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GATE 2023 EC – Question 45

Communications · Analog Communications · 2 marks · Multiple choice

Let a frequency modulated (FM) signal $x(t)=A\cos\left(\omega_ct+k_f\int_{-\infty}^tm(\lambda)d\lambda\right)$, where $m(t)$ is a message signal of bandwidth $W$. It is passed through a non-linear system with output $y(t)=2x(t)+5(x(t))^2$. Let $B_T$ denote the FM bandwidth. The minimum value of $\omega_c$ required to recover $x(t)$ from $y(t)$ is

  1. $B_T+W$
  2. $\dfrac32B_T$
  3. $2B_T+W$
  4. $\dfrac52B_T$

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Correct answer: (B) $\dfrac32B_T$

Explanation

$x^2$ contains a baseband term of width $2B_T$ (centred at 0, occupying $[-B_T,B_T]$ in frequency terms) and a term around $2\omega_c$ of width $2B_T$ ($[2\omega_c-B_T,2\omega_c+B_T]$), while $x$ itself occupies $[\omega_c-B_T/2,\ \omega_c+B_T/2]$. To filter out $x$ without overlap: $\omega_c-B_T/2\ge B_T$ gives $\omega_c\ge\tfrac32B_T$, and $2\omega_c-B_T\ge\omega_c+B_T/2$ gives the same bound. The minimum is $\tfrac32B_T$.