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GATE EC 2020 Set 1 — Question 57

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NAT+2 / -0HardAngle Modulation (FM & PM)Analog CommunicationsCommunications

Communications → Analog Communications → Angle Modulation (FM & PM)

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Question

SPM(t)S_{PM}(t) and SFM(t)S_{FM}(t) as defined below, are the phase modulated and the frequency modulated waveforms, respectively, corresponding to the message signal m(t)m(t) shown in the figure: m(t)m(t) is 0 up to t=1t = 1 s, rises linearly to 10 V at t=3t = 3 s, stays at 10 V until t=6t = 6 s, falls linearly to 0 at t=7t = 7 s, and is 0 afterwards.SPM(t)=cos⁡(1000πt+Kpm(t))S_{PM}(t) = \cos(1000\pi t + K_p m(t))andSFM(t)=cos⁡(1000πt+Kf∫−∞tm(τ)dτ)S_{FM}(t) = \cos\left(1000\pi t + K_f \int_{-\infty}^{t} m(\tau) d\tau\right)where KpK_p is the phase deviation constant in radians/volt and KfK_f is the frequency deviation constant in radians/second/volt. If the highest instantaneous frequencies of SPM(t)S_{PM}(t) and SFM(t)S_{FM}(t) are same, then the value of the ratio KpKf\frac{K_p}{K_f} is ______ seconds.
Graph of message signal m(t) versus time t showing a trapezoidal pulse starting at t=1, reaching 10V at t=3, staying at 10V until t=6, and returning to 0V at t=7
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