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Let a frequency-modulated (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 $W$. It is passed through a nonlinear 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. $\dfrac{3}{2}B_T$
3. $2B_T+W$
4. $\dfrac{5}{2}B_T$

Correct answer: $2B_T+W$

Solution

The nonlinear term $[x(t)]^2$ generates a baseband component and a component centered at $2\omega_c$. To recover the original FM signal, the FM band around $\omega_c$ must not overlap with the distortion band around $2\omega_c$, which leads to the stated minimum carrier condition. Thus the correct answer is $2B_T+W$.

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