Lecture on Fourier Transform

Introduction

• Goal: Introduction to Fourier Transform
• Idea: Decomposing frequencies from sound
• Extensions: Beyond sound into various areas of math & physics

Sound and Frequency

• Pure Tone (A, 440 Hz): Air pressure oscillates at 440 times/second
• Lower Pitch (D): Fewer oscillations per second
• Combined Notes: Resulting pressure vs. time graph is complex
• Microphone Recording: Captures final summed air pressure

Central Question

• Objective: Decompose complex signal into pure frequencies
• Analogy: Unmixing stirred paint colors

Building the Mathematical Machine

• Pure Signal: Example with 3 beats/sec, limited time graph (0 to 4.5 sec)
• Wrapping Around Circle: Graph height => Vector length
  • Vector Rotation: 2 sec = 1 rotation
  • Two Frequencies: Signal & winding frequency
• Winding Frequency Adjustment: Faster or slower rotations
• Special Case: Winding frequency matches signal frequency
• Mass Analogy: Center of mass shifts based on winding frequency
• Plot: Winding frequency vs. x-coordinate of center of mass

Simplified Almost Fourier Transform

• Graph Transformation: Keeping track of variations and spikes
• New Signal Example: 2 beats/sec signal analyzed similarly
• Multiple Frequencies: Identifying and separating complex signals

Practical Applications

• Sound Editing: Filtering unwanted frequencies using Fourier transform
• Inverse Fourier Transform: Reconstructing original signal

Mathematical Foundation

• Complex Numbers: Interpretation in the complex plane
• Euler’s Formula: Basis for winding graph rotation
• Integral Approach: Averaging points on wound up graph

Defining Fourier Transform

• Expression: Integral of function tied to frequency
• Scalability: Longer signals increase Fourier transform magnitude
• Summary: Fourier transform provides frequency vs. intensity function (ɛ→ ĝ(f))

Key Observations

• Complex Output: Includes real & imaginary components
• Infinite Integral Bounds: Considering all time intervals

Future Directions

• Upcoming Topics: Broader mathematical applications
• Subscription Prompt: Encourages staying tuned for more

Sponsorship Puzzler

• Jane Street Problem: Proving the convexity of set D from set C
• Recruitment: Highlights Jane Street’s focus on intellectual curiosity