Question

Given X [z] = z(z-0,25)/z²-0,7071z+0,25 With a Region of Convergence (ROC) of 0,5 <∣z∣, what is the structure of the response x (n)?

a. A₁ (0,5)ⁿ cos (π/4n) + A₂ (0,5)ⁿ sin (π/4n), n ≥ 0
b. A₁ (0,25)ⁿ + A₂ (-.7071)ⁿ, n ≥ 0
c. None of the given answers
d. A₁n (0,5)ⁿ + A₂ (0,5)ⁿ , n ≥ 0
e. A₁ (-0,25)ⁿ + A₂ (.7071)ⁿ, n ≥ 0

Answer 1

The structure of the response x(n) is given by option d: A₁n (0.5)ⁿ + A₂ (0.5)ⁿ, n ≥ 0.

Step-by-step explanation:

The given transfer function X(z) = \frac{{z(z-0.25)}}{{z^2-0.7071z+0.25}} has a Region of Convergence (ROC) of 0.5 < |z|. To determine the structure of the response x(n), we need to find the inverse Z-transform of X(z).

The inverse Z-transform of X(z) can be found using partial fraction decomposition and taking the inverse Z-transform of each term. After simplifying and applying the Z-transform tables, the structure of the response x(n) is given by

option d: A₁n (0.5)ⁿ + A₂ (0.5)ⁿ, n ≥ 0.