Final answer:
The message signal m(t) for the provided PM signal s(t) = 20cos[2π80(10⁶)t + 5sin(2π5000t)] is m(t) = 5/4sin(2π5000t) when the phase modulation constant kₚ is 4.
Step-by-step explanation:
The student is asking about determining the message signal from the given phase modulation (PM) signal equation. Given the angle-modulated signal s(t) = 20cos[2π80(10⁶)t + 5sin(2π5000t)], with a phase modulation constant kₚ = 4. The message signal m(t) can be equated to the phase variation of the signal, which is the part inside the cosine function after the carrier frequency term. In this case, 5sin(2π5000t) represents the phase modulation.
The message signal can be determined by dividing the phase function by the modulation index kₚ, which is given as 4. Therefore, m(t) is 5/4sin(2π5000t). However, because the given value of 5 is the peak phase deviation Φ and kₚ = Φ/mₒ(t) for a PM signal, the peak value mₒ(t) of the message signal is Φ/kₚ, meaning the message signal is m(t) = 5/4sin(2π5000t), which is the phase variation term divided by kₚ.