Answer :
In order to transmit some message over long distance, amplitude modulation is used, where a initially low frequency message signal is superimposed on a high frequency carrier wave as a result a new wave is formed having high frequency whose amplitude is varying in accordance with message signal
As shown in figure
The equation of waveform of message signal is
m(t) = Amsinωmt
where Am is the amplitude of message signal , ωm is the angular frequency of the message signal
The equation of waveform of Carrier wave is
c(t) = Acsinωct
where cis the amplitude of Carrier wave, ωc is the angular frequency of the Carrier wave
the equation of waveform of amplitude modulated wave having high frequency of carrier wave and variation in amplitude in accordance with message signal is
cm(t) = (Ac + Amsinωmt)sin ωct
or can be written as
cm(t) = Ac (1 + Am/Ac sinωmt)sin ωct
so this ratio of amplitude of original message signal to amplitude of carrier wave is called modulation index
𝜇 = Am/Ac
Where 𝜇 is the modulation index , Am is amplitude of message signal and Ac is amplitude of carrier wave
If the modulation index is greater than 1, when Am > Ac ( as can be seen from the equation) that means the amplitude of the modulating wave is higher than that of the carrier wave. This cause’s over- modulation of the carrier wave which results in distortion .i.e. some portion of original message signal will be lost in the process so a low modulation index is always required
Now putting 𝜇 = Am/Ac the equation of amplitude modulated wave becomes
cm(t) = Ac (1 + 𝜇 sinωmt)sinωct
or on rearranging we get
cm(t) = Acsinωct + 𝜇 Acsinωct sinωmt
further solving and using trigonometric identity
sinA sinB = 1/2 {(cos(A – B) – cos (A + B)}
we get
As we can see there in total three frequencies ωc , (ωc + ωm)and (ωc - ωm) called as band frequencies , i.e. 3 waveforms. now ,while producing a amplitude modulated wave some other frequencies are also produced, bandpass filter is used to reject other frequencies and allow only these three frequencies to pass through while transmission of amplitude modulated wave.
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