An audio signal given by $e_s=15\sin 2\pi \left(2000t\right)$  modulates a carrier wave $e_c=80\sin 2\pi \left(100,000t\right)$. Calculate frequency spectrum of the modulate wave.

$$\begin{gathered} By comparing equations with s\tan dard form \\ {e}_s=e_s\sin 2\pi f_{s}t=15\sin 2\pi \left(2000\right)t \\ {e}_c=e_c\sin 2\pi f_{c}t=80\sin 2\pi \left(100,000t\right) \\ So, f_s=2000Hz\text{\hspace{0.17em}, \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{f}_c=100,000Hz \\ \therefore f_{LSB}=f_c-f_s=100000-2000=98KHz \\ {f}_{USB}=f_c+f_s=100000+2000=102KHz \\ Thus, frequency spectrum of modulated wave extends from 98KHz to 102 KHz \end{gathered}$$