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- Fourier Series
- Introduction
- Inner Product
- Introduction
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- Inner Product For Functions
- Basis
- Fourier Series Proof
- Continuous Time Fourier Series Calculator
- Discrete Time Fourier Series
- Fourier Transform
- DFT
- Simple Visualizations
- Signal & System
- 1.0 Introduction
- 1.1 Continuous Time & Discrete Time Signals
- 1.2 Transformations Of The Independent Variable
- 1.2.1 Examples Of Transformations Of The Independent Variable
- 1.2.2 Periodic Signals
- 1.2.3 Even & Odd Signals
- 1.3 Exponential & Sinusoidal Signals
- 1.3.1 Continuous Time Complex Exponential & Sinusoidal Signals
- 1.3.2 Discrete Time Complex Exponential & Sinusoidal Signals
- 1.3.3 Periodicity Properties Of Discrete Time Complex Exponentials
- 1.4 The Unit Impulse & Unit Step Functions
- 1.4.1 The Discrete Time Unit Impulse & Unit Step Sequences
- 1.4.2 The Continuous Time Unit Step & Unit Impulse Functions
- 1.5 Continuous Time & Discrete Time Systems
- 1.6 Basic System Properties
- 1.6.1 Systems With & Without Memory
- 1.6.2 Invertibility & Inverse Systems
- 1.6.3 Causality
- 1.6.4 Stability
- 1.6.5 Time Invariance
- 1.6.6 Linearity
- 1.7 Summary
- 1.8 Problems
- 2.0 Introduction
- 2.1 Discrete Time LTI Systems The Convolution Sum
- 2.1.1 The Representation Of Discrete Time Signals In Terms Of Impulses
- 2.1.2 The Discrete Time Unit Impulse Response & The Convolution Sum Representation Of LTI Systems
- 2.2 Continuous Time LTI Systems The Convolution Integral
- 2.2.1 The Representation Of Continuous Time Signals In Terms Of Impulses
- 2.2.2 The Continuous Time Unit Impulse Response & The Convolution Integral Representation Of LTI Systems
- 2.3 Properties Of Linear Time Invariant Systems
- 2.3.1 The Commutative Property
- 2.3.2 The Distributive Property
- 2.3.3 The Associative Property
- 2.3.4 LTI Systems With & Without Memory
- 2.3.5 Invertibility Of LTI Systems
- 2.3.6 Causality For LTI Systems
- 2.3.7 Stability For LTI Systems
- 2.3.8 The Unit Step Response Of An LTI System
- 2.4 Causal LTI Systems Described By Differential & Difference Equations
- 2.4.1 Linear Constant Coefficient Differential Equations
- 2.4.2 Linear Constant Coefficient Difference Equations
- 2.4.3 Block Diagram Representations Of First Order Systems Described By Differential & Difference Equations
- 2.5 Singularity Functions
- 2.5.1 The Unit Impulse As An Idealized Short Pulse
- 2.5.2 Defining The Unit Impulse Through Convolution
- 2.5.3 Unit Doublets & Other Singularity Functions
- 2.6 Summary
- 3.0 Introduction
- 3.1 Historical Perspective
- 3.2 The Response Of LTI Systems To Complex Exponentials
- 3.3 Fourier Series Representation Of Continuous Time Periodic Signals
- 3.3.1 Linear Combinations Of Harmonically Related Complex Exponentials
- 3.3.2 Determination Of The Fourier Series Representation Of A Continuous Time Periodic Signal
- 3.4 Convergence Of The Fourier Series
- 3.5 Properties Of Continuous Time Fourier Series
- 3.5.1 Linearity
- 3.5.2 Time Shifting
- 3.5.3 Time Reversal
- 3.5.4 Time Scaling
- 3.5.5 Multiplication
- 3.5.6 Conjugation & Conjugate Symmetry
- 3.5.7 Parseval Relation For Continuous Time Periodic Signals
- 3.5.8 Frequency Shifting
- 3.5.9 Periodic Convolution
- 3.5.10 Differentiation
- 3.5.11 Integration
- 3.5.12 Even Odd Decomposition Of Real Signals
- 3.5.13 Examples
- 3.6 Fourier Series Representation Of Discrete Time Periodic Signals
- 3.6.1 Linear Combinations Of Harmonically Related Complex Exponentials
- 3.6.2 Determination Of The Fourier Series Representation Of A Discrete Time Periodic Signal
- 3.7 Properties Of Discrete Time Fourier Series
- 3.7.1 Linearity
- 3.7.2 Time Shifting
- 3.7.3 Time Reversal
- 3.7.4 Time Scaling
- 3.7.5 Multiplication
- 3.7.6 Conjugation & Conjugate Symmetry
- 3.7.7 Parseval Relation For Discrete Time Periodic Signals
- 3.7.8 Frequency Shifting
- 3.7.9 Periodic Convolution
- 3.7.10 First Difference
- 3.7.11 Running Sum
- 3.7.12 Even Odd Decomposition Of Real Signals
- 3.7.13 Examples
- 3.8 Frequency Response Of Fourier Series & LTI Systems
- 3.9 Filtering
- 3.10 Examples Of Continuous Time Filters Described By Differential Equations
- 3.11 Examples Of Discrete Time Filters Described By Difference Equations
- 3.12 Summary
- 4.0 Introduction
- 4.1 Representation Of Aperiodic Signals The Continuous Time Fourier Transform
- 4.1.1 Development Of The Fourier Transform Representation Of An Aperiodic Signal
- 4.1.2 Convergence Of Fourier Transforms
- 4.1.3 Examples Of Continuous Time Fourier Transforms
- 4.2 The Fourier Transform For Periodic Signals
- 4.3 Properties Of The Continuous Time Fourier Transform
- 4.3.1 Linearity
- 4.3.2 Time Shifting & Frequency Shifting
- 4.3.3 Conjugation & Conjugate Symmetry
- 4.3.4 Differentiation & Integration
- 4.3.5 Time & Frequency Scaling
- 4.3.6 Duality
- 4.3.7 Parsevals Relation
- 4.4 The Convolution Property
- 4.5 The Multiplication Property
- 4.6 Tables Of Fourier Properties & Of Basic Fourier Transforms
- 4.7 Systems Characterized By Linear Constant Coefficient Differential Equations
- 4.8 Summary
- 5.0 Introduction
- 5.1 Representation Of Aperiodic Signals The Discrete Time Fourier Transform
- 5.1.1 Development Of The Discrete Time Fourier Transform
- 5.1.2 Examples Discrete Time Fourier Transform
- 5.1.3 Convergence Issues Associated With The Discrete Time Fourier Transform
- 5.2 The Fourier Transform For Periodic Signals
- 5.3 Properties Of The Discrete Time Fourier Transform
- 5.3.1 Periodicity Of The Discrete Time Fourier Transform
- 5.3.2 Linearity Of The Fourier Transform
- 5.3.3 Time Shifting & Frequency Shifting
- 5.3.4 Conjugation & Conjugate Symmetry
- 5.3.5 Differencing & Accumulation
- 5.3.6 Time Reversal
- 5.3.7 Time Expansion
- 5.3.8 Differentiation In Frequency
- 5.3.9 Parsevals Relation
- 5.4 The Convolution Property
- 5.5 The Multiplication Property
- 5.6 Tables Of Fourier Transform Properties & Basic Fourier Transform Paris
- 5.7 Duality
- 5.8 Systems Characterized By Linear Constant Coefficient Difference Equations
- 5.9 Summary
- 6.0 Introduction
- 6.1 The Magnitude & Phase Representation Of The Fourier Transform
- 6.2 The Magnitude Phase Representation Of The Frequency Response Of Lti Systems
- 6.3 Time Domain Properties Of Ideal Frequency Selective Filters
- 6.4 Time Domain & Frequency Domain Aspects Of Nonideal Filters
- 6.5 First Order & Second Order Continuous Time Systems
- 7.0 Introduction
- 7.1 Representation Of A Continuous Time Signal By Its Samples The Sampling Theorem
- 7.2 Reconstruction Of A Signal From Its Samples Using Interpolation
- 7.3 The Effect Of Undersampling Aliasing
- 7.4 Discrete Time Processing Of Continuous Time Signals
- 7.5 Sampling Of Discrete Time Signals
- 7.6 Summary
- 8.0 Introduction
- 8.1 Complex Exponential & Sinusoidal Amplitude Modulation
- 8.1.1 Amplitude Modulation With A Complex Exponential Carrier
- 8.1.2 Amplitude Modulation With A Sinusoidal Carrier
- 8.2 Demodulation For Sinusoidal Amplitude Modulation
- 8.3 Frequency Division Multiplexing
- 8.4 Single Sideband Sinusoidal Amplitude Modulation
- 8.5 Amplitude Modulation With A Pulse Train Carrier
- 8.6 Pulse Amplitude Modulation
- 8.6.1 Pulse Amplitude Modulated Signals
- 8.6.2 Intersymbol Interference In PAM Systems
- 8.6.3 Digital Pulse Amplitude & Pulse Code Modulation
- 8.7 Sinusoidal Frequency Modulation
- 8.7.1 Narrowband Frequency Modulation
- 8.7.2 Wideband Frequency Modulation
- 8.7.3 Periodic Square Wave Modulating Signal
- 8.8 Discrete Time Modulation
- 8.9 Summary
- 9.1 The Laplace Transform
- 9.2 The Region Of Convergence For Laplace Transform
- 9.3 The Inverse Laplace Transform
- 9.4 Geometric Evaluation Of The Fourier Transform From The Pole Zero Plot
- 9.5 Properties Of The Laplace Transform
- 9.5.1 Linearity Of The Laplace Transform
- 9.5.2 Time Shifting
- 9.5.3 Shifting In The S Domain
- 9.5.4 Time Scaling
- 9.5.5 Conjugation
- 9.5.6 Convolution Property
- 9.5.7 Differentiation In The Time Domain
- 9.5.8 Differentiation In The S Domain
- 9.5.9 Integration In The Time Domain
- 9.5.10 The Initial & Final Value Theorem
- 9.7 Analysis & Characterization Of Lti Systems Using The Laplace Transform
- 9.8 System Function Algebra & Block Diagram Representations
