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Define z-transform; differ from Fourier transform

Short answer

Z-transform uses complex z=re^{iω}; Fourier transform is DTFT on |z|=1 (z=e^{iω}). Z handles growing sequences and transient analysis, DTFT needs absolute summability. X(e^{iω})=X(z) on unit circle if ROC contains it.

Define z-transform. How does it differ from Fourier transform?

Z-transform

The z-transform of a sequence x(k) is X(z) = Σ (k = −∞ to ∞) x(k) z⁻ᵏ, where z = re^(iω) is a complex variable. It converts a difference equation into an algebraic equation and is widely used for analysing discrete-time systems.

Difference from the Fourier transform

Pointz-transformFourier transform (DTFT)
VariableComplex z = re^(iω)Real frequency ω (z = e^(iω))
DefinitionΣ x(k) z⁻ᵏΣ x(k) e^(−iωk)
ConvergenceExists for z in a region of convergence (ROC)Converges if Σ|x(k)| < ∞ (the unit circle must lie in the ROC)
Signals handledAlso growing sequences (e.g. 2ᵏ)Needs absolutely summable sequences
RelationGeneralA special case: the z-transform on the unit circle
Main useTransient analysis, stability, solving difference equations with initial conditionsFrequency (steady-state) spectrum

So X(e^(iω)) = X(z) evaluated at |z| = 1, provided the unit circle lies in the ROC.

✓ Answer checked by Claude Opus 5.5 (high effort)

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