Hacker News 1:15 pm on June 10, 2024
The text discusses the negative amplitude in Fourier Series and its physical implications, emphasizing practicality by maintaining absolute value. It outlines how to represent periodic signals using sine functions with phase shifts or changing signs while highlighting equivalent expressions' equivalence. An example demonstrates a sawtooth function's representation as an infinite series of odd harmonics in trigonometric form and visualizes Fourier Series Machinery, showing frequency-related circles generating the signal.
- Negative amplitude significance: Despite negative amplitudes seeming physically improper, they are used without hindrance; phase shifts or changing signs can serve practical purposes.
- Periodic signal representation: Periodic signals in sine functions can be expressed with phase shifts (e.g., \(sin(x+\pi)\)) or altered signs without loss of equivalence, as shown through an example of sawtooth function's Fourier Series.
- Fourier Series Machinery: The series construction with circles at various frequencies and radii depict the signal generation process, providing a visual understanding of complex signals from simple components.
- Sawtooth function example: An inverse sawtooth wave is represented as an infinite sum involving odd harmonics in sine form, showcasing how Fourier Series can model various periodic functions and patterns.
https://www.andreinc.net/2024/04/24/from-the-circle-to-epicycles
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