If the two side band frequencies of a modulated wave are 1050 kHz and 950 kHz, then find the frequency of the modulating wave.

CONCEPT:

Amplitude modulation:

• In amplitude modulation, the amplitude of the carrier 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 ≤ q1 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 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.

CALCULATION:

Given fc + fm = 1050 kHz and fc - fm = 950 kHz

Where fc = frequency of the carrier wave and fm = frequency of the modulating signal

• We know that that (fc + fm) and (fc - fm) are the sidebands of a modulated signal.
• Therefore by adding the two sidebands,

⇒ (fc + fm) - (fc - fm) = (1050 - 950) kHz

⇒ 2fm = 100 kHz

⇒ fm = 50 kHz

• Hence, option 2 is correct.