Q24 of 805 Page 1

A sinusoidal carrier wave of amplitude AC and angular frequency ωC is modulated in accordance with a sinusoidal information signal of amplitude Am and angular frequency ωm. Show that the amplitude modulated signal contains three frequencies centered around ωC. Draw the frequency spectrum of the resulting modulated signal.

The carrier wave is,


c(t)=Ac sinωc t


The message signal is,


(t)=Am sinωm t


So the modulated signal cm(t) can be written as,


cm (t)=(Ac+Am sinωm t)sinωc t



we can write,


cm (t)=Ac sinωc t+μAc sinωc tsinωm t


Here,



Is called the modulation index, Using the trigonometric relation,



we can write,



Here ωc-ωmand ωc+ωmare respectively called the lower side and upperside frequencies. The modulated signal now consists of thecarrier waveof frequency ωc plus two sinusoidal waves each with a frequency slightly different from, known as side bands. The frequency spectrum of the amplitude modulated signal is shown below,



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