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 $c(t)=A_c \sin \omega_c t$ where $\omega_c = 2 \pi f_c$ And signal wave be $m(t) \sin \omega_m t$ where $\omega_m = 2 \pi f_m$ The modulated signal is $c_m(t)=(A_c + A_m \sin \omega_m t) \sin \omega_c t$ $c_m(t)=A_c \left(1+\frac{A_m}{A_c} \sin \omega_m t\right) \sin \omega_c t$ $c_m(t)=A_c \sin \omega_c t + \mu \frac{A_c}{2} \cos (\omega_c - \omega_m) t$ $- \mu \frac{A_c}{2} \cos (\omega_c + \omega_m) t$ The modulation index $\mu = \frac{A_m}{A_c}$ Lower frequency band $\omega_c - \omega_m$ Upper frequency band $\omega_c + \omega_m$