📘 Z–Transform – Definition, ROC & Basic Transform Pairs
Z-transform converts a discrete-time signal into complex frequency domain. It is the fundamental tool for analyzing discrete LTI systems.
🔹 1. Definition
X(z) = Σ x[n] z^{-n}
Summation over all n. z is complex variable:z = re^{jω}
🔹 2. Region of Convergence (ROC)
Z-transform exists only for values of z where summation converges. ROC is:- Ring in z-plane
- Does not include poles
- Determines causality & stability
🔹 3. Example – Geometric Sequence
Given:x[n] = a^n u[n]
Z-transform:X(z) = 1 / (1 − a z^{-1})
ROC:|z| > |a|
Right-sided signal → ROC outside circle.🔹 4. Important Standard Pairs
Impulse:δ[n] ↔ 1
Unit Step:u[n] ↔ 1 / (1 − z^{-1})
Exponential:a^n u[n] ↔ 1 / (1 − a z^{-1})
🔹 5. ROC Rules
Right-sided signal → ROC outside outermost pole Left-sided signal → ROC inside innermost pole Two-sided → ROC between poles🔹 6. Poles and Zeros
Poles → Values making denominator zero Zeros → Values making numerator zero System behavior depends heavily on pole location.🔹 7. Stability Condition
System stable if:ROC includes unit circle (|z| = 1)
Very important for GATE.🎯 GATE Important Points
- ROC determines signal type
- ROC never includes poles
- Stability depends on unit circle
- Know standard transform pairs
Z-Transform = Laplace Transform of Discrete Signals
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