GATE 2026 IN Q.18: solution and answer

Official answer: ASource: Official answer key, GATE 2026 IN (IIT Guwahati)

1. Fourier series of a square wave. A symmetric square wave that swings between $+A$ and $-A$ with fundamental frequency $f_0$ contains only odd harmonics: $$x(t)=\frac{4A}{\pi }\left[\sin ({\omega }_{0}t)+\frac{1}{3}\sin (3{\omega }_{0}t)+\frac{1}{5}\sin (5{\omega }_{0}t)+\cdots \right],{\omega }_0=2\pi f_0$$

2. Frequencies present. With $f_0=20\text{kHz}$, the spectral lines sit at $20\text{kHz}$, $60\text{kHz}$, $100\text{kHz},\dots$ (plus a DC term only if the wave has an offset, for example one that swings between $0$ and $A$).

3. What an ideal LPF does. An ideal low-pass filter with cut-off $f_c=21\text{kHz}$ passes every component with $f<21\text{kHz}$ unchanged, apart from at most a constant delay, and removes everything above the cut-off completely.

4. What gets through. Only the fundamental gets through, because $20\text{kHz}<21\text{kHz}$. The next harmonic is at $3f_0=60\text{kHz}>21\text{kHz}$, so the filter removes it and every higher harmonic. The conclusion does not depend on the square wave being symmetric: any periodic signal with fundamental $20\text{kHz}$ has its harmonics at $2f_0=40\text{kHz}$ and above, which are all beyond the cut-off.

5. Output. $$y(t)=\frac{4A}{\pi }\sin \left(2\pi \cdot 20\times 10^{3}t\right)$$ This is one sinusoid at $20\text{kHz}$, raised by the DC level if the square wave has one. The output is therefore a 20 kHz sine wave, which is option (A).