Difference between revisions of "BSc: Signals And Systems"

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= Differential Equations =
  
   
  +
= Signals and Systems =
−
* <span>'''Course name:'''</span> Differential Equations
  
−
* <span>'''Course number:'''</span> XYZ
 +
* '''Course name''': Signals and Systems
−
* <span>'''Subject area:'''</span> Math
 +
* '''Code discipline''': XYZ
  +
* '''Subject area''': Electric Engineering
   
−
== Course characteristics ==
 +
== Short Description ==
  +
This course covers the following concepts: 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.
   
  +
== Prerequisites ==
−
=== Key concepts of the class ===
  
   
  +
=== Prerequisite subjects ===
−
* Ordinary differential equations
  
  +
* [https://eduwiki.innopolis.university/index.php/BSc:AnalyticGeometryAndLinearAlgebraI CSE202 — Analytical Geometry and Linear Algebra]: complex numbers, vecrors and matrix operations, basis and basis decomposition, concept of eigen values and vectors.
−
* Basic numerical methods
  
  +
* [https://eduwiki.innopolis.university/index.php/BSc:MathematicalAnalysisI CSE201 — Mathematical Analysis I]: limits and absolutly summabale series, exponent function, besic integration.
  +
* [https://eduwiki.innopolis.university/index.php/BSc:Logic_and_Discrete_Mathematics CSE113 — Philosophy I - (Discrete Math and Logic)]: Algorithm time and space complexity.
   
  +
=== Prerequisite topics ===
−
=== What is the purpose of this course? ===
  
   
−
The course is designed to provide Software Engineers and Computer Scientists by knowledge of basic (core) concepts, definitions, theoretical results and techniques of ordinary differential equations theory, basics of power series and numerical methods, applications of the all above in sciences. All definitions and theorem statements (that will be given in lectures and that are needed to explain the keywords listed above) will be formal, but just few of these theorems will be proven formally. Instead (in the tutorial and practice classes) we will try these definitions and theorems on work with routine exercises and applied problems.
  
   
−
=== Course Objectives Based on Bloom’s Taxonomy ===
 +
== Course Topics ==
  +
{| class="wikitable"
  +
|+ Course Sections and Topics
  +
|-
  +
! Section !! Topics within the section
  +
|-
  +
| Complex numbers and functions, vector and Hilbert Spaces, computational aspects ||
  +
# Complex numbers and their matrix representation
  +
# Vector spaces with dot-product
  +
# Metrics and convergence, Hilbert spaces
  +
# Algorithms and their computational (space and time) complexity
  +
|-
  +
| Discrete Fourier Transform and Fast Fourier Transforms (DFT and FFT) ||
  +
# 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
  +
|-
  +
| Discrete-time signals and systems: properties and classification ||
  +
# 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
  +
|-
  +
| Convolution, Discrete-time Fourier Transformation, filtering ||
  +
# 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.
  +
|}
  +
== Intended Learning Outcomes (ILOs) ==
   
−
=== - What should a student remember at the end of the course? ===
 +
=== What is the main 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.
   
  +
=== ILOs defined at three levels ===
−
* recognize the type of the equation,
  
−
* identify the method of analytical solution,
  
−
* define an initial value problem,
  
−
* list alternative approaches to solving ordinary differential equations,
  
−
* match the concrete numerical approach with the necessary level of accuracy.
  
   
−
=== - What should a student be able to understand at the end of the course? ===
 +
==== Level 1: What concepts should a student know/remember/explain? ====
  +
By the end of the course, the students should be able to ...
  +
* 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),
   
  +
==== Level 2: What basic practical skills should a student be able to perform? ====
−
* understand application value of ordinary differential equations,
  
  +
By the end of the course, the students should be able to ...
−
* explain situation when the analytical solution of an equation cannot be found,
  
  +
* relations between analog and digital signals (sampling)
−
* give the examples of functional series for certain simple functions,
  
  +
* relations between convolution, correlation, and filtering of discrete signals
−
* describe the common goal of the numeric methods,
  
  +
* role of impulse and frequency domains of discrete signals
−
* restate the given ordinary equation with the Laplace Transform.
  
  +
* 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? ===
 +
==== Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios? ====
  +
By the end of the course, the students should be able to ...
  +
* 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.
  +
== Grading ==
   
  +
=== Course grading range ===
−
* solve the given ordinary differential equation analytically (if possible),
  
  +
{| class="wikitable"
−
* apply the method of the Laplace Transform for the given initial value problem,
  
  +
|+
−
* predict the number of terms in series solution of the equation depending on the given accuracy,
  
−
* implement a certain numerical method in self-developed computer software.
  
−  
−
=== Course evaluation ===
  
−  
−
{|
  
−
|+ Course grade breakdown
  
−
!
  
−
!
  
−
!align="center"| '''Proposed points'''
  
 
|-
  
|-
  +
! Grade !! Range !! Description of performance
−
| Labs/seminar classes
  
−
| 20
  
−
|align="center"| 20
  
 
|-
  
|-
  +
| A. Excellent || 104-130 || -
−
| Interim performance assessment
  
−
| 30
  
−
|align="center"| 70
  
 
|-
  
|-
  +
| B. Good || 84-103 || -
−
| Exams
  
−
| 50
 +
|-
  +
| C. Satisfactory || 65-83 || -
−
|align="center"| 80
  
  +
|-
  +
| D. Poor || 0-64 || -
 
|}
  
|}
   
  +
=== Course activities and grading breakdown ===
−
If necessary, please indicate freely your course’s features in terms of students’ performance assessment:
  
  +
{| class="wikitable"
−  
  +
|+
−
==== 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: ====
  
−  
−
* in-class tests up to 10 points for each test (i.e. up to 40 points in total for 2 theory and 2 practice tests),
  
−
* computational practicum assignment up to 10 points for each task (i.e. up to 30 points for 3 tasks).
  
−  
−
==== Exams: ====
  
−  
−
* mid-term exam up to 40 points,
  
−
* final examination up to 40 points.
  
−  
−
==== Overall score: ====
  
−  
−
170 points (100%).
  
−  
−
=== Grades range ===
  
−  
−
{|
  
−
|+ Course grading range
  
−
!
  
−
!
  
−
!align="center"| '''Proposed range'''
  
 
|-
  
|-
  +
! Activity Type !! Percentage of the overall course grade
−
| A. Excellent
  
−
| 90-100
  
−
|align="center"| 136-170
  
 
|-
  
|-
  +
| Labs/seminar classes || 20
−
| B. Good
  
−
| 75-89
  
−
|align="center"| 102-135
  
 
|-
  
|-
  +
| Interim performance assessment || 90
−
| C. Satisfactory
  
−
| 60-74
  
−
|align="center"| 68-101
  
 
|-
  
|-
−
| D. Poor
 +
| Exams || 20
−
| 0-59
  
−
|align="center"| 0-68
  
 
|}
  
|}
   
  +
=== Recommendations for students on how to succeed in the course ===
−
If necessary, please indicate freely your course’s grading features:
  
   
−
* A: at least 80% of the overall score;
  
−
* B: at least 60% of the overall score;
  
−
* C: at least 40% of the overall score;
  
−
* D: less than 40% of the overall score.
  
   
−
=== Resources and reference material ===
 +
== Resources, literature and reference materials ==
   
−
==== Textbook: ====
 +
=== Open access resources ===
  +
* Martin Vetterli, Jelena Kovacevic, and Vivek K Goyal.Foundations of Signal Processing.Cambridge University Press, 2014. ISBN 10703860X
  +
* 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
   
  +
=== Closed access resources ===
−
*
 
   
−
==== Reference material: ====
  
   
  +
=== Software and tools used within the course ===
−
*
 
  +
−
*
 
  +
= Teaching Methodology: Methods, techniques, & activities =
−
*
 
   
  +
== Activities and Teaching Methods ==
−
== Course Sections ==
  
  +
{| class="wikitable"
  +
|+ Activities within each section
  +
|-
  +
! Learning Activities !! Section 1 !! Section 2 !! Section 3 !! Section 4
  +
|-
  +
| Homework and group projects || 1 || 1 || 1 || 1
  +
|-
  +
| Testing (written or computer based) || 1 || 0 || 0 || 0
  +
|-
  +
| Reports || 1 || 1 || 1 || 1
  +
|-
  +
| Discussions || 1 || 1 || 1 || 1
  +
|-
  +
| Development of individual parts of software product code || 0 || 1 || 1 || 1
  +
|-
  +
| Midterm evaluation || 0 || 1 || 1 || 1
  +
|}
  +
== Formative Assessment and Course Activities ==
   
  +
=== Ongoing performance assessment ===
−
The main sections of the course and approximate hour distribution between them is as follows:
  
   
  +
==== Section 1 ====
−
{|
  
  +
{| class="wikitable"
−
|+ Course Sections
  
  +
|+
−
|align="center"| '''Section'''
  
−
| '''Section Title'''
  
−
|align="center"| '''Lectures'''
  
−
|align="center"| '''Seminars'''
  
−
|align="center"| '''Self-study'''
  
−
|align="center"| '''Knowledge'''
  
 
|-
  
|-
  +
! Activity Type !! Content !! Is Graded?
−
|align="center"| '''Number'''
  
−
|
  
−
|align="center"| '''(hours)'''
  
−
|align="center"| '''(labs)'''
  
−
|align="center"|
  
−
|align="center"| '''evaluation'''
  
 
|-
  
|-
  +
| Question || Prove that each complex number has a square root. || 1
−
|align="center"| 1
  
−
| First-order equations and their applications
  
−
|align="center"| 12
  
−
|align="center"| 6
  
−
|align="center"| 12
  
−
|align="center"| 4
  
 
|-
  
|-
  +
| Question || Prove that the neutral element is unique in a vector space. || 1
−
|align="center"| 2
  
−
| Introduction to numeric methods for algebraic and first-order differential equations
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 22
  
−
|align="center"| 1
  
 
|-
  
|-
  +
| Question || Prove that pixel (Manhattan) and Euclidean norms are equivalent in finite-dimensional real (complex) spaces. || 1
−
|align="center"| 3
  
−
| Second-order differential equations and their applications
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 8
  
−
|align="center"| 2
  
 
|-
  
|-
  +
| Question || Is the set of integers complete in the discrete metrics? || 1
−
|align="center"| 4
  
−
| Laplace transform
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 12
  
−
|align="center"| 3
  
 
|-
  
|-
  +
| Question || What is space and time complexity of dot product in a complex n-dimensional vector space? || 1
−
|align="center"| 5
  
−
| Series approach to linear differential equations
  
−
|align="center"| 8
  
−
|align="center"| 4
  
−
|align="center"| 12
  
−
|align="center"| 0
  
 
|-
  
|-
  +
| Question || Prove that each complex number but zero has the inverse. || 0
−
|align="center"| Final examination
  
−
|
 +
|-
  +
| Question || Prove that each vector of a vector space has unique opposite element. || 0
−
|align="center"|
  
  +
|-
−
|align="center"|
  
  +
| Question || Prove that pixel and the universal norms are equivalent in finite-dimensional real(complex) spaces. || 0
−
|align="center"|
  
  +
|-
−
|align="center"| 2
  
  +
| Question || Is the set of rational numbers complete in the discrete metrics? || 0
−
|}
  
  +
|-
−  
  +
| Question || What is space and time complexity of finite matrices multiplication (according to the definition)? || 0
−
=== Section 1 ===
  
  +
|}
−  
−
==== Section title: ====
 +
==== Section 2 ====
  +
{| class="wikitable"
−  
  +
|+
−
First-order equations and their applications
  
  +
|-
−  
  +
! Activity Type !! Content !! Is Graded?
−
=== Topics covered in this section: ===
  
  +
|-
−  
  +
| Question || Compute circular convolution of given two short integer signals. || 1
−
* The simplest type of differential equation
  
  +
|-
−
* Separable equation
  
  +
| Question || Explain Discrete Fourier Transform as orthogonal vector decomposition. || 1
−
* Initial value problem
  
  +
|-
−
* Homogeneous nonlinear equations, substitutions
  
  +
| Question || Compute DFT and FFT for given short integer signal. || 1
−
* Linear ordinary equations, Bernoulli &amp; Riccati equations
  
  +
|-
−
* Examples of applications to modeling the real world problems
  
  +
| Question || Prove circular impulse shift property. || 0
−
* Exact differential equations, integrating factor
  
  +
|-
−  
  +
| Question || Study commutativity, linearity and associativity of the circular convolution. || 0
−
=== What forms of evaluation were used to test students’ performance in this section? ===
  
  +
|-
−  
  +
| Question || Give matrix representation for the circular convolution for several small dimensions. || 0
−
<div class="tabular">
  
  +
|-
−  
  +
| Question || Recall 2-redex fast Fourier transform and draw its matrices for several small dimensions. || 0
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
  +
|}
−
Development of individual parts of software product code &amp; 0<br />
  
  +
==== Section 3 ====
−
Homework and group projects &amp; 1<br />
  
  +
{| class="wikitable"
−
Midterm evaluation &amp; 1<br />
  
  +
|+
−
Testing (written or computer based) &amp; 1<br />
  
  +
|-
−
Reports &amp; 0<br />
  
  +
! Activity Type !! Content !! Is Graded?
−
Essays &amp; 0<br />
  
  +
|-
−
Oral polls &amp; 1<br />
  
  +
| Question || Give examples of (infinite) absolutely summable/non-summable, bounded/unbounded,etc., signals. || 1
−
Discussions &amp; 1<br />
  
  +
|-
−  
  +
| Question || Is autocorrelation linear system? Is it shift-invariant? || 1
−  
  +
|-
−  
  +
| Question || Prove that a linear system is memoryless iff its matrix is diagonal. || 1
−
</div>
  
  +
|-
−
=== Typical questions for ongoing performance evaluation within this section ===
  
  +
| Question || Prove that a linear system is causal iff its matrix is low-triangle. || 0
−  
  +
|-
−
# What is the type of the first order equation?
  
  +
| Question || A linear system is shift-invariant iff its matrix consists (exclusively) of diagonals of some constant (individual for each diagonal). || 0
−
# Is the equation homogeneous or not?
  
  +
|-
−
# Which substitution may be used for solving the given equation?
  
  +
| Question || Prove that product of finite power series is convolution of the finite signals consisting of the coefficients of these series. || 0
−
# Is the equation linear or not?
  
  +
|}
−
# Which type of the equation have we obtained for the modeled real world problem?
  
  +
==== Section 4 ====
−
# Is the equation exact or not?
  
  +
{| class="wikitable"
−  
  +
|+
−
=== Typical questions for seminar classes (labs) within this section ===
  
  +
|-
−  
  +
! Activity Type !! Content !! Is Graded?
−
# Determine the type of the first order equation and solve it with the use of appropriate method.
  
  +
|-
−
# Find the integrating factor for the given equation.
  
  +
| Question || Do there exists a periodic function with non-commensurable periods? || 1
−
# Solve the initial value problem of the first order.
  
  +
|-
−
# Construct a mathematical model of the presented real world problem in terms of differential equations and answer for the specific question about it.
  
  +
| Question || Prove that product of two exponents is equal to the exponent with sum of powers. || 1
−  
  +
|-
−
=== Test questions for final assessment in this section ===
  
  +
| Question || Prove conjugate property for DTFT. || 1
−  
  +
|-
−
# Linear first order equation. Integrating factor.
  
  +
| Question || Prove DTFT-correspondence for impulse shift. || 0
−
# Bernoulli &amp; Riccati equations.
  
  +
|-
−
# Homogeneous nonlinear equations equations.
  
  +
| Question || Prove DTFT-correspondence for frequency shift. || 0
−
# Exact equations. Substitutions.
  
  +
|-
−  
  +
| Question || Design a low-band filter with a given spectrum consisting of a single box. || 0
−
=== Section 2 ===
  
  +
|}
−  
−
==== Section title: ====
 +
=== Final assessment ===
  +
'''Section 1'''
−  
  +
# Build if possible (or prove that it isn’t) ...
−
Introduction to numeric methods for algebraic and first-order differential equations
  
  +
'''Section 2'''
−  
  +
# Assume that a finite signal ...
−
=== Topics covered in this section: ===
  
  +
'''Section 3'''
−  
  +
# Compute cross-correlation of two box signals.
−
* Method of sections (Newton method)
  
  +
# Study properties (linearity, causality, stability, etc.) of a weighted accumulator
−
* Method of tangent lines approximation (Euler method)
  
  +
'''Section 4'''
−
* Improved Euler method
  
  +
# Show that exponent with imaginary power is a periodic function, find the smallest period.
−
* Runge-Kutta methods
  
  +
# Prove sampling and scaling properties for the Dirac Delta function.
−  
−
=== What forms of evaluation were used to test students’ performance in this section? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 1<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 1<br />
  
−
Testing (written or computer based) &amp; 0<br />
  
−
Reports &amp; 1<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# What is the difference between the methods of sections and tangent line approximations?
  
−
# What is the approximation error for the given method?
  
−
# How to improve the accuracy of Euler method?
  
−
# How to obtain a general formula of the Runge-Kutta methods?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# For the given initial value problem with the ODE of the first order implement in your favorite programming Euler, improved Euler and general Runge-Kutta methods of solving.
  
−
# Using the developed software construct corresponding approximation of the solution of a given initial value problem (provide the possibility of changing of the initial conditions, implement the exact solution to be able to compare the obtained results).
  
−
# Investigate the convergence of the numerical methods on different grid sizes.
  
−
# Compare approximation errors of these methods plotting the corresponding chart for the dependency of approximation error on a grid size.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
−
# Newton’s approximation method.
  
−
# Euler approximation method.
  
−
# Improved Euler method.
  
−
# Runge-Kutta methods.
  
−  
−
=== Section 3 ===
  
−  
−
==== Section title: ====
  
−  
−
Second-order differential equations and their applications
  
−  
−
=== Topics covered in this section: ===
  
−  
−
* Homogeneous linear equations.
  
−
* Constant coefficient homogeneous equations.
  
−
* Constant coefficient non-homogeneous equations.
  
−
* A method of undetermined coefficients.
  
−
* A method of variation of parameters.
  
−
* A method of the reduction of order.
  
−  
−
=== What forms of evaluation were used to test students’ performance in this section? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 0<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 1<br />
  
−
Testing (written or computer based) &amp; 0<br />
  
−
Reports &amp; 0<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# What is the type of the second order equation?
  
−
# Is the equation homogeneous or not?
  
−
# What is a characteristic equation of differential equation?
  
−
# In which form a general solution may be found?
  
−
# What is the form of the particular solution of non-homogeneous equation?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# Compose a characteristic equation and find its roots.
  
−
# Find the general of second order equation.
  
−
# Determine the form of a particular solution of the equation and reduce the order.
  
−
# Solve a homogeneous constant coefficient equation.
  
−
# Solve a non-homogeneous constant coefficient equation.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
−
# Homogeneous linear second order equations.
  
−
# Constant coefficient equations. A method of undetermined coefficients.
  
−
# Constant coefficient equations. A method of variation of parameters.
  
−
# Non-homogeneous linear second order equations. Reduction of order.
  
−  
−
=== Section 4 ===
  
−  
−
==== Section title: ====
  
−  
−
Laplace transform
  
−  
−
=== Topics covered in this section: ===
  
−  
−
* Improper integrals. Convergence / Divergence.
  
−
* Laplace transform of a function
  
−
* Existence of the Laplace transform.
  
−
* Inverse Laplace transform.
  
−
* Application of the Laplace transform to solving differential equations.
  
−  
−
=== What forms of evaluation were used to test students’ performance in this section? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 0<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 0<br />
  
−
Testing (written or computer based) &amp; 1<br />
  
−
Reports &amp; 0<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# What is an improper integral?
  
−
# How to compose the Laplace transform for a certain function?
  
−
# What is a radius of convergence of the Laplace transform?
  
−
# How to determine the inverse Laplace transform for a given expression?
  
−
# How to apply the method of Laplace transform for solving ordinary differential equations?
  
−  
−
=== Typical questions for seminar classes (labs) within this section ===
  
−  
−
# Find the Laplace transform for a given function. Analyze its radius of convergence.
  
−
# Find the inverse Laplace transform for a given expression.
  
−
# Solve the first order differential equation with the use of a Laplace transform.
  
−
# Solve the second order differential equation with the use of a Laplace transform.
  
−  
−
=== Test questions for final assessment in this section ===
  
−  
−
# Laplace transform, its radius of convergence and properties.
  
−
# Inverse Laplace transform. The method of rational functions.
  
−
# Application of Laplace transform to solving differential equations.
  
−  
−
=== 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? ===
  
−  
−
<div class="tabular">
  
−  
−
<span>|a|c|</span> &amp; '''Yes/No'''<br />
  
−
Development of individual parts of software product code &amp; 0<br />
  
−
Homework and group projects &amp; 1<br />
  
−
Midterm evaluation &amp; 0<br />
  
−
Testing (written or computer based) &amp; 1<br />
  
−
Reports &amp; 0<br />
  
−
Essays &amp; 0<br />
  
−
Oral polls &amp; 1<br />
  
−
Discussions &amp; 1<br />
  
−  
−  
−  
−
</div>
  
−
=== Typical questions for ongoing performance evaluation within this section ===
  
−  
−
# What are the power and functional series?
  
−
# How to find the radius of convergence of a series?
  
−
# What is a Taylor series?
  
−
# How to differentiate a functional series?
  
   
  +
=== The retake exam ===
−
=== Typical questions for seminar classes (labs) within this section ===
  
  +
'''Section 1'''
   
  +
'''Section 2'''
−
# Find the radius of convergence of a given series.
  
−
# Compose the Taylor series for a given function.
  
−
# Solve the first order differential equation with the use of Series approach.
  
−
# Solve the second order differential equation with the use of Series approach.
  
   
  +
'''Section 3'''
−
=== Test questions for final assessment in this section ===
  
   
  +
'''Section 4'''
−
# Taylor and Maclaurin series as functional series. Radius of convergence.
  
−
# Uniqueness of power series. Its differentiation.
  
−
# Application of power series to solving differential equations
  

Latest revision as of 13:11, 13 July 2022

Signals and Systems

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

Short Description

This course covers the following concepts: 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.

Prerequisites

Prerequisite subjects

Prerequisite topics

Course Topics

Course Sections and Topics
Section Topics within the section
Complex numbers and functions, vector and Hilbert Spaces, computational aspects
  1. Complex numbers and their matrix representation
  2. Vector spaces with dot-product
  3. Metrics and convergence, Hilbert spaces
  4. Algorithms and their computational (space and time) complexity
Discrete Fourier Transform and Fast Fourier Transforms (DFT and FFT)
  1. Circular convolution, eigen vectors and values of the circular convolution
  2. Discrete Fourier Transform (DFT) and its inverse
  3. Circutate filters and filtering
  4. Fast Fourier Transform (FFT),its inverse, and computational aspects of DFT and fast FFT
Discrete-time signals and systems: properties and classification
  1. Kotelnikov-Whittaker–Nyquist–Shannon sampling Theorem.
  2. Discrete signals as sequences, spaces of absolutely summable and bounded sequences.
  3. Auto- and cross-correlation; memoryless, causal and shift-invariant systems
  4. Linear systems, their matrix representation and properties
  5. Convolution and its relations to linear shift-invariant systems
Convolution, Discrete-time Fourier Transformation, filtering
  1. Math preliminaries on complex exponent and Euler formulas.
  2. Introduction of the discrete-time Fourier transform via convolution eigen values and vectors.
  3. Discrete-time Fourier transform as the frequency response of a linear shift-invariant system.
  4. Inverse discrete-time Fourier transform.
  5. DTFT properties (including convolution theorem).
  6. Elements of ideal Filter Design.

Intended Learning Outcomes (ILOs)

What is the main 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.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

  • 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),

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

  • 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)

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

  • 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.

Grading

Course grading range

Grade Range Description of performance
A. Excellent 104-130 -
B. Good 84-103 -
C. Satisfactory 65-83 -
D. Poor 0-64 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Labs/seminar classes 20
Interim performance assessment 90
Exams 20

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

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

Closed access resources

Software and tools used within the course

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section
Learning Activities Section 1 Section 2 Section 3 Section 4
Homework and group projects 1 1 1 1
Testing (written or computer based) 1 0 0 0
Reports 1 1 1 1
Discussions 1 1 1 1
Development of individual parts of software product code 0 1 1 1
Midterm evaluation 0 1 1 1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Question Prove that each complex number has a square root. 1
Question Prove that the neutral element is unique in a vector space. 1
Question Prove that pixel (Manhattan) and Euclidean norms are equivalent in finite-dimensional real (complex) spaces. 1
Question Is the set of integers complete in the discrete metrics? 1
Question What is space and time complexity of dot product in a complex n-dimensional vector space? 1
Question Prove that each complex number but zero has the inverse. 0
Question Prove that each vector of a vector space has unique opposite element. 0
Question Prove that pixel and the universal norms are equivalent in finite-dimensional real(complex) spaces. 0
Question Is the set of rational numbers complete in the discrete metrics? 0
Question What is space and time complexity of finite matrices multiplication (according to the definition)? 0

Section 2

Activity Type Content Is Graded?
Question Compute circular convolution of given two short integer signals. 1
Question Explain Discrete Fourier Transform as orthogonal vector decomposition. 1
Question Compute DFT and FFT for given short integer signal. 1
Question Prove circular impulse shift property. 0
Question Study commutativity, linearity and associativity of the circular convolution. 0
Question Give matrix representation for the circular convolution for several small dimensions. 0
Question Recall 2-redex fast Fourier transform and draw its matrices for several small dimensions. 0

Section 3

Activity Type Content Is Graded?
Question Give examples of (infinite) absolutely summable/non-summable, bounded/unbounded,etc., signals. 1
Question Is autocorrelation linear system? Is it shift-invariant? 1
Question Prove that a linear system is memoryless iff its matrix is diagonal. 1
Question Prove that a linear system is causal iff its matrix is low-triangle. 0
Question A linear system is shift-invariant iff its matrix consists (exclusively) of diagonals of some constant (individual for each diagonal). 0
Question Prove that product of finite power series is convolution of the finite signals consisting of the coefficients of these series. 0

Section 4

Activity Type Content Is Graded?
Question Do there exists a periodic function with non-commensurable periods? 1
Question Prove that product of two exponents is equal to the exponent with sum of powers. 1
Question Prove conjugate property for DTFT. 1
Question Prove DTFT-correspondence for impulse shift. 0
Question Prove DTFT-correspondence for frequency shift. 0
Question Design a low-band filter with a given spectrum consisting of a single box. 0

Final assessment

Section 1

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

Section 2

  1. Assume that a finite signal ...

Section 3

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

Section 4

  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.

The retake exam

Section 1

Section 2

Section 3

Section 4

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