Side-by-side comparison
| Parameter | Laplace | Z-Transform |
|---|---|---|
| Domain | Continuous time, complex variable s = σ + jω | Discrete time, complex variable z = re^{jω} |
| Definition | X(s) = ∫_{0}^{∞} x(t)·e^{−st} dt (unilateral) | X(z) = Σ_{n=−∞}^{∞} x[n]·z^{−n} |
| Stability condition | All poles must be in left-half s-plane (Re{s} < 0) | All poles must be inside unit circle |z| < 1 |
| Mapping | s-plane | z = e^{sT}; jω axis → unit circle |z| = 1 |
| ROC | Vertical strips or half-planes in s-plane | Annular regions in z-plane (e.g., |z| > 0.5) |
| Frequency axis | jω axis (−j∞ to +j∞) | Unit circle |z| = 1 |
| Inverse transform | Partial fractions + table lookup | Partial fractions + table, or power series expansion |
| Bilinear transform | Used to design analog prototype filters | s = (2/T)·(z−1)/(z+1) maps analog to digital design |
| Transfer function | H(s) = Y(s)/X(s) — polynomials in s | H(z) = Y(z)/X(z) — polynomials in z^{-1} |
| Real example | PID controller designed in s-domain for servo motor | IIR Butterworth filter implemented on DSP TMS320C6748 |
Key differences
The Laplace Transform is to continuous systems what the Z-Transform is to discrete systems. The critical mapping is z = e^{sT}: the left half of the s-plane (stable poles) maps to the interior of the unit circle |z| < 1 in the z-plane. A causal discrete system with H(z) = 1/(1 − 0.5z^{-1}) has a pole at z = 0.5 — inside the unit circle, so it is stable. The bilinear transform s = (2/T)(z−1)/(z+1) lets you convert an analog Butterworth prototype H(s) into a digital IIR filter H(z) directly.
When to use Laplace
Use the Laplace Transform when designing or analysing analog control systems — for example, computing the closed-loop poles of a DC motor PID controller using root locus in the s-plane.
When to use Z-Transform
Use the Z-Transform when implementing filters or controllers on a digital processor — for example, deriving the difference equation for an IIR low-pass filter from H(z) to code it on a TMS320C6748 DSP.
Recommendation
Choose Laplace for analog circuit and control problems; choose Z-Transform for anything sampled or discrete. For GATE, know the z = e^{sT} mapping and be able to map pole positions between s and z domains. In placements, if asked about digital filter design, lead with the Z-Transform and bilinear transform.
Exam tip: GATE frequently gives H(z) = (z + 1)/(z − 0.5) and asks for stability — locate the pole at z = 0.5, confirm |0.5| < 1, and state it is stable; also find the ROC and state it explicitly.
Interview tip: An interviewer at an embedded systems company will ask how you convert an analog filter to digital — explain the bilinear transform s = (2/T)(z−1)/(z+1), mention pre-warping to correct frequency distortion, and name the Texas Instruments TMS320 as a typical target DSP.