In Amplitude Modulation (AM), a sinusoidal carrier wave is modulated by a message signal (or information signal). The carrier wave has a frequency denoted by ωc\omega_c, and the message signal has a frequency denoted by ωm\omega_m.
AM Modulation and Frequency Components
When a signal is modulated, the modulated wave consists of:
1. A carrier frequency (ωc\omega_c),
2. Upper sideband and lower sideband frequencies, which are displaced by ωm\omega_m from the carrier frequency.
Mathematically, the modulated signal is given by:
s(t)=Ac(1+mcos(ωmt))cos(ωct)s(t) = A_c \left( 1 + m \cos(\omega_m t) \right) \cos(\omega_c t)
where:
• AcA_c is the amplitude of the carrier,
• mm is the modulation index,
• ωc\omega_c is the angular frequency of the carrier,
• ωm\omega_m is the angular frequency of the modulating signal.
The spectrum of the modulated signal will consist of:
• A carrier frequency at ωc\omega_c,
• A lower sideband at ωc−ωm\omega_c - \omega_m,
• An upper sideband at ωc+ωm\omega_c + \omega_m.
Bandwidth of the Modulated Signal
The bandwidth of the modulated signal is the range of frequencies that the modulated signal occupies. In the case of AM modulation, the bandwidth is given by:
Bandwidth=2ωm\text{Bandwidth} = 2\omega_m
This is because the signal spans frequencies from ωc−ωm\omega_c - \omega_m to ωc+ωm\omega_c + \omega_m.
Frequency Not Contained in the Modulated Wave
The frequency that is not contained in the modulated wave is the carrier frequency itself, ωc\omega_c, because it does not carry any information (it is a constant tone at ωc\omega_c).
Answer
The frequency not contained in the modulated wave is:
ωc\boxed{\omega_c}
Thus, the correct answer is C) ωc\omega_c.
