Final answer:
The z-transforms of the input and impulse response signals are obtained through respective series summation considering their Regions of Convergence, and the system's output is determined by multiplying these transforms and computing the inverse z-transform.
Step-by-step explanation:
The question you've asked pertains to finding the z-transform of signals in a discrete system and using the z-transform method to determine the system's output. Let's tackle each part of the question step by step:
- Find the z-transform of x(n): The signal x(n) is given by (0.6)nu(n), where u(n) is the unit step function. The z-transform of x(n) is X(z) = ∑ (from n=0 to ∞) (0.6)nz-n. This series converges when |0.6/z| < 1, so the Region of Convergence (ROC) is |z| > 0.6.
- Find the z-transform of h(n): Similarly, the impulse response h(n) is (0.9)nu(n). Its z-transform H(z) can be found using the same approach, with the resulting ROC as |z| > 0.9.
- Find the output y(n): The output y(n) can be found by computing the inverse z-transform of the product Y(z) = X(z)H(z). This involves finding the convolution of x(n) and h(n) in the time domain, which corresponds to multiplying their z-transforms.
To summarize, the z-transforms of x(n) and h(n) are found by summing their respective series, and the system output y(n) is found by multiplying these transforms and finding the inverse z-transform of the result.