Overview
This lecture introduces the Fourier transform, focusing on its use in decomposing complex signals into pure frequencies, primarily with sound, and explores its broader mathematical applications.
Decomposing Sound into Frequencies
- Pure notes (like A at 440 Hz) correspond to simple sine waves in air pressure vs. time graphs.
- When multiple notes play, their waveforms sum, creating complex, non-sinusoidal signals.
- The challenge is to decompose this complicated signal back into its original pure frequencies.
The Winding Machine (Visualizing Frequencies)
- Decompose a signal by "winding" its graph around a circle at various winding frequencies.
- When the winding frequency matches a component frequency of the signal, peaks align, resulting in a distinctive pattern.
- The center of mass of the wound-up graph reveals the presence and strength of frequency components.
Properties and Linearity
- For signals composed of several pure frequencies, the winding process produces spikes at each original frequency.
- The process is linear: transforming the sum of two signals equals the sum of their transforms.
Applications: Filtering and Editing
- Fourier transforms help identify and isolate frequency components, such as filtering out unwanted noise in audio editing.
- The inverse Fourier transform allows for reconstructing the original signal after modifying its frequency components.
Mathematical Formalism of the Fourier Transform
- "Winding" is elegantly expressed using complex exponentials (Euler's formula): ( e^{-2\pi i f t} ).
- Multiply the signal ( g(t) ) by the complex exponential, then integrate to find the "center of mass" for each frequency.
- The Fourier transform does not divide by the time interval—its value scales with the length of the signal.
Interpretation and Notation
- The Fourier transform turns a time-based signal into a frequency-based function (commonly denoted as ( \hat{g}(f) )).
- Output values are complex numbers; the real and imaginary parts correspond to different aspects of frequency contributions.
- In theory, Fourier transforms are defined over infinite intervals, though practical applications use finite samples.
Key Terms & Definitions
- Frequency — Number of cycles per second (Hz) in a wave.
- Fourier Transform — Mathematical operation converting a time signal into its constituent frequencies.
- Complex Exponential — ( e^{ix} ) describes rotation in the complex plane, key to Fourier analysis.
- Center of Mass (of wound graph) — Geometric representation of frequency strength at a given winding frequency.
- Inverse Fourier Transform — Operation reconstructing the original signal from its frequency components.
Action Items / Next Steps
- Review the concept of Euler’s formula and complex exponentials.
- Consider how the Fourier transform can apply beyond sound (to be explored in upcoming lectures).
- Optional: Watch related videos for additional visual intuition.