Side-by-side comparison
| Parameter | Laplace | Fourier Transform |
|---|---|---|
| Variable | s = σ + jω; complex frequency plane | jω (imaginary axis only); real frequency ω |
| Definition | X(s) = ∫_{-∞}^{∞} x(t)·e^{−st} dt | X(jω) = ∫_{-∞}^{∞} x(t)·e^{−jωt} dt |
| Region of Convergence | ROC is a vertical strip or half-plane in s-plane | Exists only if ROC includes the jω axis |
| Stability analysis | Poles in left-half s-plane → BIBO stable | Cannot directly determine stability from poles |
| Handles growing signals? | Yes — e^{at}u(t) with a > 0 converges for Re{s} > a | No — e^{at}u(t) with a > 0 has no Fourier Transform |
| Relationship | FT is LT evaluated on jω axis: X(jω) = X(s)|_{s=jω} | Special case of LT when ROC includes jω axis |
| Bode plot | H(s) → substitute s = jω to get H(jω) for Bode plot | Directly gives magnitude/phase vs ω for stable systems |
| Initial/final value theorem | Applies: lim x(t)_{t→∞} = lim s·X(s)_{s→0} | Not directly available |
| Typical use | Transient analysis of RLC, control systems (PID, root locus) | Frequency response of audio filters, communication channel |
| Inverse transform | Partial fraction + table; contour integral in s-plane | Inverse FT integral; use symmetry and transform pairs |
Key differences
The Laplace Transform uses the complex variable s = σ + jω, allowing it to converge for signals that the Fourier Transform cannot handle — specifically signals that grow exponentially. The Fourier Transform is a special case: X(jω) = X(s)|_{s=jω}, valid only when the ROC of X(s) includes the imaginary axis. For stability: if all poles of H(s) are in the left-half plane (Re{s} < 0), the system is BIBO stable and its Fourier Transform exists. A pole at s = 2 means the signal grows as e^{2t} — no FT, but LT converges for Re{s} > 2.
When to use Laplace
Use the Laplace Transform when the problem involves transient response, stability, or signals that are not absolutely integrable — for example, finding the step response of a series RLC circuit with R = 50 Ω, L = 10 mH, C = 100 µF.
When to use Fourier Transform
Use the Fourier Transform when you need the steady-state frequency response of a stable system — for example, computing the −3 dB bandwidth of a 2nd-order Butterworth filter with f0 = 1 kHz from its H(jω).
Recommendation
For GATE, choose Laplace for any problem mentioning poles, stability, transient, or ROC. Choose Fourier when the problem asks for frequency response or spectrum. In interviews, explain that FT is LT on the jω axis and watch the interviewer nod — it shows you understand the connection.
Exam tip: GATE tests the ROC condition: for a right-sided signal, ROC is Re{s} > σ0; the FT exists only if σ0 < 0 — know this and you can answer 3-mark ROC problems in under 90 seconds.
Interview tip: Interviewers at control-system companies ask you to find the transfer function of a second-order system from its differential equation — take LT with zero initial conditions, factorise the denominator, and identify poles; do not mix up s and jω.