Laplace Transform and System Analysis

The Laplace transform extends the Fourier transform to signals that grow or start at t = 0. It turns differential equations into algebra and is the main tool for circuits and control.

Definition

X(s) = ∫₀^∞ x(t) e^(−st) dt (unilateral), s = σ + jω. The region of convergence (ROC) is where the integral converges; for right-sided signals it is to the right of the rightmost pole.

Common pairs

Properties

• Time shift: x(t − t₀)u(t − t₀) → e^(−st₀) X(s).
• Frequency shift: e^(−at) x(t) → X(s + a).
• Differentiation: dx/dt → sX(s) − x(0⁻). Integration: ∫x dt → X(s)/s.
• Convolution in time → product X(s)H(s).
• Initial value theorem: x(0⁺) = lim_{s→∞} s X(s).
• Final value theorem: x(∞) = lim_{s→0} s X(s), valid only if all poles of sX(s) are in the left half-plane.

Inverse transform

Use partial fractions: write X(s) = Σ A_i/(s + p_i), with A_i = (s + p_i)X(s) evaluated at s = −p_i, and invert term by term. Repeated and complex poles give t e^(−at) and damped sinusoid terms.

Transfer functions

H(s) = Y(s)/X(s) with zero initial conditions = Laplace transform of h(t).

• Poles (roots of the denominator) set the natural response; zeros shape it.
• A causal system is stable if all poles have negative real parts (left half s-plane). Poles on the jω axis (simple) → marginally stable.

Worked example

X(s) = 5/(s(s + 5)).

• Partial fractions: 1/s − 1/(s + 5), so x(t) = (1 − e^(−5t)) u(t).
• Final value: lim s × 5/(s(s + 5)) = 1; initial value 0.

Practice