DSP (digital signal processing) is the field of applied mathematics and engineering dedicated to transforming and manipulating digital signals.

Examples of real digital signals include audio files, image files, video files, and digitized recordings of various physical quantities by computers like the configuration of a robot as it moves in time, measurements of the processes in a factory, the trajectory of a spacecraft — almost anything that can be periodically sampled and take on a finite set of values [1] can be seen as a digital signal.

DSP includes using tools like the Discrete Fourier Transform (DFT), the Z-transform, wavelet analysis, probability, statistics, and linear algebra to do things such as filter a signal (example: audio equalizer), predict future values (example: weather forecasting), data compression (example: JPEGs), system identification (example: fit a model of the earth to predict seismic activity), control (example: make a DC motor to respond to position commands), and stabilization (example: keep plane from "wanting" to smash into the ground). Particularly, it requires a careful consideration of the effect of sampling a signal (example: if done carelessly, you can make the sampled system unstable [read: explode]), as well as an interpolation process of some kind if you plan on using that signal outside your computer (example: you want to hear an audio signal stored on your computer).

I got into DSP because I was an audio engineer and musician [2], and I wanted to design my own audio plugins. IMO I think almost everyone would benefit from some knowledge of DSP, but the math is really intense. Personally, I found out late in life that I have a nearly infinite appetite for math, so it's a good fit for me.

Here's a playlist about DSP if you're interested.

[1] Actually, a lot of basic DSP books don't restrict the signal to be in a finite set because it makes the math easier if the signal could be any real number. However, certain structures that would be exactly equivalent in theory are not equivalent on a real computer because ordinary computer arithmetic is approximate.

[2] I still play music, but not as much as before engineering school.

Singing!