Derive expression for amplitude modulated wave.
Deducing expression for lower and upper side bands.
Obtaining expression for modulation index.
Let a carrier wave be given by
$\mathit{c}\left(\mathit{t}\right)={\mathit{A}}_{\mathit{c}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{c}}\mathit{t}$ where ${\mathit{\omega }}_{\mathit{c}}=2\mathit{\pi }{\mathit{f}}_{\mathit{c}}$
And signal wave be
$\mathit{m}\left(\mathit{t}\right)\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{m}}\mathit{t}$ where ${\mathit{\omega }}_{\mathit{m}}=2\mathit{\pi }{\mathit{f}}_{\mathit{m}}$
The modulated signal is
${\mathit{c}}_{\mathit{m}}\left(\mathit{t}\right)=\left({\mathit{A}}_{\mathit{c}}+{\mathit{A}}_{\mathit{m}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{m}}\mathit{t}\right)\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{c}}\mathit{t}$
${\mathit{c}}_{\mathit{m}}\left(\mathit{t}\right)={\mathit{A}}_{\mathit{c}}\left(1+\frac{{\mathit{A}}_{\mathit{m}}}{{\mathit{A}}_{\mathit{c}}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{m}}\mathit{t}\right)\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{c}}\mathit{t}$
${\mathit{c}}_{\mathit{m}}\left(\mathit{t}\right)={\mathit{A}}_{\mathit{c}}\mathit{s}\mathit{i}\mathit{n}{\mathit{\omega }}_{\mathit{c}}\mathit{t}+\mathit{\mu }\frac{{\mathit{A}}_{\mathit{c}}}{2}\cos \left({\mathit{\omega }}_{\mathit{c}}-{\mathit{\omega }}_{\mathit{m}}\right)\mathit{t}$ $-\mathit{\mu }\frac{{\mathit{A}}_{\mathit{c}}}{2}\cos \left({\mathit{\omega }}_{\mathit{c}}+{\mathit{\omega }}_{\mathit{m}}\right)\mathit{t}$
The modulation index $\mathit{\mu }=\frac{{\mathit{A}}_{\mathit{m}}}{{\mathit{A}}_{\mathit{c}}}$
Lower frequency band ${\mathit{\omega }}_{\mathit{c}}-{\mathit{\omega }}_{\mathit{m}}$
Upper frequency band ${\mathit{\omega }}_{\mathit{c}}+{\mathit{\omega }}_{\mathit{m}}$