EE 265.3

Discrete-Time Signals and Systems Department of Electrical and Computer Engineering

Fall 2014

Description: This course introduces the fundamental concepts and techniques for modeling and analysis of

discrete-time signals and linear systems. Topics include sinusoids and complex exponential

representation, Fourier series, sampling and reconstruction of continuous-time signals, discrete-

time representation of signals and systems, linear time-invariant (LTI) systems, finite impulse

response (FIR) filters, and frequency response of FIR filters. MATLAB is introduced and used

in all simulation-based laboratories that explore analysis tools and their applications. (Official

Description from the Course and Program Catalogue)

Prerequisites: MATH123.3 Calculus I for Engineers

MATH124.3 Calculus II for Engineers

Students are expected to be able to communicate in spoken and written English.

Prerequisite or

Corequisite:

CMPT 116 Computing I (for Engineers)

Instructor: Brian Daku, Ph.D., P.Eng.

Professor and Department Head, Electrical and Computer Engineering (ECE)

Office: Room 3B48.2E (Engineering Building)

Phone: (306) 966-5421

Email: brian.daku@usask.ca

Class Sessions: Tuesday - Thursday, 1:00–2:20PM, Room 2C44E

Laboratory: Section L01: Tuesday, 11:30AM–12:50PM, Room 2B04E (Delta Lab)

Section L03: Thursday, 2:30–3:50PM, Room 2B04E (Delta Lab)

Website: The course website is on the ECE edX server at edx.engr.usask.ca . You will need to register an

account and then register in EE265. This can be done in one step by selecting the link EE265

Discrete-Time Signals and Systems on the edx.engr.usask.ca site. Then select the button

Register for EE265. Fill in the form that appears. Make sure you use your NSID for your Public

Username and your University email in the form “NSID”@mail.usask.ca for your email

address. You need to accurately fill in this form to receive credit for the online marked

activities. Note that any registration without a valid name, NSID email and associated

NSID public username will be removed.

Course Reference

Numbers (CRNs):

89349 (lectures), 89350 and 89352 (laboratory)

Textbook: No Required Textbook

Recommended Reference Textbook: James H. McClellan, Ronald W. Schafer and Mark A.

Yoder, “Signal Processing First”, Pearson Prentice Hall, 2003.

Office Hours: Students are welcome and encouraged to drop by my office at any time for help with the course

material. Though, as Department Head my schedule can be challenging, so it is advisable to

schedule a meeting time by email. Email is the best way to contact me, and I will usually be able

to respond to emails within one business day. Be advised that I generally don't respond to emails

in the evenings or on weekends.

Reading List: None

Course Structure: This course operates on the principle that you will learn best when you are actively working on

a task rather than passively listening. Both passive listening and active work have their places in

the learning process, but since active work is generally harder and requires more assistance than

passive listening, we will be spending the bulk of class time together working actively on

problems, where you can work at your own pace and ask questions freely of the professor and

Teaching Assistants (TA’s).

To make time and space for this amount of active work in class, most of the lectures for this

class are videos. These lectures have the same content and are roughly the same length as

lectures that would normally be given in class. Rather than doing extensive homework problem

sets outside of class, your outside of class time will largely be spent watching the lectures

(generally around 45 to 60 minutes worth of lecture for each 75 minutes of meeting time, so

something like 1.5-2 hours a week) and answering structured study questions about what you

are watching. Then we will work on homework in class in the form of problems that will

challenge you and help you learn to apply the basic material to new and interesting problems.

Please be assured that you are not being asked to "teach yourself". In fact, you will be receiving

a much higher quantity and quality of professor help in this format that you would in a

traditional format, because the professor is available in the same room as you exactly when you

are encountering the hardest work. Also, multiple means of asking questions and getting help

outside of class have been set up so that you are never really working on your own (except on

timed assessments, of course).

The basic workflow of the course consists of:

Before the Class

o Guided Practice

Videos, reading and exercises

During the Class

o Entrance Quiz

Evaluate basic learning outcomes covered in Guided Practice

o Class Activity

Apply knowledge to more complex problems

After the Class

o Homework

Online homework and solution write-up in a homework journal

Assessment: The methods of assessment and their respective weightings are given below:

Guided Practice Exercises 5%

Class Participation (primarily Entrance Quiz) 5%

Homework 6%

Homework Journal 4%

Lab Assessments (2) 15%

In-Class Assessments (2) 30%

Final Assessment 35%

Final Grades: The final grades will be consistent with the system specified in the university’s grading system.

http://students.usask.ca/current/academics/grades/grading-system.php

For information regarding appeals of final grades or other academic matters, please consult the

University Council document on academic appeals.

http://www.usask.ca/university_secretary/honesty/StudentAcademicAppeals.pdf

Course Content: See the course content on the edx.engr.usask.ca site under the Courseware tab.

Homework: A homework assignment will be posted periodically throughout the term.

Tutorials: To be arranged when needed

Quizzes: An Entrance Quiz is associated with each Guided Practice

Assessments: Two in-class assessments during the term and one final assessment on Dec 4, 2014.

Important Dates: Thursday, September 4, 2014, 1:00-2:20PM EE265 class begins

Tuesday, October 9, 2014, 1:00-2:20PM In-Class Assessment 1 (may change)

Thursday, November 6, 2014, 1:00-2:20PM In-Class Assessment 2 (may change)

Thursday, December 4, 2014, 1:00-4:00PM Final Assessment (may change)

Note that these dates and times depend progress through the course material and on scheduling

appropriate computer lab space and thus could change.

Student Conduct: Ethical behaviour is an important part of engineering practice. Each professional engineering

association has a Code of Ethics, which its members are expected to follow. Since students are

in the process of becoming Professional Engineers, it is expected that students will conduct

themselves in an ethical manner.

The APEGS (Association of Professional Engineers and Geoscientists of Saskatchewan) Code

of Ethics states that engineers shall “conduct themselves with fairness, courtesy and good faith

towards clients, colleagues, employees and others; give credit where it is due and accept, as well

as give, honest and fair professional criticism” (Section 20(e), The Engineering and Geoscience

Professions Regulatory Bylaws, 1997).

The first part of this statement discusses an engineer’s relationships with his or her colleagues.

One of the ways in which engineering students can demonstrate courtesy to their colleagues is

by helping to maintain an atmosphere that is conducive to learning, and minimizing disruptions

in class. This includes arriving on time for lectures, turning cell phones and other electronic

devices off during lectures, not leaving or entering the class at inopportune times, and refraining

from talking to others while the instructor is talking. However, if you have questions at any time

during lectures, please feel free to ask (chances are very good that someone else may have the

same question as you do).

For more information, please consult the University Council Guidelines for Academic Conduct.

http://www.usask.ca/university_secretary/council/reports_forms/reports/guide_conduct.php

Academic Honesty: The latter part of the above statement from the APEGS Code of Ethics discusses giving credit

where it is due. At the University, this is addressed by university policies on academic integrity

and academic misconduct. In this class, students are expected to submit their own individual

work for academic credit, properly cite the work of others, and to follow the rules for

examinations. Academic misconduct, plagiarism, and cheating will not be tolerated. Copying of

assignments and lab reports is considered academic misconduct. Students are responsible for

understanding the university’s policies on academic integrity and academic misconduct. For

more information, please consult the University Council Regulations on Student Academic

Misconduct and the university’s examination regulations.

http://www.usask.ca/university_secretary/honesty/StudentAcademicMisconduct.pdf

http://www.usask.ca/university_secretary/council/academiccourses.php

Safety: The APEGS Code of Ethics also states that Professional Engineers shall “hold paramount the

safety, health and welfare of the public and the protection of the environment and promote

health and safety within the workplace” (Section 20(a), The Engineering and Geoscience

Professions Regulatory Bylaws, 1997).

Safety is taken very seriously by the Department of Electrical and Computer Engineering.

Students are expected to work in a safe manner, follow all safety instructions, and use any

personal protective equipment provided. Students failing to observe the safety rules in any

laboratory will be asked to leave.

Laboratory Learning

Outcomes:

See the attached Lab Learning Outcomes document.

Course Learning

Outcomes:

See the attached Course Learning Outcomes document.

Attribute Mapping: Level of Performance*

Learning

Outcome

Attribute**

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12

All 1 1 1

**Attributes:

A1 Knowledge base for engineering

A2 Problem analysis

A3 Investigation

A4 Design

A5 Use of engineering tools

A6 Individual and team work

A7 Communication skills

A8 Professionalism

A9 Impact of engineering on society

and the environment

A10 Ethics and equity

A11 Economics and project

management

A12 Life-long learning

*Levels of Performance:

1 - Knowledge of the skills/concepts/tools but not using them to solve

problems.

2 - Using the skills/concepts/tools to solve directed problems.

(“Directed” indicates that students are told what tools to use.)

3 - Selecting and using the skills/concepts/tools to solve non-directed,

non-open-ended problems. (Students have a number of S/C/T to

choose from and need to decide which to employ. Problems will

have a definite solution.)

4 - Applying the appropriate skills/concepts/tools to solve open-ended

problems. (Students have a number of S/C/T to choose from and

need to decide which to employ. Problems will have multiple

solution paths leading to possibly more than one acceptable

solution.)

Accreditation Unit (AU) Mapping: (% of total class AU)

Math Natural Science

Complementary

Studies

Engineering

Science Engineering Design

0 0 0 45.8 0

Assessment Mapping:

Component Weighting Methods of Feedback***

Learning Outcomes Evaluated

Exercises and Quizzes 10% S All

Homework 10% S All

Lab Assessments 15% S All

In-Class Assessments 30% S All

Final Assessment 35% S All

***Methods of Feedback:

F – formative (written comments and/or oral discussions)

S – summative (number grades)

EE 265 Lab Learning Outcomes Fall 2014

1. Lab 1: Basic Calculations

Once you have completed this lab you should be able to:

(a) Start and quit MATLAB.

(b) Generate and execute MATLAB expressions that use the basic arithmetic operators:addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^).

(c) Define variables that can be used in other MATLAB calculations.

(d) Control MATLAB’s displayed numerical output: suppressing the output completely oraltering the number of decimal places displayed.

(e) Generate and execute MATLAB expressions using basic trigonometric, exponential, andlogarithmic functions.

2. Lab 2: Working with Vectors

Once you have completed this lab you should be able to:

(a) Generate variables containing a row or column vector.

(b) Apply the basic arithmetic operators to vector variables: addition (+), subtraction (-).

(c) Use the array operators: element by element multiplication (.*), element by elementdivision (./), element by element exponentiation (.^).

(d) Calculate the dot (scalar) product.

(e) Access individual elements in vector variables.

(f) Use some basic trigonometric, exponential, and logarithmic functions with vector vari-ables.

3. Lab 3: Basic Plotting

Once you have completed this lab you should be able to:

(a) Generate long vectors for use in plotting.

(b) Produce the x and y vectors used in plotting.

(c) Place two plots on one figure.

(d) Label (annotate) plots.

4. Lab 4: M-File Script Programs

Once you have completed this lab you should be able to:

(a) Display, save, load and clear the contents of the MATLAB workspace.

(b) Load the contents of the MATLAB workspace.

(c) Generate a script using the MATLAB Editor.

(d) Add comments and help information to a script file.

(e) Save the script file in a user created folder.

(f) Tell MATLAB where the script file is located.

(g) Execute a script file.

5. Lab 5: M-File Functions, Relational Operators and If Statements

Once you have completed this lab you should be able to:

(a) Write an M-file function.

(b) Use an M-file function.

(c) Compare arrays using relational operators (<, >, <=, >=, ==, ~=).

(d) Write a MATLAB program using an if statement to make a decision.

6. Lab 6: Solving Equations

Once you have completed this lab you should be able to:

(a) Generate a variable containing a matrix (two-dimensional array).

(b) Use MATLAB’s left division operator to solve a set of linear equations.

7. Lab 7: Working with Matrices

Once you have completed this lab you should be able to:

(a) Multiply two matrices (two-dimensional arrays).

(b) Calculate the determinant, rank, and inverse of a matrix.

8. Lab 8: Sinusoidal Signals

Once you have completed this lab you should be able to:

(a) Use vectors to approximate a continuous-time sinusoid that can be displayed in a figurewindow using the command plot.

(b) Use vectors to generate a discrete-time sinusoid plot using the command stem.

(c) Annotate figures with axes labels and a title.

(d) Generate and execute a MATLAB program, which is referred to as an m-file script.

(e) Generate multiple figures in the same figure window using subplot.

(f) Generate multiple line plots on the same figure using hold.

9. Lab 9: Digital/Discrete-Time Signals

Once you have completed this lab you should be able to:

(a) Load a digital audio signal from a .wav file into MATLAB using wavread.

(b) Play discrete-time signals using MATLAB commands sound.

(c) Use the gtext or text command to place labels on a figure.

10. Lab 10: Generating and Playing Discrete-Time Signals

Once you have completed this lab you should be able to:

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(a) Generate discrete-time sinusoids and play them on a sound card using various samplingfrequencies.

11. Lab 11: Frequency Response of a FIR System

Once you have completed this lab you should be able to:

(a) Determine an expression for the frequency response of an FIR system from the unitsample response, h[n] = b[n]. The frequency response is generated using the DTFT(Discrete-Time Fourier Transform).

(b) Generate a discrete-time signal, x[n], consisting of the sum of two sinusoids.

(c) Determine the frequency representation of a discrete-time signal using the DTFS todetermine the DTFT at specific discrete frequencies.

(d) Use the MATLAB command filter(b, a, x) to determine the output of the system definedby h[n] = b[n], given the input x[n].

(e) Determine the effect of an FIR filter in the frequency domain by comparing the frequencyrepresentation of the input signal, x[n], and frequency response H(ejω) of the FIR filterto justify the frequency representation of the output signal, y[n].

12. Lab 12: FIR Filter

Once you have completed this lab you should be able to:

(a) Determine the unit sample response for an FIR filter from the filter difference equation.

(b) Determine the type of filter (lowpass, highpass, bandpass) from the frequency response,H(ejω), of the FIR filter calculated using the DTFT.

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EE 265 Course Learning Outcomes Fall 2014

1. Guided Practice 1: Radians, Sinusoids and the Unit Circle

• Generate a definition for radian.

• Generate and use the relationship between arc length, angle and radius.

• Generate the plot of a sine waveform using a unit circle.

• Demonstrate that radian is a unitless measure.

• Generate the plot of a sampled sinusoidal waveform using the unit circle.

• Generate the plot of a delayed (shifted) sinusoidal waveform using the unit circle.

2. Guided Practice 2: Working with Sinusoids

• Determine the amplitude, period and midline of a sinusoid function from a graphicalplot.

• Generate the mathematical equation of a sinusoid function using the amplitude, periodand midline.

• Determine the frequency and phase of a continuous-time sinusoid.

• Generate and use Euler’s formula: ejθ = cos θ + j sin θ.

• Calculate the principal phase of a time-shifted continuous-time sinusoid.

• Determine the frequency of a complex exponential, ejΩt from unit circle vector plots.

• Generate the the inverse Euler expressions for cos θ and sin θ from the Euler’s formulafor ejθ and e−jθ.

3. Guided Practice 3: Complex Numbers Part I

• Generate the real part, Rez, and imaginary part, Imz, of a complex number z.

• Add and subtract complex numbers.

• Plot complex numbers as vectors on an Argand diagram.

• Add and subtract complex numbers geometrically using vector addition on an Arganddiagram.

• Calculate the distance between two complex numbers.

• Calculate the midpoint coordinates between two complex numbers.

• Determine the complex conjugate, z∗, of a complex number z.

• Multiply and divide complex numbers.

• Graphically show that z + z∗ = 2Rez, where z∗ is the complex conjugate of z.

• Graphically show that z − z∗ = 2jImz, where z∗ is the complex conjugate of z.

• Provide a geometrically interpretation for the multiplication of a complex number by j.

• Generate the vector addition or subtraction of two or more complex numbers representedas complex exponentials or as real and imaginary components.

4. Guided Practice 4: Complex Numbers Part II

• Generate the magnitude (absolute value, modulus) of a complex number.

• Generate the principal value of the angle (argument) of a complex number.

• Convert between rectangular and polar forms of a complex number.

• Divide two complex numbers using the complex exponential form.

• Determine zc using complex exponentials, where z is a complex number and c is a realnumber.

• Derive trigonometric identities, such as cos (α + β) = cos α cos β−sin α sinβ, using com-plex exponentials.

• Generate the phasor for a sinusoidal signal.

• Add two phasors.

• Derive the product of two sinusoids using the inverse Euler’s formulas.

5. Guided Practice 5: Spectrum Representation Using Phasors

• Generate the phasor for an arbitrary sinusoid.

• Use the MATLAB command abs to determine the magnitude of a phasor.

• Use the MATLAB command angle to determine the phase of a phasor.

• Determine the sinusoid function representing the addition of two sinusoid functions, bothhaving the same frequency, using the phasor representation for the sinusoids.

• Generate a spectrum plot for a sinusoid.

• Determine the fundamental frequency and fundamental period for a sum of two harmon-ically related sinusoids.

• Use the inverse Euler formula to generate the magnitude and phase plots representingthe spectrum of the sum of N arbitrary sinusoids, where N is an integer.

• Generate the time-domain equation from the plot that represents the spectrum of a sumof sinusoids.

• Extend the basic learning outcome of determining the fundamental frequency and fun-damental period for a sum of two harmonically related sinusoids to a set of N sinusoids.

6. Guided Practice 6: Basic DT Signals

• Generate a unit sample sequence.

• Generate a unit step sequence.

• Generate and plot, using MATLAB, a real exponential sequence.

• Generate and plot, using MATLAB, a sinusoid sequence.

• Determine the period of a sinusoid sequence.

• Generate and plot the magnitude and phase, using MATLAB, of a complex exponentialsequence.

• Generate an expression for an arbitrary DT sequence, using shifted and weighted unitsample sequences.

• Determine the frequency and the fundamental period, N , of a DT sinusoid sequence.

7. Guided Practice 7: DT Signal Properties: Part I

• Identify if a DT signal is periodic.

• Determine the period N of a periodic DT signal.

• Generate a signal that is a time shifted, time reversed or both time shifted and reversed.

• Determine whether a DT signal is even, odd or neither.

• Determine the even and odd components of a DT signal.

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• Given a sequence x[n] and h[n], evaluate the expression

y[n] =

∞∑

k=−∞

x[k]h[n − k]

This expression, which makes use of time shifting and time reversal, is used to determinethe output of a system. The equation represents an operation that is referred to asconvolution, a topic covered in detail later.

8. Guided Practice 8: DT Signal Properties: Part II

• Calculate the energy in a DT signal.

• Calculate the average power in a DT signal.

• Determine the magnitude of a complex function z[n] using |z[n]| = z[n]z∗[n] where ∗

represents complex conjugate.

• Calculate the sum of the geometric series, αn for n = 0, 1, · · · , N , where α is a realnumber, using the closed form of the sum.

• Calculate the sum of the more general geometric series, zn for n = N1, · · · , N2, where z

can be complex and N2 ≥ N1, using the closed form of the sum (derived in the CourseHandout document: Closed Form of a Sum), where z is either:

– a real number,

– a complex number, ejω.

• Given a periodic sequence x[n] with period N , evaluate the expression

c[k] =∑

n=<N>

x[n]e−jkn2π/N

where the notation n =< N > indicates the sum is taken over one period. This expres-sion is used to determine the frequency components of x[n]. The equation represents anoperation that is referred to as the Fourier series, a topic covered in detail later.

9. Guided Practice 9: DT System Properties

• Define the system properties: memoryless, time invariance, linearity, causality, and sta-bility.

• Determine whether a system is memoryless, time-invariant (shift-invariant), linear, causal,and stable.

• Determine whether more complex systems, such as those listed below, are memoryless,time-invariant (shift-invariant), linear, causal, and stable.

– y[n] = nx[n]

– y[n] = x[n] − x[n − 1]

– y[n] = x[n]x[n + 1]

10. Guided Practice 10: Convolution

• Determine the output of a simple LTI system with impulse response h[n] for a giveninput x[n], using convolution.

• Determine the output of more complex systems and input signals, such as those listedbelow:

– h[n] and x[n] are both rectangular pulses

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– h[n] and x[n] are both exponential sequences

– h[n] and x[n] are both periodic sequences

11. Guided Practice 11: Fourier Series

• Determine the DTFS for sinusoidal signals.

• Use MATLAB to determine the DFT (which is directly related to the DTFS) of adiscrete-time signal.

• Determine the DTFS of more complex signals, such as those listed below:

– Periodic square wave

– Rectified sine wave

– Periodic triangle wave

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