GATE 2025 EC – Question 14
Consider a frequency-modulated (FM) signal
$$f(t)=A_c\cos\big(2\pi f_ct+3\sin(2\pi f_1t)+4\sin(6\pi f_1t)\big),$$
where $A_c$ and $f_c$ are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency $f_1$ is in Hz, and assume that $f_c>100f_1$.
The peak frequency deviation of the FM signal in Hz is ________.
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Correct answer: (A) $15f_1$
Explanation
The phase deviation is $\theta(t)=3\sin(2\pi f_1t)+4\sin(2\pi\cdot3f_1t)$. The frequency deviation is $\frac1{2\pi}\frac{d\theta}{dt}=3f_1\cos(2\pi f_1t)+12f_1\cos(6\pi f_1t)$. Both cosines equal 1 at $t=0$, so the peak deviation is $3f_1+12f_1=15f_1$.