Q. 6 5.0( 2 Votes )

Define modulation index. Why is it kept low? What is the role of a bandpass filter?

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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