If the amplitude of modulating signal is kept constant then the modulation index is proportional to the:

CONCEPT:

Amplitude modulation:

• In amplitude modulation, the amplitude of the carrier wave is varied in accordance with the information signal.
• Here we explain the amplitude modulation process using a sinusoidal signal as the modulating signal.
• Let c(t) = Acsin(ωct) represent carrier wave and m(t) = Amsin(ωmt) represent the message or the modulating signal.
• So the modulated signal can be represented as,

⇒ cm(t) = [Ac + Am.sin(ωmt)].sin(ωct)

⇒ cm(t) = [Ac + μAc.sin(ωmt)].sin(ωct)

Where Ac = amplitude of the carrier wave, Am = amplitude of the modulating signal, ωc = angular frequency of the carrier wave, ωm = angular frequency of the modulating signal, and μ = modulation index

• The modulation index is given as,

• In practice, μ is kept  to avoid distortion.
• Using the trignomatric relation 2sinA.sinB = cos(A - B) - cos(A + B), we can represent the modulated signal as,

• Here (ωc - ωm) and (ωc + ωm) are respectively called the lower side and upper side frequencies.
• The modulated signal now consists of the carrier wave of frequency ωc plus two sinusoidal waves each with a frequency slightly different from ωc, known as sidebands.

• As long as the broadcast frequencies (carrier waves) are sufficiently spaced out so that sidebands do not overlap, different stations can operate without interfering with each other.

EXPLANATION:

• We know that if c(t) = Acsin(ωct) represent carrier wave and m(t) = Amsin(ωmt) represent message signal, then the modulation index is equal to the,

Where Ac = amplitude of the carrier wave, and Am = amplitude of the modulating signal

• If the amplitude of modulating signal is kept constant then,

• Hence, option 2 is correct.