GATE 2020 EC – Question 57
$S_{PM}(t)$ and $S_{FM}(t)$ as defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal $m(t)$ shown in the figure.
$$S_{PM}(t)=\cos\left(1000\pi t+K_pm(t)\right)\quad\text{and}\quad S_{FM}(t)=\cos\left(1000\pi t+K_f\int_{-\infty}^tm(\tau)\,d\tau\right)$$
where $K_p$ is the phase deviation constant in radians/volt and $K_f$ is the frequency deviation constant in radians/second/volt. If the highest instantaneous frequencies of $S_{PM}(t)$ and $S_{FM}(t)$ are same, then the value of the ratio $\frac{K_p}{K_f}$ is ________ seconds.
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Correct answer: 2
Explanation
For PM the instantaneous angular frequency is $1000\pi+K_p\,m'(t)$. The steepest upward slope of $m(t)$ is $\frac{10}{3-1}=5$ V/s, so its maximum is $1000\pi+5K_p$. For FM it is $1000\pi+K_fm(t)$, with maximum $1000\pi+10K_f$. Equating, $5K_p=10K_f$, so $\frac{K_p}{K_f}=2$ s.