Identify the mathematical expression for amplitude modulated wave:

Options

A. `A_(c )sin[{omega_(c )+k_(1)V_(m)(t)}t+phi]`
B. `A_(c )sin{omega_(c )t+phi+k_(2)V_(m)(t)}`
C. `{A_(c ) + k_(2)V_(m)(t)}sin(omega_(c )t +phi)`
D. `A_(c) V_(m)(t) sin (omega_(c ) t+phi)`

Answer

C

Explanation

Consider a sinusoidal modulating signal represented by
`A_(c) V_(m)(t) sin (omega_(c ) t+phi)`
where, `A_(m)` = Amplitude of modulating signal `omega_(m)` = Angular frequency `= 2pi V_(m) = phi V_(m)`
Also consider a sinusoidal carrier wave represented by `C(t) = A_(c ) sin omega_(c )t" .....(ii)"`
Thus, modulated wave is given by
`C_(m) (t) = (A_(c ) +A_(m)sin omega_(m)t)sin omega_(c ) t`
`=A_(c)[1+(A_(m))/(A_(c))sin omega_(m)t)sinomega_(c)t`
Here, `(A_(m))/(A_(c ))=M`
`implies C_(m)(t) = (A_(c ) +A_(c)xxmusinomega_(m)t)sin omega_(c )t" .....(iii)"`
Now, we know thta `A_(c) xx mu = K` [wave constant]
and `sin omega_(m)t = V_(m)` [wave velocity]
Thus, Eq. (iii) becomes
`C_(m) (t) = (A_(c) +K xx V_(m))sin omega_(c ) t`
Now, consider a change in phase angle by `phi` then `sin omega_(c ) t rarr sin (omega_(c ) t + phi)`
Thus, `C_(m)(t) = (A_(c)+KV_(m))(sin omega_(c ) +phi)`