Question

Determine all possible signal x(n) associated with the z transform

x(z) = ((1 - 2z⁻¹ + z⁻²)/(1 - 4z⁻¹ + 4z⁻²))

Answer 1

To determine all possible signals x(n) associated with the given z-transform x(z) = ((1 - 2z⁻¹ + z⁻²)/(1 - 4z⁻¹ + 4z⁻²)), we need to find the inverse z-transform of the given expression. The possible signals x(n) are: x(n) = 2^n and x(n) = n*2^n.

Step-by-step explanation:

The student's question is about determining all possible signals x(n) associated with the given z-transform x(z) = ((1 - 2z⁻¹ + z⁻²)/(1 - 4z⁻¹ + 4z⁻²)). However, to carry out this process, we need clarity on whether the z-transform represents a causal or non-causal system, as it can affect the region of convergence (ROC) and, consequently, the time-domain sequence. To determine all possible signals x(n) associated with the given z-transform x(z) = ((1 - 2z⁻¹ + z⁻²)/(1 - 4z⁻¹ + 4z⁻²)), we need to find the inverse z-transform of the given expression. The inverse z-transform can be found by partial fraction decomposition and using the properties of the z-transform inverse. By performing the calculations, we find that the possible signals x(n) are:

1. x(n) = 2^n
2. x(n) = n*2^n