To calculate the modulation index, we need to determine the peak frequency deviation (Δf) caused by the modulating signal and the maximum frequency deviation (Δf_max) allowed by the FM system.
Given: Modulating signal: Vm(t) = 4sin(2π 5000t) Deviation sensitivity: 2 kHz/V Carrier signal: Vc(t) = 110sin(2π 90Mt)
a. Peak Frequency Deviation (Δf): The peak frequency deviation is given by the formula: Δf = Δv × deviation sensitivity
Since the modulating signal Vm(t) has an amplitude of 4, the peak voltage deviation (Δv) is 4. Therefore: Δf = 4 V × 2 kHz/V = 8 kHz
b. Modulation Index (β): The modulation index is the ratio of the peak frequency deviation (Δf) to the maximum frequency deviation (Δf_max) allowed by the FM system. For narrowband FM, Δf_max is typically equal to the maximum modulating frequency (Wm).
Given that the modulating frequency is 5000 Hz, we can assume that Δf_max = Wm = 5000 Hz. Therefore: β = Δf / Δf_max = 8 kHz / 5 kHz = 1.6
c. Amplitudes of Sidebands: For FM modulation, the amplitude of each sideband can be calculated using the Carson's rule, which states that most of the signal power is contained within a bandwidth of 2Δf + Wm, where Δf is the peak frequency deviation and Wm is the modulating frequency.
In this case, the bandwidth will be 2Δf + Wm = 2(8 kHz) + 5000 Hz = 16 kHz + 5000 Hz = 16.005 MHz.
The amplitudes of the sidebands can be calculated using the Bessel function of the first kind, Jn(x), where n represents the order of the sideband. The sidebands occur at frequencies that are integer multiples of the modulating frequency (Wm).
For the given carrier frequency of 90 MHz (90 × 10^6 Hz), the sidebands will occur at frequencies:
Sideband 1: 90 MHz - 16.005 MHz = 73.995 MHz Sideband 2: 90 MHz + 16.005 MHz = 106.005 MHz
The amplitudes of the sidebands can be determined using Bessel functions as follows:
Sideband 1: Amplitude = 2 × J1(β) = 2 × J1(1.6) ≈ 1.27
Sideband 2: Amplitude = 2 × J1(β) = 2 × J1(1.6) ≈ 1.27
Therefore, the amplitudes of both sidebands will be approximately 1.27.