The z-transform is a technique for design and analysis of discrete-time systems and signals.

The z-transform is a technique for design and analysis of discrete-time systems and signals.

• A. A tool for continuous-time systems
• B. A type of time-domain sampling
• C. A technique for design and analysis of discrete-time systems and signals
• D. A method to convert signals to the time domain

Answer: (C)

The z-transform is a tool for representing discrete-time signals and systems in the z-domain so you can analyze and design them with algebraic methods. It converts a sequence x[n] into a complex function X(z) = sum x[n] z^{-n}, allowing you to work with transfer functions, poles, and zeros to understand how a system responds to inputs. For a linear time-invariant discrete-time system, the relationship between input and output in the z-domain is Y(z) = H(z) X(z) with H(z) = Y(z)/X(z). The region of convergence tells you where the transform exists and ties directly to stability: for a causal, stable system, the ROC includes the unit circle. This makes the z-transform the go-to technique for both analyzing how discrete-time signals behave and designing systems to achieve desired responses. It’s not about continuous-time systems (that uses the Laplace transform), nor is it a time-domain sampling method, nor a way to convert signals back to the time domain—the z-transform moves you into the z-domain for powerful analysis.