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  3. A signal of low frequency is to be transmitted using a carrier wave of frequency . Derive
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A signal of low frequency f_{m} is to be transmitted using a carrier wave of frequency f_{c}. Derive the expression for the amplitude modulated wave and deduce expressions for the lower and upper sidebands produced. Hence, obtain the expression for modulation index.

 

 

Answers (1)
S Safeer PP

Let's derive an expression for amplitude modulated wave:-

Let a carrier wave be given by;

C(t)=A_{c}\sin \; \omega _{c}t

where, \omega _{c}= 2\pi fc and

A signal wave be -

m(t)=A_{m}\sin \; \omega_{m}t

where, \omega _{m}=2\pi f_{m}

The modulated signal is given by :

C_{m}(t)= (A_{c}+A_{m}\sin \omega _{m}t)\sin \omega _{c}t

C_{m}(t)=A_{c} (1+A_{m}/A_{c}\sin \omega _{m}t)\sin \omega _{c}t

C_{m}(t)=A_{c} \sin \omega _{c}t+ \frac{\mu}{2} \; \cos (\omega _{c}-\omega _{m})t-\mu \frac{A_{c}}{2} \; \cos (\omega _{c}+\omega _{m})t

Hence, the lower sideband frequency is given as, (\omega _{c}-\omega _{m}) and the upper sideband frequency is given as, (\omega _{c}+\omega _{m}).

The modulation index, \mu =\frac{A_{m}}{A_{c}}

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