Unit 2: Signals and Systems

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This unit focuses on analyzing the behaviors of systems. General principles for developing mathematical models of systems are developed in the context of using feedback in control systems, with tangible applications in robotic steering.

 Diagram of a vehicle attempting to follow a straight line, and its responses to steering.

Topics difference equations, system functions, controllers
Lab Exercises robotic steering
Theme modeling complex systems, analyzing behaviors

Looking for something specific in this course? The Resource Index compiles links to most course resources in a single page.

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Session Overview

Diagram showing behavior of the robot in a hallway. 

In this session, we’ll learn about poles: what they are, how to find them, and what they tell us about the long-term behavior of an LTI system.

The overview handout provides a more detailed introduction, including the big ideas of the session, key vocabulary, what you should understand (theory) and be able to do (practice) after completing this session, and additional resources.

Session Content

Readings

Read sections 5.4-5.7 of the course notes.

Lecture Video

Watch the lecture video. The handout and slides present the same material, but the slides include answers to the in-class questions.

About this Video

Introduction to poles, which provide a way to characterize the behavior of a system in terms of a mathematical description as a system function.

Recitation Video

These videos have been developed for OCW Scholar, and are designed to supplement the lecture videos.

Session Activities

The problems in the tables below are taken from the 6.01 Online Tutor, an interactive environment that is not available on OCW. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

Software Lab

PROBLEM # QUESTIONS
5.1.1 System function class (PDF)
5.1.2 Cascading systems (PDF)
5.1.3 Cascade of system functions (PDF)
5.1.4 Feedback subtract expression (PDF)
5.1.5 FeedbackSubtract code (PDF)

Design Lab

Do problem 5.2.1 before the design lab.

PROBLEM # QUESTIONS
5.2.1 Wall finder system function (PDF)
5.3.2 Wall follower system function (PDF)
5.3.4 Wall follower model (PDF)

Additional Exercises

PROBLEM # QUESTIONS
5.4.1 System behavior (PDF)

Check Yourself

Nano-Quiz

Nano-quiz problems and solutions are taken from a previous version of the 6.01 Online Tutor. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

Homework

PROBLEM # QUESTIONS
5.5.1 Modeling the controller/sensor (PDF)
5.5.2 Modeling the plant (PDF)
5.5.3 Putting it together (PDF)
5.5.4 Analyzing the system (PDF)

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Session Overview

Diagram showing the flow of water through a branching point. 

In this session, we look at how to put all the components of the Signals and Systems unit together. We’ll model complex systems by breaking them down into smaller systems and combining them. Then we’ll solve for the poles of those complex systems and use them to predict long-term behavior. We can also look at a plot of poles in our system to discover how changing parameters of a small part of a complex system affects the overall behavior of that system.

The overview handout provides a more detailed introduction, including the big ideas of the session, key vocabulary, what you should understand (theory) and be able to do (practice) after completing this session, and additional resources.

Session Content

Readings

Review sections 5.1-5.7 of the course notes, which were assigned in the last two sessions. The worked examples in section 5.8 may also be useful.

Lecture Video

Watch the lecture video. The handout and slides present the same material, but the slides include answers to the in-class questions.

About this Video

System functions provide a summary of information that help optimize the design of a control system. Poles are discussed further, based on their location on the unit circle.

Session Activities

The problems in the tables below are taken from the 6.01 Online Tutor, an interactive environment that is not available on OCW. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

Design Lab

PROBLEM # QUESTIONS
6.1.1 Proportional plus delay (PDF)
6.1.3 Proportional plus angle (PDF)

Optional Exercises

PROBLEM # QUESTIONS
6.2.1 Control system analysis (PDF)
6.2.2 System representations (PDF)

Check Yourself

Nano-Quiz

Nano-quiz problems and solutions are taken from a previous version of the 6.01 Online Tutor. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

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Session Overview

 Diagram of feedback addition composition.

In the last session, we demonstrated the versatility of state machines and introduced signals and systems. In this session, we will focus on linear time-invariant (LTI) systems. We’ll be able to represent LTI systems using state machines, and introduce other ways to represent LTI systems. Once we know that a system is LTI, we can use what we know about linear time-invariance to analyze and predict the behavior of the system.

The overview handout provides a more detailed introduction, including the big ideas of the session, key vocabulary, what you should understand (theory) and be able to do (practice) after completing this session, and additional resources.

Session Content

Readings

Read sections 5.1-5.4 of the course notes.

Recitation Video

These videos have been developed for OCW Scholar, and are designed to supplement the lecture videos.

Session Activities

The problems in the tables below are taken from the 6.01 Online Tutor, an interactive environment that is not available on OCW. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

Software Lab

PROBLEM # QUESTIONS
4.1.1 Constructing signals (PDF)
4.1.2 Step signal (PDF)
4.1.3 Subclasses of signals (PDF)
4.1.4 Additional subclasses (PDF)
4.1.5 PolyR on signals (PDF)

Design Lab

Do problem 4.2.1 before the design lab.

PROBLEM # QUESTIONS
4.2.1 Difference equations (PDF)
4.3.1 Wall finder (PDF)
4.3.3 State machine composition (PDF)
4.3.5 Wall finder state machine (PDF)
4.3.6 Wall finder gains (PDF)

Additional Exercises

PROBLEM # QUESTIONS
4.4.1 LTISM (PDF)
4.4.2 Implement LTISM (PDF)
4.4.3 TransducedSignal (PDF)

Check Yourself

Nano-Quiz

Nano-quiz problems and solutions are taken from a previous version of the 6.01 Online Tutor. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

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Session Overview

 Block diagram of a discrete-time system.

Any model of the world we want to study, that we address as a system, we also probably want to observe in some way (this is probably the reason we made the model in the first place). The information we choose to observe is the signal. Based on what we know about different kinds of signals and systems, we can choose what information to collect as a signal, and make effective long-term predictions about our system based on that signal.

The overview handout provides a more detailed introduction, including the big ideas of the session, key vocabulary, what you should understand (theory) and be able to do (practice) after completing this session, and additional resources.

Session Content

Readings

Read section 4.2 of the course notes.

Lecture Video

Watch the lecture video. The handout and slides present the same material, but the slides include answers to the in-class questions.

About this Video

Introduction to signals and systems, focusing on multiple representations of discrete-time systems: difference equations, block diagrams, and operator representations.

Session Activities

The problems in the tables below are taken from the 6.01 Online Tutor, an interactive environment that is not available on OCW. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

Software Lab

PROBLEM # QUESTIONS
3.1.1 Simulating cascade (PDF)
3.1.2 Cascading machines (PDF)
3.1.3 Function machines (PDF)
3.1.4 Combining accounts (PDF)
3.1.5 Sequential combinations (PDF)
3.1.6 Feedback SM (PDF)

Design Lab

Additional Exercises

PROBLEM # QUESTIONS
3.3.1 Map (PDF)
3.3.2 Indexing nested lists (PDF)
3.3.3 Finding systems (PDF)
3.3.4 Representations (PDF)

Check Yourself

Nano-Quiz

Nano-quiz problems and solutions are taken from a previous version of the 6.01 Online Tutor. Do not try to answer these questions in the PDF files; answers will not be checked, and cannot be submitted.

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