Question

Find the 2-D Z-transform and region of convergence of each of the following:

(a) u₊₊​(n1​,n2​)
(b) rho(ⁿ¹⁺ⁿ²​),∣rho∣<1
(c) b(ⁿ¹⁺²ⁿ²​)u(n1​,n2​)
(d) u₋₊​(n1​,n2​)

Answer 1

To find the 2-D Z-transform and region of convergence for each expression: (a) For u₊₊(n₁, n₂), the Z-transform is 1 / (1 - z₁⁻ⁿ₁ - z₂⁻ⁿ₂), excluding the origin. (b) For rho(ⁿ¹⁺ⁿ²), |rho| < 1, the Z-transform is rho / (1 - rho * z₁⁻ⁿ¹ - rho * z₂⁻ⁿ²), excluding the origin. (c) For b(ⁿ¹⁺²ⁿ²)u(n₁, n₂), the Z-transform is b / (1 - z₁⁻ⁿ¹ - z₂⁻ⁿ² - 4 * z₁⁻ⁿ¹⁺²ⁿ²), excluding the origin. (d) For u₋₊(n₁, n₂), the Z-transform is 1 / (1 - z₁⁻⁽ⁿ¹⁺ⁿ²) - z₂⁻⁽ⁿ¹⁺ⁿ²), excluding the origin.

Step-by-step explanation:

To find the 2-D Z-transform and region of convergence for each expression:

(a) For u₊₊(n₁, n₂), the Z-transform is 1 / (1 - z₁⁻ⁿ₁ - z₂⁻ⁿ₂), and the region of convergence is the entire z-plane excluding the origin.

(b) For rho(ⁿ¹⁺ⁿ²), |rho| < 1, the Z-transform is rho / (1 - rho * z₁⁻ⁿ¹ - rho * z₂⁻ⁿ²), and the region of convergence is the entire z-plane excluding the origin.

(c) For b(ⁿ¹⁺²ⁿ²)u(n₁, n₂), the Z-transform is b / (1 - z₁⁻ⁿ¹ - z₂⁻ⁿ² - 4 * z₁⁻ⁿ¹⁺²ⁿ²), and the region of convergence is the entire z-plane excluding the origin.

(d) For u₋₊(n₁, n₂), the Z-transform is 1 / (1 - z₁⁻⁽ⁿ¹⁺ⁿ²) - z₂⁻⁽ⁿ¹⁺ⁿ²), and the region of convergence is the entire z-plane excluding the origin.