Final answer:
The FM modulator's modulation index is 1.5, calculated as the ratio of frequency deviation to modulating frequency. The PM modulator's modulation index is also 1.5, but it's measured in radians as the product of deviation sensitivity and the signal amplitude. Both modulation types produce sidebands around the carrier frequency, affecting the frequency spectrum.
Step-by-step explanation:
To calculate the modulation index for both the FM and PM modulators, we utilize the given parameters. For the FM modulator, the modulation index (β) is given by the ratio of the frequency deviation to the modulating frequency. The deviation sensitivity (k) for the FM modulator is 1.5 kHz/V, and the amplitude of the modulating signal (vᵤ) is 2 V. Therefore, the frequency deviation (Δf) is k × vᵤ, which gives us Δf = 1.5 kHz/V × 2 V = 3 kHz. If the modulating frequency (fᵤ) is 2 kHz (since the modulating signal is sin(2π × 2kHz t)), the modulation index for the FM modulator is β = Δf / fᵤ = 3 kHz / 2 kHz = 1.5.
For the PM modulator, the modulation index (μ) is given by the peak phase deviation, which depends on the product of the deviation sensitivity (kᵰ) and the amplitude of the modulating signal (vᵤ). The deviation sensitivity for the PM modulator is 0.75 rad/V, and thus, the peak phase deviation is μ = kᵰ × vᵤ = 0.75 rad/V × 2 V = 1.5 radians.
The output frequency spectrum from both modulators will contain the carrier frequency and sidebands that are generated around the carrier frequency. In the case of FM modulation, the number and amplitude of the sidebands depend on the modulation index and the modulating signal. Higher modulation indexes lead to a wider bandwidth. In PM modulation, the sidebands' amplitude also depends on the modulation index, but since phase modulation is directly related to frequency modulation through the derivative/integral relationship between phase and frequency, the FM and PM spectra can be similar for a given modulation index and modulating signal.