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Week 1
Definition of Signals and Systems Additivity
Additivity Examples Homogeneity Definition
Homogeneity Examples Shift invariance Definition
Real Life Example The RC Circuit
The RC Circuit Variations for Superposition and Shift Invariance
Two Challenging Problems for the Student
Week 2
The Professor Responds: Week 1
Constructing Continuous Function with Narrow Pulses
Sifting Property of Narrow Pulses
Combining Narrow Pulses into a Function
Impulse Response and the Consequence of Additivity Homogeneity and Shift Invariance
Invoking all three together Additivity Homogeneity Shift Invariance
Relation Between Unit Step and Unit Impulse
Taking the Derivative of a Unit Step Actually
Unit Impulse Response of RC Circuit
Interpretation of Unit Impulse Response of RC Circuit
Week 3
Additivity in Discrete Systems
Homogeneity and Shift Invariance
Causal and Stable Discrete Systems
The Unit Impulse and the Unit Impulse Response in Discrete Systems
The Impulse Response for Discrete LSI System Characterization
Consequence of Associativity and Commutativity Permuting LSI Sytems
Causality of a Linear Shift Invariant System from its Impulse Response
Investigating Stability from the Impulse Response
Absolute Summability and Absolute Integrability as a Sufficient Condition for Stability
Absolute Summability as a Necessary Condition of Stability
Week 4
Input Output Relationship in LSI Systems the Impulse Response
Illustrating One Output Point Calculation in Discrete LSI Systems
The Train Platform Analogy for Discrete Convolution
Commutativity of Continuous and Discrete Convolution Setting up a Continuous Convolution Example
Evaluating a Continuous Time Example Convolution
Sketching the Example Convolution and Setting up a More Difficult Exercise
Commutativiy and Associativity of Convolution
Completing the Proof of Associativity of Convolution
Consequences of Commutativity and Associativity of Convolution
The Typical LSI System Description and a Conclusion of Module One
Week 5
Sinusoids as Naturally Generated Voltages
Mathematical Convenience of Sinusoid
The Problem of Dealing with a Phase Change
How we Solve it with Rotating Complex Numbers or Phasors
Phase Changes in a Sinusoid and in a Phasor
One Phasor going through a Linear Shift Invariant System
One Sinusoid going through a Linear Shift Invariant System
Periodic Inputs to Shift Invariant Systems
Recapitulating Finite Dimensional Vectors
Examples of Dot Product Calculations
Inner Products of Infinite Length Discrete and Continous Signals
Sinusoids with the Same Period
Week 6
Reflection on What we are Doing
Orthogonal Unit Vectors at a Particular Angular Frequency
Finding the two Components at a Particular Frequency
Decomposition of a Symmetric Square Wave
Exercises of Decomposition to be Worked Out by the Student
Complex Exponential Fourier Series Decomposition
Going from Complex Exponential to Sinusoidal Decomposition
A Periodic Input to a Simple RC Circuit
Two Small Points about Fourier Series Decomposition
Periodic Inputs to a General Linear Shift Invariant System
Week 7
Introducing the Fourier Transform
Seeking an Inverse for the Fourier Transform
Completing Reconstruction of h(t)
Orthogonality of Infinite Length Phasors
The Fourier Transform and its Inverse
Fourier Transform of a Rectangular Pulse
Linearity of the Fourier Transform
Time Shift and Modulation with the Fourier Transform
The Convolution Property of the Fourier Transform
Week 8
Interpreting Transforms as a Change of Paradigm
Introducing Duality in the Fourier Transform
Using Duality to give an Example of How Convolution can be Simplified
The Multiplication Theorem or The Fourier Transform of a Product of Two Functions
Deriving Parsevals Theorem from The Multiplication Theorem
The Parsevals Theorem Interpreted as an Inner Product Invariance
More on the Interpretation of Parsevals Theorem
Parseval’s Theorem Interpreted for Energy Spectral Density
The Dual Pair of Differentiation and Multiplication by the Independent Variable
More on the Differentiation Property of The Fourier Transform