EE 120: SIGNALS AND SYSTEMS
Department:
Electrical Engineering and Computer Sciences
Instructor:
Dr. Babak Ayazifar (EECS Faculty)
Credit Units:
Prerequisites:
4
EE 20N (Introductory Signals and Systems)
Math 53 (Multivariate Calculus)
Math 54 (Linear Algebra and Differential Equations)
Course Structure:
 Lecture hours per week (Lec):
 Discussion (recitation) section
hours per week (D):
 Instructor office hours, focused on groupbased problem solving (OH):
4
1 (not mandatory)
2 (not mandatory)
Course Components:
 Problem Sets/Homework (HW)
 Pop quizzes (Q)
 Exams (E)
Textbooks:
 Required:
o E. A. Lee and P. Varaiya, Structure and Interpretation of Signals
and Systems, Addison-Wesley, 2003.
 Review of:
 Appendix A (sets and functions)
 Appendix B (complex numbers)
 Chapter 1 (Signals and Systems)
 Chapter 2 (Defining Signals and Systems)
 Extended coverage of:
 Chapter 7 (Frequency Domain)
 Chapter 8 (Frequency Response)
 Chapter 9 (Filtering)
 Chapter 10 (The Four Fourier Transforms)
 In-depth first-time coverage of:
 Chapter 11 (Sampling and Reconstruction)
 Chapter 12 (Stability)
 Chapter 13 (Laplace and Z Transforms)
 Chapter 14 (Composition and Feedback Control)
 Miscellaneous:
 Chapter 5 (Linear Systems/State-Space Analysis)
 Recommended:
o A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and
Systems, Prentice-Hall, 1997.
o H. P. Hsu, Schaum’s Outline of Signals and Systems, McGraw-Hill,
1995.
o S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications
to Physics, Biology, Chemistry, and Engineering, Addison-Wesley,
1994.
RELATIONSHIP TO ABET PROGRAM OUTCOMES:
Students who complete this course successfully meet the following ABET
program outcomes: (a), (b), (c), (e), (g), (i), (k).
EE 120 continues the tradition of its lower-division prerequisite course EE 20N by
requiring students to apply a fundamental knowledge of mathematics, science
and engineering to not only solve electrical and computer engineering problems,
but also to recognize the multidisciplinary reach of the topics and techniques
emphasized in the course.
Students learn modern skills, techniques and engineering tools. They refine their
skills in back-of-the-envelope and design-oriented analysis. They learn to model
signals and systems, with an eye toward design.
Problem sets and exams are designed to probe a thorough understanding of
fundamental concepts and to de-emphasize rote algebraic manipulation.
Exam problems insist on refined and to-the-point responses; limited space is
allocated for each problem to encourage students to think more clearly and
logically, and to articulate their responses efficiently and without clutter. The
various components of the course encourage students to think graphically and to
draw, before they reach for mathematical formulae.
Lectures, discussion sections, and instructor’s office hours promote group work
by engaging the students in a collaborative learning environment where they
divide into groups of 3-5 students and discuss the solutions to various problems.
In discussion sections and office hours, students present their solutions to their
peers on the board; they generate peer discussion by asking each other
questions and assisting each other toward solutions.
Recognizing that team work is an integral part of engineering practice, the
homework policy encourages collaborative groups of up to five students.
Lectures also cultivate group learning. For almost every example problem the
students encounter in lecture, they take time to discuss the solution among
themselves, explaining concepts and learning about alternative solutions from
their peers.
COURSE LEARNING OBJECTIVES AND OUTCOMES:
This course trains students for an intermediate level of fluency with signals and
systems in both continuous time and discrete time, in preparation for more
advanced subjects in digital signal processing (including audio, image and video
processing), communication theory, and system theory, control, and robotics.
Upon successful completion, a student should:

Classify systems based on their properties: in particular, understand and
exploit the implications of linearity, time-invariance, causality, memory,
and bounded-input, bounded-out (BIBO) stability.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Excellent

Know the principles of vector spaces, including how to relate the concepts
of basis, dimension, inner product, and norm to signals.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Learn to treat signals as vectors in a vector space and ascribe geometry
to that space by defining an appropriate inner product—in both discretetime and continuous-time, and for both periodic and aperiodic signals. For
example, treat the set of periodic discrete-time signals having period p as
a vector space, and define an inner product for that space.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Excellent

Know how to analyze, design, approximate, and manipulate signals using
vector-space concepts. For example,
o know how to project a signal onto another signal, and exploit signal
orthogonality to develop
 Fourier series expansions of periodic discrete-time and
continuous-time signals—that is, decompose a signal in
terms of complex exponentials in the frequency domain;
 know how to interpret and plot the Fourier series coefficients
of a signal and understand what they mean in terms of the
frequency content of the signal;
 find a best approximation for a signal using a strict subset of
the basis signals in the appropriate vector space—that is,
approximate the signal when the number of basis signals is
smaller than the dimensionality of the signal space.

understand that a signal’s energy is the 2-norm of a signal,
that the 2-norm is defined according to an appropriate inner
product, and that signal energy can be expressed in the time
and frequency domains via Parseval’s relation.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Determine Fourier transforms for continuous-time and discrete-time
signals (or impulse-response functions), and understand how to interpret
and plot Fourier transform magnitude and phase functions.
o Understand the discontinuous nature of the Fourier transform of a
signal that is not absolutely summable or integrable.
o Understand the impulsive nature of the Fourier transform of a
periodic signal, and how the Fourier series coefficients relate to the
Fourier transform of the signal.
o Know how to draw the frequency-domain magnitude and phase
plots using geometric reasoning.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Excellent

Understand the sampling theorem and how it links continuous-time signals
to discrete-time signals. In particular, know
o how to derive the sampling theorem from first principles—from the
basic properties of the Fourier transform;
o how the spectrum of a sampled signal relates to the spectrum of
the original signal;
o how to use the sampling theorem to understand aliasing
phenomena in the real-world (e.g., the carriage wheel effect), and
how to reduce or prevent aliasing; and
o how to perform discrete-time processing of continuous-time signals,
and vice versa, using C/D and D/C converters.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Understand the need to define two new transforms—the Laplace and Z
transforms—to treat a class of signals broader than what the Fourier
transform can handle.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Excellent

Understand the combined implications of linearity and time invariance in
the Laplace and Z transform domains. In particular,
o know how to represent the response of an LTI system to a more
general form of complex exponential—est in continuous time or zn in
discrete-time—and understand that complex exponentials are
eigenfunctions of LTI systems;
o use the Laplace transform to determine the transfer function of a
continuous-time LTI system; solve for a response given the input,
system description and initial conditions; and answer questions
related to BIBO stability, including the central role of the i-axis in
the transform-domain representations of continuous-time signals
and systems;
o use the Z transform to determine the transfer function of a discretetime LTI system; solve for a response given the input, system
description and initial conditions; and answer questions related to
BIBO stability, including the central role of the unit circle in the
transform-domain representations of discrete-time signals and
systems;
o represent an LTI system by its transfer function.
 Understand the implications of causality, BIBO stability,
invertibility and other system properties on the region of
convergence of the transform.
 Understand the relationship between the locations of the
poles and zeros with system properties such as causality,
BIBO stability, invertibility, and frequency response.
o determine the input-output behavior of an LTI system entirely in the
transform domain, using relationships between time-domain and
the frequency-domain (e.g., convolution in the time domain
corresponds to multiplication in the frequency domain);
o understand the conditions under which the transfer function of a
system (or the Laplace or Z transform of a signal) is rational, and
know that
 a continuous-time LTI system with a rational transfer function
can be represented by a linear, constant-coefficient
differential equation; and
 a discrete-time LTI system with a rational transfer function
can be represented by a linear, constant-coefficient
difference equation.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Excellent

Understand the relationships among the various representations of LTI
systems—linear constant-coefficient difference or differential equation,
frequency response, transfer function, and impulse response—and infer
one representation from another (e.g., determine the impulse response
from the difference equation, etc.)
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Understand the conditions for a time-domain function to have a Fourier
transform, and know how to relate the Fourier transform to its Laplace or Z
transform:
o Evaluate the Laplace transform on the i-axis for continuous-time
functions;
 Learn how to draw Bode magnitude and phase diagrams.
o Evaluate the Z transform on the unit circle for discrete-time
functions;
o Infer the properties of the Fourier transform magnitude and phase
plots from the locations of the poles and zeros;
o Design low-pass, band-pass, high-pass, notch, anti-notch, and allpass filters by appropriate distribution of poles and zeros on the
complex plane.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Excellent

Understand the various properties of the four Fourier transforms, the
Laplace transform, and the Z transform—including time-shift, modulation
(frequency shift), duality, symmetry and anti-symmetry—and exploit them
to analyze and design signals and systems.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good.

Understand the properties, as well the analysis and design implications, of
interconnections of LTI systems—parallel, series (cascade), and
feedback—in the time and transform domains.
o Know the effect of poles and zeros, as well as associated system
properties such as causality and BIBO stability, on the existence
and qualitative nature of steady-state behavior;
o Know how to use feedback to move system poles to desired
locations to extract specified behaviors from filters.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Know how to derive and exploit basic concepts in communication theory,
including amplitude modulation and frequency modulation.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Understand how to use the unilateral Laplace or Z transform to
o decompose the response of an LTI system into a zero-state
component and a zero-input component.
o solve linear, constant-coefficient differential or difference equations,
with possibly non-zero initial conditions.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Develop reasonably-accurate mathematical models for physical systems,
find LTI approximations to the models, produce block-diagram
implementations of the mathematical models, and analyze the block
diagram realizations with a view toward designing more complex systems
or more sophisticated models.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Very Good

Learn to develop and analyze state-space models of linear and nonlinear
systems. This includes
o drawing qualitative plots of state trajectories;
o determining internal stability including the stability of equilibrium
points;
o determining the modes of LTI systems, especially second-order
systems, by performing eigenanalysis of the state transition matrix;
o developing an aptitude for modeling a multidisciplinary array of
systems in state-space form.
Coverage: Lec, D, HW, Q, E, OH
Student Success: Good
TOPICS COVERED:
 Signals and Systems
o Linearity, causality, BIBO stability, time invariance, memory,
invertibility
 Linear Time-Invariant Systems
o Convolution integral and convolution summation
o Impulse response, frequency response
o Differential equations, homogeneous and particular solutions
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o Difference equations
Fourier Series (vector-space interpretation)
o Discrete-time Fourier series
o Continuous-time Fourier series including convergence issues
Fourier Transform
o Discrete-time Fourier transform (vector-space interpretation)
o Continuous-time Fourier transform
o Relation of the four Fourier series/transforms
Sampling
o Sampling theorem and the Nyquist rate
o Continuous-time-to-discrete-time (C/D) or analog-to-digital (A/D)
conversion
o Aliasing
o Upsampling and downsampling
o Discrete-time-to-continuous-time (C/D) or digital-to-analog (D/A)
conversion, zero-order hold and first-order hold
o Continuous-time processing of discrete-time signals and discretetime processing of continuous-time signals
o Bandpass sampling
Communication
o Pulse amplitude modulation, Nyquist pulses, synchronization
o Frequency modulation, narrowband approximation
o Discrete tone modulation
Control
o Laplace Transform, region of convergence
o Feedback systems, pole-zero plots, stability, root locus
o Geometric evaluation of Fourier transform
o Bode Plots
Z Transform
o Two-sided z-transform, region of convergence, relation of the Z
transform to the discrete-time Fourier transform
o BIBO Stability of discrete-time systems
o Unilateral Z transform, application to solving difference equations
Filter Design
o Analog prototype, Bilinear transform
o Stability, causality, selection of poles and zeros
o Using geometric reasoning to design filters.
State-Space Modeling and Analysis of Linear and Nonlinear Systems
Prepared by:
Babak Ayazifar
Date:
8 February 2009