Question

When the input to a causal LTI system is

x[n]= -1/3 (1/2)ⁿ u(n) - 4/3(2)ⁿ u(-n-1)
the z-transform of the output is Y(z)= 1+z⁻¹/(1-z⁻¹) (1+1/2z⁻¹) (1-2z⁻¹)
Find the z-transform of x[n]?

Answer 1

The z-transform of the input signal x[n] is X(z) = -1/3 * (1 / (1 - 1/2z⁻¹)) - 4/3 * (1 / (1 - 2z⁻¹)).

Step-by-step explanation:

To determine the z-transform of the given input signal x[n], which comprises two terms, we break down the signal into its constituent parts:

x[n] = -1/3 * (1/2)ⁿ u[n] - 4/3 * (2)⁻ⁿ u[-n-1]

The z-transform of each term is calculated individually. The first term's z-transform is found using the geometric series formula, resulting in -1/3 * (1 / (1 - 1/2z⁻¹)). The z-transform of the second term is similarly derived, yielding -4/3 * (1 / (1 - 2z⁻¹)).

Combining these z-transforms, the overall z-transform of the input signal x[n] is:

X(z) = -1/3 * (1 / (1 - 1/2z⁻¹)) - 4/3 * (1 / (1 - 2z⁻¹))

This final expression represents the input signal x[n] in the z-domain, demonstrating the transformation from the time-domain signal x[n] to its z-transform X(z) in terms of their individual components' z-transforms.