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The Z-Transform, Its ROC & the Relationship to the DTFT
Inside this lesson
- Define the (bilateral) Z-transform X(z) = Σx[n]z^(-n) and identify it as the discrete-time counterpart of the Laplace transform
- Apply the standard pair a^n·u[n] ↔ 1/(1−a·z^(-1)) with ROC |z| > |a|
- Explain why the Z-transform ROC is always an annulus or disk in the z-plane, not a half-plane
- State the relationship X(e^(jω)) = X(z)|_(z=e^(jω)) and the condition under which the DTFT exists
- Draw the parallel between the s-plane jω-axis (Laplace→Fourier) and the z-plane unit circle (Z→DTFT)
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Educational content covering topics typical of a first- and second-year electrical engineering curriculum. Not a substitute for accredited coursework, and not suitable for real antenna, transmission-line, or signal-processing system design without review by a licensed professional engineer.