Fourier Transform

1. Introduction to Fourier Transformation

• Definition and Importance

• Analogy with Concert Sounds

2. Fourier Series

• Basics

• Analogy with Smoothie Ingredients :

• Prerequisites : Complex Numbers

• Sine and Cosine Waves

• Integrals

3. Fourier Transform

• Basics

• Formula Breakdown

• Example with Simple Signal

• Inverse Fourier Transform

4. Practical Applications

• Signal Processing

• Image Processing

• Communications

5. Discrete Fourier Transform (DFT)

• Definition and Formula

• Inverse DFT

• Example

6. Fast Fourier Transform (FFT)

• Efficiency and Algorithm

• Practical Example

7. Advanced Concepts

• Windowing

• Spectral Leakage

• Zero Padding

• Sampling Theorem

8. Further Applications and Nuances

• Audio Signal Enhancement

• Mathematical Concepts:

• Lebesgue Integrable Functions

• Square Integrable Functions

• Unitarity

• Angular Frequency

9. Summary of Popular Forms of Fourier Transform

• Continuous Fourier Transform (CFT)

• Discrete Fourier Transform (DFT)

• Fast Fourier Transform (FFT)

• Short-Time Fourier Transform (STFT)

10. Generalizations and Alternatives

• Fourier Transform on Euclidean Space

• Uncertainty Principle

• Sine and Cosine Transforms

• Spherical Harmonics

• Restriction Problems

• Fourier Transform on Function Spaces

• Tempered Distributions

• Fourier–Stieltjes Transform

• Locally Compact Abelian Groups

• Gelfand Transform

• Compact Non-Abelian Groups

11. Applications

• Differential Equations

• Fourier-Transform Spectroscopy

• Quantum Mechanics

• Signal Processing

Introduction to Fourier Transformation

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