Applied Physics · Signal Processing
Digital Signal Processing
Every sound you hear is a sum of pure sine waves. The Fourier Transform decomposes that mixture into its constituent frequencies — the same math behind music EQs, MRI scanners, voice recognition, and Wi-Fi.
Key Concepts
- Time Domain — amplitude vs. time (the waveform you see)
- Frequency Domain — amplitude vs. frequency (the spectrum)
- Fourier Transform — the bridge between the two domains
- Filters — boost or cut specific frequency bands
Live Visualizer
Additive Synthesis
Drag amplitudes to zero to hear how each harmonic shapes the timbre. A square wave ≈ odd harmonics only. A sawtooth ≈ all harmonics.
Graphic Equalizer
Boost or cut frequency bands (±12 dB). Watch the spectrum change in real time.
Joseph Fourier (1807)
Proved that any periodic function can be decomposed into a sum of sines and cosines — one of the most powerful ideas in all of mathematics.
FFT Algorithm
The Fast Fourier Transform (Cooley–Tukey, 1965) computes the frequency decomposition in O(n log n) time — making real-time audio analysis possible.
Nyquist–Shannon Theorem
To capture a frequency, you must sample at least twice per cycle. CD audio samples at 44.1 kHz, capturing up to ~22 kHz — the limit of human hearing.
Applications
MP3 compression, noise cancellation, medical imaging (MRI/CT), radar, seismology, and speech recognition all rely on the same DSP fundamentals.