The baseband signal m(t) shown in the figure is phasemodulated to generate the PM signal \phi(t)=\cos \left(2 \pi f_{c} t+\right. km(t)). The time t on the x-axis in the figure is in milliseconds. If the carrier frequency is f_{c} = 50 kHz and k = 10π, then the ratio of the minimum instantaneous frequency (in kHz) is to the maximum instantaneous frequency (in kHz) is _____ (rounded off to 2 decimal places).
Chapter 1
Q. 1.1.1
Step-by-Step
Verified Solution
\begin{aligned}\phi(t) &=\cos \left(2 \Pi f_{c} t+k m(t)\right) \\\theta_{i}(t) &=2 \Pi f_{c} t+k m(t) \\f_{i} &=\frac{1}{2 \Pi} \frac{d}{d t} \theta_{i}(t) \\&=\frac{1}{2 \Pi} \frac{d}{d t} \theta_{i}(t) \\&=\frac{1}{2 \Pi} \frac{d}{d t}\left[2 \Pi f_{c} t+k m(t)\right] \\&=\frac{1}{2 \Pi}\left[2 \Pi f_{c}+k\right] \frac{d}{d t} m(t) \\&=f_{c}+\frac{k}{2 \Pi} \frac{d}{d t} m(t) \\f_{c} &=50 KHz \\k &=10 \Pi\end{aligned}
\begin{aligned}&\therefore f_{\max }=50 k +\frac{10 \Pi}{2 \Pi} \cdot \frac{2}{1 ms }=60 k \left[\left.\frac{d m(t)}{d t}\right|_{\max }=2\right] \\&t_{\min }=50 k -\frac{10 \Pi}{2 \Pi} \cdot 1 \times 10^{3}=45 k \left[\left.\frac{d m(t)}{d t}\right|_{\max }=-1\right] \\&\frac{t_{\max }}{f_{\min }}=\frac{60 k }{45 k }=0.75\end{aligned}