Introduction
The one-sided z-transform is a mathematical tool used to analyze discrete-time signals in the frequency domain. It is particularly useful for causal signals, which are signals that do not have any components before time zero. In this answer, we will discuss why the one-sided z-transform is unique for causal signals.

Explanation
The one-sided z-transform is defined as:

X(z) = ∑[x(n) * z^(-n)], n = 0 to ∞

Here, X(z) represents the z-transform of the signal x(n), and z is a complex variable. The one-sided z-transform considers only the non-negative values of n, which means it only takes into account the present and future values of the signal.

Properties of Causal Signals
Causal signals are defined as signals that do not have any components before time zero. In other words, for a causal signal x(n), x(n) = 0 for n < 0.="" this="" property="" is="" important="" when="" considering="" the="" uniqueness="" of="" the="" one-sided="" />

Uniqueness of the One-Sided Z-Transform for Causal Signals
The one-sided z-transform is unique for causal signals because it only considers the non-negative values of n. Since a causal signal has no components before time zero, there are no negative values of n to consider. Therefore, the one-sided z-transform captures all the relevant information of the signal.

If we were to consider the two-sided z-transform, which includes both positive and negative values of n, it would introduce redundancy for causal signals. The negative values of n would be zero for causal signals, and including them in the z-transform would not provide any additional information.

Therefore, to avoid redundancy and capture only the necessary information, the one-sided z-transform is used for causal signals. It provides a unique representation of the frequency domain characteristics of the signal.

Conclusion
In conclusion, the one-sided z-transform is unique for causal signals because it only considers the non-negative values of n. Causal signals do not have any components before time zero, and including them in the z-transform would introduce redundancy. The one-sided z-transform captures all the relevant information of the signal and provides a unique representation in the frequency domain.