These videos have been developed to support EGR 433: Transforms and Systems Modeling taught in the Engineering Department at the Polytechnic Campus of Arizona State University. Videos on engineering topics (currently circuit analysis and statics) are available at http://www.engineeringvideos.org.

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Contents

1. 1 Introduction to Signals and Systems
2. 2 Signals
   1. 2.1 Continuous-time Signals
   2. 2.2 Discrete-time Signals
3. 3 System Properties
   1. 3.1 System State
   2. 3.2 Linearity
   3. 3.3 Time Invariance
   4. 3.4 Stability
   5. 3.5 System Response
   6. 3.6 Discrete-time System Properties
4. 4 LTI Systems and Convolution
   1. 4.1 Continuous-time
   2. 4.2 Discrete-time
5. 5 Laplace Transform
   1. 5.1 Overview of the Laplace Transform
   2. 5.2 Computing the Inverse Transform
   3. 5.3 Laplace Transform Domain Circuit Analysis
   4. 5.4 System Analysis Example
   5. 5.5 Analysis of Feedback Control Systems
6. 6 Fourier Analysis
   1. 6.1 Continuous-time Fourier Series
   2. 6.2 Discrete-time Fourier Series
   3. 6.3 Fourier Transforms
   4. 6.4 Discrete-time Fourier Transform
   5. 6.5 The Sampling Theorem
7. 7 Modeling and Simulation
8. 8 Other Engineering Topics
9. 9 Acknowledgment

 

Introduction to Signals and Systems

• Signals and Systems Introduction An introduction to the basic ideas of systems and signals.
• Signals and Systems HVAC Example An example of considering a heating/cooling system from the perspective of systems and signals.
• Signals and Systems Car Example An example of considering the cruise control system of a car from the perspective of systems and signals.
• System Properties: Stability and Causality An introduction to SISO/MIMO, stability, and causality as properties of a system.
• System Properties: Linearity and Time Invariance An introduction to linearity and time invariance as properties of a system.
• Discrete-Time Systems and Signals Introduction A conceptual introduction to discrete-time systems and signals

Signals

Continuous-time Signals

• Signals: Continuous-Time Unit Step and Delta Introduces the continuous-time unit step function and the delta function.
• Integrals with Continuous-Time Delta Functions How to work integrals that contain delta functions
• Signals: Sinusoids and Real Exponentials Introduces sinusoids and real exponentials.
• Signals: Complex Exponentials Introduces complex exponentials.
• Operations on Signals: Scaling and Addition Demonstrates how scaling and addition of signals works.
• Operations on Signals: Time Shifting Demonstrates how time shifting a signal works.
• Signal Properties: Periodic An introduction to periodic signals
• Signal Properties: Even and Odd An introduction to even and odd signals
• Signal Properties: Energy and Power An introduction to energy and power signals

Discrete-time Signals

• DT Signals-Unit Step and Delta Introduces the discrete-time unit step and delta functions.
• DT Signals-Real Exponentials Introduces discrete-time real exponential signals.
• DT Signals-Real Sinusoids Introduces discrete-time sinusoidal (sine and cosine) signals.
• DT Signals-Complex Exponentials Introduces discrete-time complex exponential signals.
• DT Signal Property-Periodic Introduces the definition of periodic for a discrete time signal and shows six examples of determining whether a signal is periodic, and if it is, determining its fundamental period.
• DT Signal Property-Even and Odd Introduces the definitions of even and odd for a discrete time signal and shows seven examples of determining whether a signal is even, odd, or neither. Signals that are neither even nor odd are decomposed into their even and odd components.
• DT Signal Property-Energy and Power Introduces the energy and average power for a discrete-time signal and shows four examples of computing energy and/or power.
• DT Signals-Time Shifting and Reversal Shows how to time shift and reverse discrete-time signals.
• Summing Geometric Series Shows how to sum a geometric series.

System Properties

System State

• System State Introduction Introduction to the concept of system state and how you might figure out what the state of a system is.

Linearity

• Linearity Definition Explains how homogeneity and additivity, two attributes of a system, determine whether or not the system is linear.
• Linearity of a Gain Example An example of determining whether a gain (a system that scales the amplitude of a signal) is a linear system.
• Linearity of a Squarer Example An example of determining whether a system whose output is the square of its input is a linear system.
• Linearity and Time Invariance of an RC Circuit Example An example of determining whether an RC circuit is linear and time invariant. This provides some conceptual ideas about the role of initial conditions in systems whose dynamics are described by differential equations.

Time Invariance

• Time Invariance Introduction An introduction to the difference between time varying and time invariant systems.
• Time Invariance Mathematics How to mathematically determine whether a system is time varying or time invariant.

Stability

• System Stability How to determine whether a system is BIBO stable.

System Response

• System Response Characterization How we characterize the response of a system to a unit step function input in terms of steady-state error, rise and fall time, overshoot, and settling time.

Discrete-time System Properties

• DT System Properties Example: y[n] = x[-n] Shows how to determine whether the system defined by the equation y[n] = x[-n] is 1) memoryless, 2) time invariant, 3) linear, 4) causal, and 5) stable.
• DT System Properties Example: y[n] = nx[n] Shows how to determine whether the system defined by the equation y[n] = nx[n] is 1) memoryless, 2) time invariant, 3) linear, 4) causal, and 5) stable.
• DT System Properties Example: y[n] = x[n] - x[n-1] Shows how to determine whether the system defined by the equation y[n] = x[n] - x[n-1] is 1) memoryless, 2) time invariant, 3) linear, 4) causal, and 5) stable.
• DT System Properties Example: y[n] = x[n]x[n+1] Shows how to determine whether the system defined by the equation y[n] = x[n]x[n+1] is 1) memoryless, 2) time invariant, 3) linear, 4) causal, and 5) stable.

LTI Systems and Convolution

Continuous-time

• LTI Systems Introduction An explanation of how an LTI (Linear Time-Invariant) system is completely specified in terms of its impulse response, transfer function, or frequency response.
• Convolution and LTI Systems Shows how the response of an LTI system to an arbitrary input is obtained as the convolution of the impulse response of the system with the input.
• Convolution Properties A short explanation that convolution is commutative, associative, and distributive. In addition, an explanation of what happens when you convolve a signal with a delta function.
• Convolution Example: Unit Step with Exponential An example of computing the continuous time convolution of a unit step and an exponential signal.
• Convolution Example: Two Rectangular Pulses An example of computing the continuous-time convolution of two rectangular pulses.

Discrete-time

• DT LTI System Response: Convolution Shows how the response of a discrete-time LTI (Linear Time-Invariant) system to an arbitrary input is obtained as the convolution of the impulse response of the system with the input.
• DT Convolution-Simple Example Shows how to compute the discrete-time convolution of two simple waveforms.
• DT Convolution-Two Rectangular Pulses Shows how to compute the discrete-time convolution of two rectangular pulse waveforms.
• DT Convolution-Two Exponentials Shows how to compute the discrete-time convolution of two exponential signals.
• DT Convolution-Periodic Signals Shows how two discrete-time periodic signals are convolved through an example of convolving a square wave with itself.

Laplace Transform

Overview of the Laplace Transform

• Laplace Transform Introduction An introduction to the unilateral and bilateral Laplace transforms.
• Laplace Transform Properties Introduction to the following properties of the Laplace transform: linearity, time delay, time derivative, time integral, and convolution.
• Laplace Transform Example: Unit Step Computing the Laplace transform of the unit step function using the integral definition of the Laplace transform. Part 2 of the video uses the unit step function to illustrate different regions of convergence.
• Laplace Transform Example: Exponential Computing the Laplace transform of an exponential function using the integral definition of the Laplace transform.
• Pole-Zero Plots An introduction to pole-zero plots.
• Stability and Pole Locations Describes how the stability of an LTI-system can be determined from the pole locations of its transfer function.

Computing the Inverse Transform

• Inverse Laplace Transform Introduction Introduces the process of computing an inverse Laplace transform using a partial fraction expansion.
• Inverse Laplace Transform How to compute inverse Laplace transforms

Laplace Transform Domain Circuit Analysis

• Laplace Domain Circuit Analysis Introduces analysis of circuits with capacitors and inductors in the Laplace domain.

System Analysis Example

• Laplace Transform System Analysis: DC Motor Example Using the Laplace transform to analyze a simple DC motor model.

Analysis of Feedback Control Systems

• Feedback Control Introduction Presents the basic structure of a feedback control system and its transfer function.
• Simple Feedback Control Example Uses the transfer function of a feedback control system to investigate the effect of feedback on system behavior.
• Feedback and Waveform Parameters Illustrates how feedback can change waveform parameters including steady-state error, rise time, settling time, and overshoot.

Fourier Analysis

• Fourier Analysis Introduction Introduction to some applications and concepts associated with frequency domain (Fourier) analysis.
• Fourier Analysis Overview Introduces the different transforms used in frequency domain analysis.
• Frequency Spectrum Graphing with Matlab How to use Matlab to compute and graph the frequency spectrum of a sampled time signal.

Continuous-time Fourier Series

• Fourier Series Introduction An introduction to the concepts behind Fourier series
• Fourier Series with Complex Exponentials Definition of the complex exponential Fourier Series and properties of its coefficients
• Fourier Series with Sines and Cosines Definition of the trigonometric Fourier Series, properties of its coefficients, and its relation to the complex exponential Fourier Series
• Fourier Series Square Wave Example Computing the complex exponential Fourier series coefficients for a square wave.
• Fourier Series Example-Arbitrary Square Wave Computes the Fourier series coefficients of a square wave with arbitrary period T, amplitude A, and duty cycle D.
• Fourier Series-Rectified Sine Wave Computes the Fourier series coefficients of a rectified sine wave; the computation is done entirely using Fourier series properties and Fourier series coefficients computed in previous videos. The DTFS properties used include  multiplication, time shifting, linearity, and frequency shifting.

Discrete-time Fourier Series

Fourier Transforms

• Introduction to the Fourier Transform Introduces the mathematical definition of the Fourier transform as well as magnitude and phase spectra.
• Fourier Transform Rectangular Pulse Example Computing the Fourier transform of a rectangular pulse using integration.
• AM Modulation and Demodulation Example This video uses properties of the Fourier transform to explain modulation and demodulation inside a simple AM radio system.
• Fourier Characterization of Pulse Width Modulation Example This video investigates pulse width modulation in the frequency domain.

Discrete-time Fourier Transform

• Introduction to the DT Fourier Transform Introduces the discrete-time Fourier Transform and shows two simple examples of computing the DT Fourier Transform.
• DT Fourier Transform-Exponential Computes the discrete-time Fourier transform of an exponential signal.
• DT Fourier Transform-Rectangular Pulse Computes the discrete-time Fourier transform of a rectangular pulse.
• DT Fourier Transform-Ideal Filters Computes the impulse response of ideal low-pass and high-pass discrete-time filters using the frequency shifting property.
• DT Fourier Transform-Triangle Wave Computes the discrete-time Fourier transform of a triangle wave using the convolution property.
• DT Fourier Transform-Rectangular Window Computes the discrete-time Fourier transform of a cosine wave that has been windowed by a rectangular window. This is done using the multiplication property.
• DT Fourier Transform-Filter Output Computes the output of a filter in response to a periodic square wave input by using the frequency response and the discrete-time Fourier transform. The relationship between DT Fourier series coefficients and DT Fourier transform is also used.

The Sampling Theorem

• Sampling Theorem Introduction A conceptual introduction to the sampling theorem that gives the minimum sampling rate necessary for a signal.
• Sampling Theorem Derivation Deriving the sampling theorem using the properties of Fourier transforms.

Modeling and Simulation

• Simulink Simulation of a Simple System An example of how to use Simulink to simulate a simple system whose dynamics are described by a first-order constant coefficient differential equation.
• Simulink Modeling of a DC Motor An example of the process of modeling a dynamical system consisting of a DC motor and a rotating load and then implementing the model in Simulink.
• Simulink: Checking Linearity of a Model An example of determining whether a Simulink system model is linear using simulation.
• Simulink: Checking Time Invariance of a Model An example of determining whether a Simulink system model is time invariant using simulation.

Other Engineering Topics

Videos on circuit analysis and static mechanics are available at http://www.engineeringvideos.org/circuit-analysis.

Acknowledgment

The idea for these videos came from the magnificent work of Salman Khan; in this case, imitation truly is the sincerest form of flattery.