Fast Fourier Transform (FFT) Lecture Notes

Introduction to FFT

Importance: Used in various applications including video streaming, radar, sonar, 5G, and Wi-Fi.
Historical Context: Discovered during attempts to detect covert nuclear tests; could have influenced nuclear arms race.

Assumptions: Initial belief that arms race was inevitable after Hiroshima and Nagasaki bombings; turned out to be more complex.
Baruch Plan: Proposal by the US to decommission nuclear weapons in exchange for a commitment from other nations to refrain from developing nukes; rejected by the Soviets.

Testing Locations: Conducted in remote areas like the Arctic and South Pacific; Nevada became a key site for US testing.
Thermonuclear Weapons: Transition from fission to thermonuclear bombs, significantly increasing power.
Bikini Atoll Incident (1954): Unexpectedly high yield of 15 megatons led to widespread radioactive fallout; public outcry led to calls for a ban on nuclear testing.

Public and Political Response: Modern peace sign emerged for nuclear disarmament; negotiations began in Geneva for a comprehensive test ban.
Detection Challenges: Difficulty in verifying underground nuclear tests versus natural earthquakes led to partial bans.

Detection Methods: Atmospheric tests easily detectable; underwater tests via hydrophones; underground tests difficult due to radiation containment.
Seismometer Use: Scientists proposed using seismometers to detect vibrations from nuclear explosions; challenge was differentiating between explosions and earthquakes.
Fourier Transform's Role: Necessary to analyze seismometer signals to determine frequency components, requiring FFT for practicality.

Basic Concept: Decomposing signals into sine waves; correlation with sine waves indicates presence in the original signal.
Discrete Fourier Transform (DFT): Real-world signals are finite; necessitates a discrete approach to Fourier transform.
Calculating DFT: Requires significant computational resources; traditional methods become impractical for large datasets.

Discovery: Richard Garwin and John Tukey developed the FFT algorithm in 1963, reducing computation time significantly.
How it Works: Exploits the periodic nature of sine waves; reduces calculations from O(n^2) to O(n log n).
Impact: Enabled rapid analysis of seismic data, leading to better detection of underground tests.

Publication: Cooley and Tukey’s 1965 paper led to widespread adoption in various fields.
Nuclear Testing Continuation: Despite the FFT's introduction, nuclear testing continued into the late 20th century, exacerbating the arms race.
Gauss's Early Discovery: Gauss had previously discovered the principles of FFT in 1805 but did not publish his findings, delaying its application.

Wide Use: Basis for image and sound compression, signal processing, solving differential equations, radar, and sonar technologies.
Real-World Example: Used in image compression by transforming pixel values to identify significant frequency components, allowing reduction of data size while preserving image quality.

Impact of Career: Discusses the potential impact of one’s career and contributions; highlights the work of 80,000 Hours nonprofit, which aims to help individuals find impactful careers.
Call to Action: Encouragement to explore fulfilling careers that make a difference, with resources provided by 80,000 Hours.