transfer function

A transfer function is the fingerprint of a linear system: a single algebraic expression G(s) that tells you exactly how the system turns any input into an output. Think of it as a recipe written in the Laplace variable s, where multiplying input by G(s) does what convolving with the impulse response does in time — but as plain algebra instead of calculus. It is defined as the ratio of output to input, G(s) = Y(s) / X(s), assuming the system starts at rest.

Its real value is what's hiding in the numerator and denominator. The roots of the denominator are the poles — they set how the system naturally rings, decays, or blows up. The roots of the numerator are the zeros — they shape which frequencies get blocked or boosted. A second-order plant like ω_n² / (s² + 2ζω_n s + ω_n²) packs an entire personality — its speed (ω_n), its damping (ζ), whether it overshoots — into four symbols. Once you have G(s), the whole toolbox opens: you can predict the step response, draw the Bode plot, and judge stability without ever solving a differential equation by hand.