GATE 2023 EC Q.29: solution and answer

Official answer: BSource: Official answer key, GATE 2023 EC (IIT Kanpur)

1. Duality rule between the two domains: periodic in one domain $\iff$ discrete in the other, and discrete (sampled) in one domain $\iff$ periodic in the other. The complementary pairs also hold: aperiodic in one domain $\iff$ continuous in the other, and continuous in one domain $\iff$ aperiodic in the other.

2. (i) Continuous-time, aperiodic $x(t)$: the transform is the CTFT $X(j\omega )={\int }_{-\infty }^{\infty }x(t)e^{-j\omega t}dt$. Aperiodic in time gives a continuous spectrum, and continuous time gives an aperiodic spectrum, so (i) $\to$ (a).

3. (ii) Continuous-time, periodic $x(t)$ with period $T_0$: the tool is the continuous-time Fourier series, whose spectrum has lines only at $k{\omega }_0$, where ${\omega }_0=\frac{2\pi }{T_0}$. Periodic in time gives a discrete spectrum, and continuous time gives an aperiodic spectrum, so (ii) $\to$ (c).

4. (iii) Discrete-time, aperiodic $x[n]$: the transform is the DTFT $X(e^{j\Omega })={\sum }_{n=-\infty }^{\infty }x[n]e^{-j\Omega n}$, which satisfies $X(e^{j(\Omega +2\pi )})=X(e^{j\Omega })$ because $e^{-j2\pi n}=1$ for every integer $n$. Discrete time gives a periodic spectrum, and aperiodic time gives a continuous spectrum, so (iii) $\to$ (b).

5. (iv) Discrete-time, periodic $x[n]$ with period $N$: the tool is the discrete-time Fourier series (the DFT form). Its coefficients $c_k$ exist only at integer $k$ and repeat, $c_{k+N}=c_k$, so the spectrum is discrete and periodic: (iv) $\to$ (d).

6. The matching is (i) $\to$ (a), (ii) $\to$ (c), (iii) $\to$ (b), (iv) $\to$ (d), which is option (B).