What is the purpose of the inverse Z-transform?

• A. To compute X(z) from x[n].
• B. To recover x[n] from X(z) using partial fractions or long division.
• C. To delay a signal.
• D. To convert between Z-domain and Laplace-domain representations.

Answer: (B)

The inverse Z-transform recovers the original time sequence from its Z-domain representation. It undoes the forward Z-transform, turning X(z) back into x[n]. In practice you decompose X(z) into a sum of simple terms whose inverse Z-transform you know from tables or standard pairs, using partial fraction expansion or long division. Each term corresponds to a time-domain sequence, and summing those contributions gives x[n]. This is why those decomposition methods are used: they break X(z) into building blocks that map directly to x[n]. The other ideas are not the aim: the forward transform, delaying a signal, or switching between Z-domain and Laplace-domain representations are unrelated to the purpose of the inverse Z-transform.