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Home / Signals and Systems Convolution Problems and Solutions

Signals and Systems Convolution Problems and Solutions

Signals and Systems

Signals and SystemsThis course introduces students to mathematical descriptions of signals & systems, and mathematical tools for analyzing and designing systems that can operate on signals to achieve a desired effect. The focus of the course is on the class of systems called linear time invariant systems. Significant emphasis will be place both on time domain analysis of systems through the operation of convolution and on frequency domain analysis of systems using the Fourier and Laplace transforms. Both continuous-time and discrete-time signals will be considered. Several examples from engineering practice will be used throughout the course.

Learning Outcomes and Objectives

It is the intent of this course that students will:

  1. be able to describe signals mathematically and understand how to perform mathematical operations on signals.
  2. be familiar with commonly used signals such as the unit step, ramp, impulse function, sinusoidal signals and complex exponentials, and be able to classify signals as continuous-time or discrete-time, as periodic or non-periodic, as energy or power signals, and as having even or odd symmetry.
  3. be able to describe linear time invariant systems either using linear constant coefficient differential equations or using their impulse response and be able to find a state space representation of a system from a block diagram and vice versa.
  4. understand various system properties such as linearity, time invariance, presence or absence of memory, causality, bounded-input bounded-output stability and invertibility and be able to identify whether a given system exhibits these properties and its implication for practical systems.
  5. understand the process of convolution between signals, its implication for analysis of linear time invariant systems and the notion of an impulse response.
  6. be able to solve a linear constant coefficient differential equation using Laplace transform techniques.
  7. understand the intuitive meaning of frequency domain and the importance of analyzing and processing signals in the frequency domain.
  8. be able to compute the Fourier series or Fourier transform of a set of well-defined signals from first principles, and further be able to use the properties of the Fourier transform to compute the Fourier transform (and its inverse) for a broader class of signals.
  9. understand the application of Fourier analysis to ideal filtering, amplitude modulation and sampling.
  10. be able to process continuous-time signals by first sampling and then processing the sampled signal in discrete-time.
  11. develop basic problem solving skills and become familiar with formulating a mathematical problem from a general problem statement.
  12. be able to use basic mathematics including calculus, complex variables and algebra for the analysis and design of linear time invariant systems used in engineering.
  13. develop a facility with MATLAB programming to solve linear systems and signal problems.

Text Book and Relevant Sections

These course resources make use of the following text:

Simon Haykin and Barry Van Veen,Signals and Systems 2005 JustAsk! Edition, John Wiley & Sons, Inc.

The following text book sections are covered.

  • Chapter 1: 1.4, 1.5, 1.6 (1.6.2, 1.6.3, 1.6.5, 1.6.6, 1.6.8), 1.8
  • Chapter 2: 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.14
  • Chapter 3: 3.2, 3.3, 3.5, 3.7, 3.8
  • Chapter 4: 4.2
  • Chapter 3: 3.9, 3.10, 3.11, 3.12, 3.14, 3.15, 3.16, 3.17, 3.18
  • Chapter 8: 8.2
  • Chapter 4: 4.5, 4.6

Supplemental Notes

  • Complex Numbers Review
  • Discrete-Time Sinusoids
  • Fourier Magnitude and Phase
  • System Properties

Problem Sets and Solutions

Relevant Text Sections Problem Set Solutions
1.4 1.1(c, d), 1.5 (a,c,d,e,f,g), 1.6, 1.7, 1.8, 1.9 (a,b,c,f,h), 1.43, 1.44, 1.45, 1.49 PDF
1.5 1.10, 1.11, 1.12, 1.13, 1.14 (a,b,c,d,e,f), 1.15, 1.51, 1.52(f), 1.53,
1.56 (b,c,d,f,k)		 PDF
1.6 1.17, 1.18, 1.22, 1.57, 1.58, 1.59, 1.61 PDF
1.8 1.26, 1.27, 1.28, 1.29, 1.31, 1.32, 1.33, 1.34, 1.35, 1.36, 1.64 (a,b,d,e,f,i,j), 1.68, 1.72, 1.73, 1.75, 1.76 PDF
2.2, 2.3, 2.4, 2.5 2.1, 2.2, 2.3, 2.5, 2.6, 2.32, 2.33 (a,c), 2.34(a,e,k), 2.39(a,b,n), 2.40(a,k,p) PDF, errata
2.6, 2.7, 2.8, 2.9 2.44, 2.47, 2.48, 2.49, 2.50(a,b,h), 2.51 PDF
3.2, 3.3, 3.5 3.7, 3.8, 3.9, 3.10, 3.50(a,b,f), 3.51, (a,b,c,e) PDF
3.7 3.14, 3.15, 3.54, 3.55 PDF
4.2, 3.9, 3.10 4.1 (a,b), 4.16 (a,d), 3.16 (a), 3.17 (a,d), 3.18(a), 3.20(b), 3.21(a) PDF
3.11, 3.12, 3.14, 3.15, 3.16, 3.17, 3.18 3.22, 3.23, 3.25, 3.26, 3.29, 3.34, 3.40(b), 3.41 3.43, (a), 3.44, 3.58 (a,b,e,g), 3.59 (a,c,e) PDF
8.2, 4.5, 4.6 8.1, 4.10, 4.12, 4.25(a) PDF

Note: Problem set solutions are courtesy of Sanjay Nair.

Signals and Systems Convolution Problems and Solutions

Source: https://www.comm.utoronto.ca/~dkundur/course/signals-and-systems/