Signal Classes vs. Fourier Representations Match each time-domain class to the nature of its Fourier representation. List I (Time-Domain Signal) A. Continuous and aperiodic B. Continuous and periodic C. Discrete and aperiodic D. Discrete and periodic List II (Fourier Representation) Continuous and aperiodic Discrete and aperiodic Continuous and periodic Discrete and periodic

Introduction / Context:
Transform-domain intuition helps engineers choose the right analysis tool. This problem checks the standard correspondences between time-domain signal classes and the structure of their spectra (Fourier transform or series/DTFT forms).

Given Data / Assumptions:

• Continuous-time aperiodic signals have well-defined Fourier transforms (under usual conditions).
• Continuous-time periodic signals have Fourier series line spectra.
• Discrete-time aperiodic sequences use the DTFT, which is periodic in frequency.
• Discrete-time periodic sequences produce discrete line spectra in frequency and are periodic.

Concept / Approach:

Map each time-domain class to the familiar spectral structure: CT aperiodic → continuous aperiodic spectrum; CT periodic → discrete line spectrum (often listed as “discrete and aperiodic” here); DT aperiodic → continuous periodic DTFT; DT periodic → discrete, periodic set of spectral lines.

Step-by-Step Solution:

Verification / Alternative check:

Examples: A time-limited pulse (CT) has a sinc-like continuous spectrum; a CT sinusoid yields impulses in frequency; a DT rectangular window has a periodic sinc DTFT; a periodic DT sequence yields impulses at harmonics modulo 2π.

Why Other Options Are Wrong:

Swapping continuous/discrete or periodic/aperiodic labels contradicts theorems relating signal periodicity and transform domain structure.

Common Pitfalls:

Confusing the “periodic in frequency” property of DTFT with the “discrete in frequency” property of Fourier series; both can occur but under different time-domain conditions.

Final Answer:

A-1, B-2, C-3, D-4