# 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.

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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