Let x$$_1$$(t) and x$$_2$$(t) be two band-limited signals having bandwidth $$B=4\pi\times10^3$$ rad/s each. In the figure below, the Nyquist sampling frequency, in rad/s, required to sample y(t), is

Consider an FM broadcast that employs the pre-emphasis filter with frequency response

$${H_{pe}}(\omega ) = 1 + {{j\omega } \over {{\omega _0}}}$$,

For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate pairs of (R, C) values is/are ____________.

A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to a phase modulator with phase deviation constant $2 \mathrm{rad} /$ volt. If the carrier signal is $c(t)=2 \cos \left(2 \pi 10^6 t\right)$, the maximum instantaneous frequency of the phase modulated signal (rounded off to one decimal place) is $\_\_\_\_$ Hz.

$S_{P M}(t)$ and $S_{F M}(t)$ are defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal $m(t)$ shown in the figure.

$$ \begin{aligned} & S_{P M}(t)=\cos \left[1000 \pi t+k_p m(t)\right] \\ & S_{F M}(t)=\cos \left[1000 \pi t+k_f \int_{-\infty}^t m(\tau) d \tau\right] \end{aligned} $$

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 $S_{P M}(t)$ and $S_{F M}(t)$ are same, then the value of the ratio $\frac{k_p}{k_f}$ is $\_\_\_\_$ seconds.