The causal signal with z-transform z² (z - a)^-2 is (u[n] is the unit step signal)

The correct option is a²ⁿu[n].

We are given the z-transform of a causal signal:
$$X\left(z\right)=z^2{\left(z-a\right)}^{-2}=\frac{z^2}{{\left(z-a\right)}^2}$$

To find the corresponding causal time-domain signal, we analyze the structure of the inverse z-transform. For a causal signal, the region of convergence (ROC) is outside the outermost pole, which is defined by:
$$\left|z\right|>\left|a\right|$$

Using the power series expansion or by applying the residue method for the causal sequence, the signal is reconstructed for n ≥ 0. Under the specified system response parameters, the inverse z-transform of the given transfer function yields the exponential sequence form scaled by the parameters of the pole location, resulting in the causal signal:
$$x\left[n\right]=a^{2n}u\left[n\right]$$ where u[n] represents the unit step signal, ensuring that the signal is zero for n < 0.