How it works
The unilateral Laplace transform is X(s) = ∫₀^∞ x(t)e^(−st)dt where s = σ + jω. The region of convergence (ROC) is the set of s values for which the integral converges — for a right-sided signal e^(−at)u(t), the ROC is Re(s) > −a. Key pairs: δ(t) ↔ 1 (ROC: entire s-plane), u(t) ↔ 1/s (ROC: Re(s) > 0), e^(−at)u(t) ↔ 1/(s+a) (ROC: Re(s) > −a), tⁿe^(−at)u(t) ↔ n!/(s+a)^(n+1). The convolution property L{x*h} = X(s)H(s) makes transfer function analysis straightforward — poles of H(s) determine natural frequencies of the system.
Key points to remember
The ROC must be specified with every Laplace transform; different ROCs for the same X(s) correspond to different time-domain signals. For a causal stable system, all poles of H(s) must lie in the left half of the s-plane (Re(s) < 0). A repeated pole at s = −a gives time-domain terms of the form tⁿe^(−at). Partial fraction expansion is mandatory when inverting a rational X(s): factorise the denominator, assign A/(s+p₁) + B/(s+p₂) terms, multiply through and equate coefficients. Initial value theorem: x(0⁺) = lim(s→∞) sX(s); final value theorem: x(∞) = lim(s→0) sX(s), valid only when the limit exists and the system is stable.
Exam tip
The examiner always asks you to state and apply the initial and final value theorems for a given X(s) — always check the final value theorem's stability condition first, because applying it to an unstable X(s) is a guaranteed mark-losing error.