identify, formulate and solve engineering problems. *
The homework involves solving signal processing problems
identified by the assignments and exemplified by class discussion. (H)

(h) *The broad education necessary to understand the impact of engineering solutions in a global, economic, environmental and societal context. *
Motivating examples are drawn from a range of applications, including
both traditional EE applications of filtering as well as such topics as
music synthesis, medical signal processing, speech processing, image
compression, etc. (L)

(j) *Knowledge of contemporary issues.*
Applications of analysis tools are used to explain fundamental aspects
of communication theory, image processing and signal processing.
Motivating examples are drawn from applications such as digital music
and sound effects, image processing and compression, and consumer
electronics. (L)

(k) *An ability to use the techniques, skills and modern engineering tools necessary for engineering practice. *
Students use Matlab and associated data acquisition/display tools to
solve homework problems on signal analysis and filter design. (H)

**Prepared By:** Eve Riskin, Laura Vertatschitsch

**Last Revised: **12/8/12