Concept:

Causal Signals: Causal Signals are signals that are zero for all negative time.

Causality in a system determines whether a system relies on future information of a signal x [n+1].

Talking about “causality” in signals, it means whether they are zero to the left of t = 0 or zero to the right of t = 0.

A causal signal is zero for t < 0.

A system is said to be causal if its output depends upon present and past inputs, and does not depend upon future input.

Calculation:

Given a casual signal,

X (Z) = z2 (z - a)-2

$X(Z)=\frac{z^2}{{\left(a-z\right)}^2}$

$\mathrm{From\; standard\; Z-Transform,}$

$n{\left(a\right)}^{n}u\left(n\right)\leftrightarrow \frac{az}{{\left(z-a\right)}^2}$

$n{\left(a\right)}^{n-1}u\left(n\right)\leftrightarrow \frac{z}{{\left(z-a\right)}^2}$

$\mathrm{From\; the\; time-shifting\; property,}$

$x\left(n-n_0\right)\leftrightarrow z^{-n_0}X\left(z\right)$

$\left(n+1\right){\left(a\right)}^{n+1-1}u\left(n+1\right)\leftrightarrow z\left[\frac{z}{{\left(z-a\right)}^2}\right]$

$\left(n+1\right){\left(a\right)}^{n}u\left(n+1\right)\leftrightarrow \frac{z^2}{{\left(z-a\right)}^2}$

Since,

$X\left(z\right)=\frac{z^2}{{\left(z-a\right)}^2}$

Therefore,

$x\left(n\right)=\left(n+1\right){\left(a\right)}^{n}u\left(n+1\right)$

The given signal is causal, therefore,

$x\left(n\right)=\left(n+1\right){\left(a\right)}^{n}u\left(n\right)$