Topic: Communication Systems
The signal $m(t)$ as shown is applied both to a phase modulator (with $k_{p}$ as the phase constant) and a frequency modulator with ($k_{f}$ as the frequency constant) having the same carrier frequency. The ratio $k_{p}/k_{f}$ (in rad/Hz) for the same maximum phase deviation is
(Assuming $m(t)$ is a square wave alternating between +2 and -2 with a period of 4 seconds as per the source).
- A. $8\pi$
- B. $4\pi$
- C. $2\pi$
- D. $\pi$
Answer: B. $4\pi$
Explanation: For phase modulation, the phase deviation is $k_p m(t)$. For frequency modulation, the phase deviation is $2\pi k_f \int m(t) dt$. For the same maximum phase deviation: $k_p \cdot |m(t)|_{\max} = 2\pi k_f \cdot |\int m(t) dt|_{\max}$. The maximum value of $m(t)$ is 2. The maximum value of its integral (which forms a triangular wave) is $2 \times (T/2) = 2 \times 2 = 4$. So, $k_p(2) = 2\pi k_f(4) \Rightarrow k_p = 4\pi k_f \Rightarrow \frac{k_p}{k_f} = 4\pi$.