Q. 22

A message signal

Answer :

In order to transmit some message over a long distance, amplitude modulation is used, where an 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 by the message signal


As shown in figure



The equation of waveform of the message signal is


m(t) = Amsinωmt


where Amis the amplitude of the message signal, ωm is the angular frequency of the message signal


ωm = 2πfm


where fm is the frequency of the message signal or modulating signal.


Here,


Am = 20 V


fm = 20 KHz = 2 × 103 Hz = 2000 Hz


The equation of waveform of the Carrier wave is


c(t) = Acsinωct


where Ac is the amplitude of Carrier wave, ωc is the angular frequency of the Carrier wave


ωc = 2πfc


where fc is the frequency of the Carrier Wave


Here,


Ac = 40 V


fc = 2 MHz = 2 × 106 Hz = 2000 × 103 Hz = 2000 KHz


(i) The ratio of the amplitude of the original message signal to the amplitude of carrier wave is called modulation index


𝜇 = Am/Ac


Where 𝜇 is the modulation index, Am is athe mplitude of message signal and Ac is athe mplitude of the arrier wave


Here we have


Am = 20 V


Ac = 40 V


So here the modulation index is


𝜇 = 20V/40V = 1/2 = 0.5


(ii) The Equation of amplitude modulated wave is



So we can see it consist of three waveforms, Original Carrier wave with amplitude Ac and frequency fc, Lower side Band with amplitude 𝜇Ac/2 and frequency (fc - fm) i.e. difference between carrier wave frequency and modulating signal frequency, and upper side side Band with amplitude 𝜇Ac/2 and frequency (fc + fm) i.e. sum of carrier wave frequency and modulating signal frequency.


Here,


Modulating signal Frequency is


fm = 20 KHz


Carrier Wave Frequency is


fc= 2000 KHz


so the lower side band frequency is


fc - fm = 2000 KHz - 20 KHz = 1980 KHz


and the Upper side band frequency is


fc + fm = 2000 KHz + 20 KHz = 2020 KHz


Amplitude of both the side band is



So the frequency spectrum consisting of carrier wave and side bands, is as shown



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