Signals, media and vision
Audio, images, video and radio as functions to be sampled, transformed, filtered and compressed — Fourier analysis and derivatives at industrial scale.
7 topics
Topics
Signal processing
Analysing and transforming signals — sound, radio, sensor data — mostly through linear time-invariant systems, convolution and the Fourier transform. The mathematics inside every phone, headset and modem.
Sampling theorem (Nyquist–Shannon)
A signal with no frequencies above is completely determined by samples taken at more than per second, and can be rebuilt exactly with sinc interpolation. Sample slower and high frequencies masquerade as low ones (aliasing). That is why CDs use 44.1 kHz for 20 kHz hearing.
Fast Fourier transform (FFT)
Computes the -point DFT in instead of by splitting it recursively into even and odd samples (Cooley–Tukey, 1965, anticipated by Gauss). One of the most important algorithms of the 20th century.
Digital filters
Difference equations that keep some frequencies and remove others. FIR filters are convolutions; IIR filters feed back previous outputs and are analysed with the Z-transform (stable iff poles inside the unit circle).
Media compression (JPEG, MP3, video)
Transform the signal into a basis where most coefficients are tiny (cosine transforms, wavelets), quantize them coarsely where the eye or ear will not notice, and entropy-code the rest. Video adds motion compensation between frames.
Image processing and computer vision
An image is a function . Blurring is convolution, edges are large gradients , corners come from second derivatives, motion between frames from the optical-flow equation. Classical vision is applied calculus; modern vision learns the filters (CNNs).
Telecommunications (modulation, OFDM)
Sending bits over the air by modulating sinusoidal carriers in amplitude and phase (complex numbers). OFDM splits the band into many orthogonal subcarriers using an inverse FFT at the transmitter and an FFT at the receiver — Wi-Fi, 4G/5G and DSL.
The mathematics this domain runs on
ƒ Foundations ★★★★★
≈ Numerical methods ★★★★★
∇ Multivariable calculus ★★★★★
ℱ Transforms ★★★★★
- Fourier series★★★★★→Signal processing, Media compression (JPEG, MP3, video)
- Fourier transform★★★★★→Signal processing, Sampling theorem (Nyquist–Shannon), Fast Fourier transform (FFT), Image processing and computer vision, Telecommunications (modulation, OFDM)
- Convolution★★★★★→Signal processing, Digital filters, Image processing and computer vision
- Z-transform★★★★★→Digital filters
- Wavelets★★★★★→Media compression (JPEG, MP3, video), Image processing and computer vision