What is it?
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.
Formulas
- an LTI system in time and in frequency
Why does it matter?
Noise cancellation, voice codecs, equalizers, radar, ECG analysis and seismic exploration are signal processing.
The mathematics behind it
Pure tones are sinusoids; every signal is analysed as a sum of them.
Sinusoids are handled as phasors : amplitude and phase in one number.
Periodic signals are described and filtered through their harmonic content.
Spectral analysis, filtering and modulation are defined in the frequency domain.
The output of any LTI system is the input convolved with the impulse response (reverb = convolution with a room).
Signal energy, correlation and convolution are integrals (sums, once sampled).
Resampling audio between rates interpolates between samples (ideally with a sinc kernel).
Delaying a signal is ; speeding it up is , which stretches its spectrum.
Orthogonality lets a filter bank or a DFT pick out one frequency and ignore the rest.
Where is it used?
Computing topics reachable from here, through the chain of ideas that leads to them:
What depends on it
This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.