Task A1 – Finite Sequences

• X(z) for x[n] = {1,2,5} → 1 + 2z^{-1} + 5z^{-2}
• X(z) for x[n] = {0,3,0,4} → 3z^{-1} + 4z^{-3}
• ROC is entire z-plane except possibly z = ∞.

Task A2 – Infinite Sequences & ROC

• a = 0.6 → ROC: |z| > 0.6
• a = -0.8 → ROC: |z| > 0.8
• Left-sided → ROC: |z| < 0.9
• ROC excludes poles and defines convergence region.

Task A3 – Linearity & Shifting

• Linear combination: Z{2x1 - 3x2} computed symbolically.
• Time-shift: Z{x1[n−3]} shows delay by 3 samples.

Task A4 – Inverse Z-Transform

• Xa → right-sided exponential: x[n] = 0.7^n u[n]
• Xb → difference of exponentials: x[n] = 0.8^n - 0.5*0.8^n

Task A5 – Filter Analysis

• Poles: complex conjugates inside unit circle.
• Zeros: near z = 1.
• Filter Type: Low-pass.
• Magnitude plot confirms suppression of high frequencies.

Reflections

• Learned how ROC shapes the behavior of Z-transforms.
• Symbolic tools in MATLAB simplify verification.
• Frequency response plots reveal filter characteristics clearly.