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) = A_{m}sinω_{m}t

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_{c}sinω_{c}t

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_{m}sinω_{m}t)sin ω_{c}t

or can be written as

c_{m}(t) = A_{c} (1 + A_{m}/A_{c} sinω_{m}t)sin ω_{c}t

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_{m}t)sinω_{c}t

or on rearranging we get

c_{m}(t) = A_{c}sinω_{c}t + 𝜇 A_{c}sinω_{c}t sinω_{m}t

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.

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