Fourier tools and their signal/spectrum properties — match each to the appropriate description List I A. Fourier Series (FS) B. Fourier Transform (FT) C. Discrete-Time Fourier Transform (DTFT) D. Discrete Fourier Transform (DFT) List II (Property) 1. Discrete, periodic 2. Continuous, periodic 3. Discrete, aperiodic 4. Continuous, aperiodic

Introduction / Context:
Each Fourier representation pairs a class of signals with characteristic spectra. Knowing whether the spectrum is discrete or continuous, and periodic or aperiodic, helps choose the right tool for analysis and implementation (e.g., spectral lines vs. continuous bands).

Given Data / Assumptions:

• FS models continuous-time periodic signals.
• FT models continuous-time aperiodic signals.
• DTFT models discrete-time aperiodic sequences and yields periodic spectra.
• DFT computes sampled spectra of finite-length sequences, producing discrete spectra that repeat periodically.

Concept / Approach:
We map as follows: FS → discrete, periodic spectrum (sums of harmonics); FT → continuous, aperiodic spectrum; DTFT → continuous spectrum periodic in frequency (period 2π in rad/s); DFT → discrete spectrum that is periodic due to circularity.

Step-by-Step Solution:

Verification / Alternative check:
Standard signals texts depict FS with spectral lines at integer multiples; DTFT magnitude plots repeat every 2π; DFT bins are discrete and circularly periodic; FT spectra fill a continuum for aperiodic signals.

Why Other Options Are Wrong:

• Assigning “TEM”-like periodicity to FT or discreteness to DTFT contradicts definitions.

Common Pitfalls:
Confusing “periodic in time” with “periodic in frequency”. FS: time periodic → discrete spectrum; DTFT: time discrete → frequency periodic.

Final Answer:
A-1, B-4, C-2, D-3