Define modulation

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) = A_msinω_mt

where A_m is the amplitude of message signal , ω_m is the angular frequency of the message signal

The equation of waveform of Carrier wave is

c(t) = A_csinω_ct

where cis the amplitude of Carrier wave, ω_c 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

c_m(t) = (A_c + A_msinω_mt)sin ω_ct

or can be written as

c_m(t) = A_c (1 + A_m/A_c sinω_mt)sin ω_ct

so this ratio of amplitude of original message signal to amplitude of carrier wave is called modulation index

𝜇 = A_m/A_c

Where 𝜇 is the modulation index , A_m is amplitude of message signal and A_c is amplitude of carrier wave

If the modulation index is greater than 1, when A_m > A_c ( 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 = A_m/A_c the equation of amplitude modulated wave becomes

c_m(t) = A_c (1 + 𝜇 sinω_mt)sinω_ct

or on rearranging we get

c_m(t) = A_csinω_ct + 𝜇 A_csinω_ct sinω_mt

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 ω_c , (ω_c + ω_m)and (ω_c - ω_m) 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.