How is the Laplace transform used in signal processing?

Laplace transform analyzes continuous-time systems: X(s) = integral(x(t)*e^(-st)dt), where s = sigma + jw. The s-plane reveals system behavior: Pole locations determine stability (left half-plane = stable), Natural frequencies and damping from pole positions, and s = jw (imaginary axis) gives frequency response. Uses: Solving differential equations (convert to algebra), Transfer function analysis H(s), Stability analysis (root locus, Routh-Hurwitz), and System design (compensator design). Related to Fourier transform when sigma = 0. Foundation for control theory and analog filter design.