GATE EC 2026 Set 1 Q18 — Analog Communications

Consider a real, narrowband signal $x(t)=A(t) \cos \left[2 \pi f_{c} t+\theta(t)\right]$ where the maximum frequency components of $A(t)$ and $\theta(t)$ are $f_{M}$ and $f_{c}\left(=1000 f_{M}\right)$, respectively. Which of the following statements is/are correct for $-\infty < t < \infty ?$

Consider a frequency-modulated (FM) signal $$f(t)=A_{c} \cos \left(2 \pi f_{c} t+3 \sin \left(2 \pi f_{1} t\right)+4 \sin \left(6 \pi f_{1} t\right)\right)$$ where $A_{c}$ and $f_{c}$ are, respectively, the amplitude and frequency (in $\mathrm{Hz}$ ) of the carrier waveform. The frequency $f_{1}$ is in $\mathrm{Hz}$ , and assume that …

Consider a message signal $m(t)$ which is bandlimited to $[-W, W]$, where $W$ is in $\mathrm{Hz}$ . Consider the following two modulation schemes for the message signal: Double sideband-suppressed carrier (DSB-SC): $$f_{D S B}(t)=A_{c} \: m(t) \cos \left(2 \pi f_{c} t\right)$$ Amplitude modulation (AM): …

An amplitude modulator has output (in Volts) $$s(t)=A \cos (400 \pi t)+B \cos (360 \pi t)+B \cos (440 \pi t) .$$ The carrier power normalized to $1 \Omega$ resistance is $50$ Watts. The ratio of the total sideband power to the total power is $1 / 9$. The value of $\text{B}$ (in Volts, rounded off to two decimal places) is $\_\_\_\_\_\_\_$.

Let a frequency modulated $\text{(FM)}$ signal $x(t)=A \cos \left(\omega_{c} t+k_{f} \int_{-\infty}^{t} m(\lambda) d \lambda\right)$, where $m(t)$ is a message signal of bandwidth $\text{W.}$ It is passed through a non-linear system with output $y(t)=2 x(t)+5(x(t))^{2}$. Let $B_{T}$ denote the $\text{FM}$ bandwidth. The minimum value of …

Consider a carrier signal which is amplitude modulated by a single-tone sinusoidal message signal with a modulation index of $50\%$. If the carrier and one of the sidebands are suppressed in the modulated signal, the percentage of power saved (rounded off to one decimal place) is ______________