Question

What is the primary difference between CTFT and DTFT?

• A. CTFT is periodic and DTFT is non-periodic
• B. CTFT transforms a discrete-time signal and DTFT transforms a continuous-time signal
• C. CTFT is non-periodic and DTFT is periodic
• D. CTFT is defined for periodic signals and DTFT is for non-periodic signals

Hint:

Remember the core Fourier duality rule: - Continuous Time $\leftrightarrow$ Non-periodic Spectrum (CTFT) - Discrete Time $\leftrightarrow$ Periodic Spectrum (DTFT)

Answer 1

The Correct Option is C

Concept: The duality relationships between time-domain characteristics and their corresponding frequency-domain characteristics dictate that:
• Discretization in one domain leads directly to periodicity in the alternate domain.
• Continuity in one domain yields a non-periodic behavior in the alternate domain.

Step 1: Structural analysis of the Continuous-Time Fourier Transform (CTFT).
The CTFT applies to continuous-time, non-periodic signals $x(t)$. Its mathematical definition is: \[ X(\Omega) = \int_{-\infty}^{\infty} x(t) e^{-j\Omega t} \, dt \] Because the time-domain signal $x(t)$ is continuous and non-periodic, its spectral representation $X(\Omega)$ is a continuous, non-periodic function across the continuous frequency variable $\Omega \in (-\infty, \infty)$.

Step 2: Structural analysis of the Discrete-Time Fourier Transform (DTFT).
The DTFT applies to discrete-time sequences $x[n]$. Its mathematical formulation is given by: \[ X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \] Let us examine the periodicity by evaluating the spectrum at a shifted frequency $(\omega + 2\pi)$: \[ X(e^{j(\omega + 2\pi)}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j(\omega + 2\pi)n} = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} e^{-j2\pi n} \] Since $n$ is an integer, $e^{-j2\pi n} = 1$ for all valid values of $n$. Therefore: \[ X(e^{j(\omega + 2\pi)}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} = X(e^{j\omega}) \] This mathematically proves that the DTFT is inherently periodic with a fundamental period of $2\pi$. Thus, CTFT spectra are non-periodic, whereas DTFT spectra are periodic.