Final answer:
The z-transform of x(n) = (0.6)ⁿ u(n) is X(z) = 1 / (1 - 0.6z^{-1}), with the ROC being |z| > 0.6. The method used involves the sum of an infinite geometric series.
Step-by-step explanation:
Finding the z-Transform and Region of Convergence (ROC)
The task is to find the z-transform of the discrete input signal x(n) = (0.6)ⁿ u(n) and indicate its Region of Convergence (ROC). In this case, u(n) represents the unit step function, which is 1 for n >= 0 and 0 otherwise. The z-transform of x(n) is a function of the complex variable z and is defined by the series sum of x(n)z^{-n} from n=0 to infinity.
To find the z-transform of x(n), we can use the formula for the sum of an infinite geometric series:
X(z) = sum((0.6)ⁿ z^{-n}, n=0 to infinity) = 1 / (1 - 0.6z^{-1}), for |0.6z^{-1}| < 1. Therefore, the ROC is |z| > 0.6.
Similarly, the z-transform of the impulse response h(n) = (0.9)ⁿ u(n) can be found using the same approach, which would result in:
H(z) = sum((0.9)ⁿ z^{-n}, n=0 to infinity) = 1 / (1 - 0.9z^{-1}), for |0.9z^{-1}| < 1, therefore the ROC is |z| > 0.9.