Bilinear transform

Okay, so imagine you have a piece of paper with a bunch of dots on it. You want to draw a line that goes through those dots in a curved way instead of just a straight line. But drawing a curved line is hard, so instead you decide to turn those dots into squares, and then you can draw the line between the corners of those squares.

That's kind of like what bilinear transform does. When you have some complicated math problem that involves things going up and down and spinning around (which is called a continuous-time system), it can be hard to solve. So we turn that problem into a bunch of boxes with just a few corners (which is called a discrete-time system).

The bilinear transform changes the way the boxes are set up, and basically stretches and twists things around so that the curved line from the dots on the paper matches up as closely as possible with the line you drew between the corners of the squares. This makes the math easier to solve, because now we're working with just a few points instead of a bunch of wiggles and curves.