📘 Z–Transform Properties – Complete Explanation
Z-transform properties simplify complex algebraic manipulations. These are frequently used in solving difference equations and system analysis.
🔹 1. Linearity
a x₁[n] + b x₂[n] ↔ aX₁(z) + bX₂(z)
Most basic property.🔹 2. Time Shifting
If:x[n] ↔ X(z)
Then:x[n − k] ↔ z^{-k} X(z)
Very important for difference equations.🔹 3. Time Reversal
x[−n] ↔ X(1/z)
ROC changes accordingly.🔹 4. Multiplication by aⁿ
aⁿ x[n] ↔ X(z/a)
Used for exponential scaling.🔹 5. Convolution Property
x[n] * h[n] ↔ X(z) H(z)
Convolution in time → Multiplication in z-domain.🔹 6. Differentiation Property
n x[n] ↔ -z (dX(z)/dz)
Advanced but useful in some problems.🔹 7. Initial Value Theorem
x[0] = lim (z→∞) X(z)
Used to check first sample quickly.🔹 8. Final Value Theorem
If stable:lim (n→∞) x[n] = lim (z→1) (z−1) X(z)
Very important for steady-state value.🔹 9. Example
Given:X(z) = z / (z − 0.5)
Initial value:x[0] = lim (z→∞) X(z) = 1
Final value:lim (z→1) (z−1) X(z)
Evaluate for steady state.🎯 GATE Important Points
- Time shift property very common
- Convolution → multiplication shortcut
- Final value theorem requires stability
- Always consider ROC carefully
Z Properties = Fast Tools for Discrete Analysis
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