An Interactive Guide to the Fourier Transform
9 days ago
- #Fourier Transform
- #Signal Processing
- #Mathematics
- The Fourier Transform is a mathematical tool that decomposes a signal into its constituent frequencies, similar to finding the recipe of a smoothie by filtering out each ingredient.
- It changes the perspective from analyzing a signal in the time domain to understanding it in the frequency domain, making it easier to analyze, compare, and modify.
- The process involves measuring every possible cycle (frequency) in the signal, capturing their amplitude, offset, and rotation speed.
- Filters used in the Fourier Transform must be independent (not affecting each other) and complete (capturing all ingredients/frequencies).
- The Fourier Transform has wide applications, including earthquake engineering, sound processing (like bass and treble adjustments), data compression (JPEG, MP3), and radio signal filtering.
- A key insight is that any signal can be represented as a combination of circular paths (cycles), not just one-dimensional sinusoids.
- Euler's formula provides a compact way to represent these circular paths using complex numbers, though the underlying concept is about 2D circular motion.
- The Fourier Transform can be visualized by combining multiple cycles (each with their own amplitude, frequency, and phase) to reconstruct the original signal.
- Time spikes (signals with a single non-zero value) can be created by carefully aligning and canceling cycles at specific time points.
- Phase shifts allow delaying cycles, which is essential for aligning signals at different times.
- The full Fourier Transform is built by summing the contributions of each frequency component, adjusted for their phase delays.