BSc: Signals And Systems

Signals and Systems

• Course name: Signals and Systems
• Course number: XYZ
• Subject area: Electric Engineering

Course characteristics

Key concepts of the class

• discrete(-time) signals, their impulse and frequency domains
• classification of (discrete) systems (bound-input-bound-output, linear and shift-invariant)
• filters and filtering, finite and infinite impulse response filters
• discrete(-time) Fourier transform and fast Fourier transform

What is the purpose of this course?

The goal of the course is to present mathematical foundations of digital signal processing altogether with practical experience to design finite and infinite impulse response filters. The course is aimed to provide basic mathematical knowledge and practical skills needed for further studies of applied signal processing and digital signal processing from engineering as well as from mathematical perspective.

- What should a student remember at the end of the course?

• discrete (time) signals and systems, their classification
• linear shift-invariant systems, filters and filtering
• Discrete Fourier Transformation (DFT)
• Fast discrete Fourier Transformation (FFT)
• Discrete-Time Fourier Transformation (DTFT),

- What should a student be able to understand at the end of the course?

• relations between analog and digital signals (sampling)
• relations between convolution, correlation, and filtering of discrete signals
• role of impulse and frequency domains of discrete signals
• differences between infinite and finite discrete signals
• role of discrete time Fourier transform and its inverse
• role of discrete Fourier transform (DFT) and fast DFT (FFT)

- What should a student be able to apply at the end of the course?

• basic numerical tools from mathematical package SciLab/Octave
• classify discrete signals and systems
• design and implement infinite and finite impulse response filters
• implement and use discrete time Fourier transform,
• implement and use discrete Fourier transform and fast DFT.

Course evaluation

Course grade breakdown

		Proposed points
Labs/seminar classes	20	20
Interim performance assessment	30	90
Exams	50	20

If necessary, please indicate freely your course’s features in terms of students’ performance assessment:

Labs/seminar classes:

• In-class participation 1 point for each individual contribution in a class but not more than 1 point a week (i.e. 14 points in total for 14 study weeks),
• overall course contribution (to accumulate extra-class activities valuable to the course progress, e.g. a short presentation, book review, very active in-class participation, etc.) up to 6 points.

Interim performance assessment:

• Each of 4 home-made individual lab (computational) assignments costs 15 points (i.e. 60 points for all 4 assignments).
• Each of 3 home-made individual written theory tests cost 10 points (i.e. 30 points for all 3 tests).

Exam:

• final examination costs 20 points.

Overall score:

130 points (100%).

Grades range

Course grading range

		Proposed range
A. Excellent	80-100	104-130
B. Good	65-79	84-103
C. Satisfactory	50-64	65-83
D. Poor	0-49	0-64

If necessary, please indicate freely your course’s grading features:

• A: at least 80% of the overall score;
• B: at least 65% of the overall score;
• C: at least 50% of the overall score;
• D: less than 50% of the overall score.

Resources and reference material

Textbook:

• Martin Vetterli, Jelena Kovacevic, and Vivek K Goyal.Foundations of Signal Processing.Cambridge University Press, 2014. ISBN 10703860X

Reference material:

• Oppenheim, Alan V., and A. S. Willsky. Signals and Systems (2nd ed.) Prentice Hall, 1996. ISBN 0-13-814757-4.
• Richard G. Lyons.UnderstandingDigitalSignalProcessing. Prentice Hall, 2010. ISBN 978-0137027415

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Course Sections

Section	Section Title	Lectures	Seminars	Self-study	Knowledge
Number		(hours)	(labs)		evaluation
1	Complex numbers and functions, vector and Hilbert Spaces, computational aspects	4	4	4	1
2	Discrete Fourier Transform and Fast Fourier Transforms (DFT and FFT)	6	6	6	2
3	Discrete-time signals and systems: properties and classification	6	6	6	2
4	Convolution, Discrete-time Fourier Transformation, filtering	6	6	6	2
Final examination					2

Section 1

Section title:

Complex numbers and functions, vector and Hilbert Spaces, computational aspects

Topics covered in this section:

• Complex numbers and their matrix representation
• Vector spaces with dot-product
• Metrics and convergence, Hilbert spaces
• Algorithms and their computational (space and time) complexity

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	0
Homework and group projects	1
Midterm evaluation	0
Testing (written or computer based)	1
Reports	1
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Prove that each complex number has a square root.
2. Prove that the neutral element is unique in a vector space.
3. Prove that pixel (Manhattan) and Euclidean norms are equivalent in finite-dimensional real (complex) spaces.
4. Is the set of integers complete in the discrete metrics?
5. What is space and time complexity of dot product in a complex n-dimensional vector space?

Typical questions for seminar classes (labs) within this section

1. Prove that each complex number but zero has the inverse.
2. Prove that each vector of a vector space has unique opposite element.
3. Prove that pixel and the universal norms are equivalent in finite-dimensional real(complex) spaces.
4. Is the set of rational numbers complete in the discrete metrics?
5. What is space and time complexity of finite matrices multiplication (according to the definition)?

Test questions for final assessment in this section

(TBD)

1. Build if possible (or prove that it isn’t) ...

Section 2

Section title:

Discrete Fourier Transform and Fast Fourier Transforms (DFT and FFT)

Topics covered in this section:

• Circular convolution, eigen vectors and values of the circular convolution
• Discrete Fourier Transform (DFT) and its inverse
• Circutate filters and filtering
• Fast Fourier Transform (FFT),its inverse, and computational aspects of DFT and fast FFT

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	0
Reports	1
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Compute circular convolution of given two short integer signals.
2. Explain Discrete Fourier Transform as orthogonal vector decomposition.
3. Compute DFT and FFT for given short integer signal.

Typical questions for seminar classes (labs) within this section

1. Prove circular impulse shift property.
2. Study commutativity, linearity and associativity of the circular convolution.
3. Give matrix representation for the circular convolution for several small dimensions.
4. Recall 2-redex fast Fourier transform and draw its matrices for several small dimensions.

Test questions for final assessment in this section

TBD

1. Assume that a finite signal ...

Section 3

Section title:

Discrete-time signals and systems: properties and classification

Topics covered in this section:

• Kotelnikov-Whittaker–Nyquist–Shannon sampling Theorem.
• Discrete signals as sequences, spaces of absolutely summable and bounded sequences.
• Auto- and cross-correlation; memoryless, causal and shift-invariant systems
• Linear systems, their matrix representation and properties
• Convolution and its relations to linear shift-invariant systems

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	0
Reports	1
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Give examples of (infinite) absolutely summable/non-summable, bounded/unbounded,etc., signals.
2. Is autocorrelation linear system? Is it shift-invariant?
3. Prove that a linear system is memoryless iff its matrix is diagonal.

Typical questions for seminar classes (labs) within this section

1. Prove that a linear system is causal iff its matrix is low-triangle.
2. A linear system is shift-invariant iff its matrix consists (exclusively) of diagonals of some constant (individual for each diagonal).
3. Prove that product of finite power series is convolution of the finite signals consisting of the coefficients of these series.

Test questions for final assessment in this section

TBD

1. Compute cross-correlation of two box signals.
2. Study properties (linearity, causality, stability, etc.) of a weighted accumulator

Section 4

Section title:

Convolution, Discrete-time Fourier Transformation, filtering

Topics covered in this section:

• Math preliminaries on complex exponent and Euler formulas.
• Introduction of the discrete-time Fourier transform via convolution eigen values and vectors.
• Discrete-time Fourier transform as the frequency response of a linear shift-invariant system.
• Inverse discrete-time Fourier transform.
• DTFT properties (including convolution theorem).
• Elements of ideal Filter Design.

What forms of evaluation were used to test students’ performance in this section?

	Yes/No
Development of individual parts of software product code	1
Homework and group projects	1
Midterm evaluation	1
Testing (written or computer based)	0
Reports	1
Essays	0
Oral polls	0
Discussions	1

Typical questions for ongoing performance evaluation within this section

1. Do there exists a periodic function with non-commensurable periods?
2. Prove that product of two exponents is equal to the exponent with sum of powers.
3. Prove conjugate property for DTFT.

Typical questions for seminar classes (labs) within this section

1. Prove DTFT-correspondence for impulse shift.
2. Prove DTFT-correspondence for frequency shift.
3. Design a low-band filter with a given spectrum consisting of a single box.

Test questions for final assessment in this section

TBD

1. Show that exponent with imaginary power is a periodic function, find the smallest period.
2. Prove sampling and scaling properties for the Dirac Delta function.

Section 5

Section title:

Series approach to linear differential equations

Topics covered in this section:

• Functional series.
• Taylor and Maclaurin series.
• Differentiation of power series.
• Series solution of differential equations.

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 1
Midterm evaluation & 0
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. What are the power and functional series?
2. How to find the radius of convergence of a series?
3. What is a Taylor series?
4. How to differentiate a functional series?

Typical questions for seminar classes (labs) within this section

1. Find the radius of convergence of a given series.
2. Compose the Taylor series for a given function.
3. Solve the first order differential equation with the use of Series approach.
4. Solve the second order differential equation with the use of Series approach.

Test questions for final assessment in this section

1. Taylor and Maclaurin series as functional series. Radius of convergence.
2. Uniqueness of power series. Its differentiation.
3. Application of power series to solving differential equations