What is the primary purpose of the z-transform in discrete-time signal analysis?

• A. It maps discrete-time sequences to the complex z-plane for stability and frequency analysis.
• B. It converts continuous-time signals to the frequency domain.
• C. It eliminates noise from signals.
• D. It provides time-domain representation without frequency information.

Answer: (A)

The z-transform maps discrete-time sequences to the complex z-plane, enabling stability assessment and frequency analysis. It turns time-domain difference equations into algebraic ones in z, so you can study how an LTI system will respond to inputs by looking at poles and zeros. The locations of these poles and zeros tell you about the system’s stability (for causal systems, poles inside the unit circle) and how different frequencies are amplified or attenuated. To see the frequency content, you evaluate the z-transform on the unit circle, z = e^{jω}, which gives the frequency response via the discrete-time Fourier transform. The region of convergence also matters, as it indicates which signals produce a finite transform and relates to causality and stability. This framework is distinct from converting continuous-time signals (that uses the Laplace transform) and isn’t about noise elimination by itself.