Single tone DSB-FC AM in time and frequency domain

Time-domain expression

Let the message and carrier be

$$m(t) = A_m\cos2\pi f_mt, \qquad c(t) = A_c\cos2\pi f_ct, \quad f_c \gg f_m$$

In DSB-FC AM the carrier amplitude varies linearly with $m(t)$:

$$\begin{aligned} s(t) &= [A_c + A_m\cos2\pi f_mt]\cos2\pi f_ct \\ &= A_c[1 + \mu\cos2\pi f_mt]\cos2\pi f_ct, \quad \mu = \frac{A_m}{A_c} \end{aligned}$$

Using $\cos A\cos B = \tfrac12[\cos(A-B) + \cos(A+B)]$:

$$s(t) = A_c\cos2\pi f_ct + \frac{\mu A_c}{2}\cos2\pi(f_c+f_m)t + \frac{\mu A_c}{2}\cos2\pi(f_c-f_m)t$$

Three components: carrier, upper side frequency (USF), lower side frequency (LSF).

 s(t):   envelope Ac(1 + u cos wm t)
       Ac(1+u) _      ____
              / \/\/\/    \/\
   ..........|..............|....  0
              \ /\/\/\    /\/
       -Ac(1+u) -      ----

Frequency-domain expression

Taking the Fourier transform, using $\cos2\pi f_0t \leftrightarrow \tfrac12[\delta(f-f_0) + \delta(f+f_0)]$:

$$\begin{aligned} S(f) = {} & \frac{A_c}{2}[\delta(f-f_c) + \delta(f+f_c)] \\ & + \frac{\mu A_c}{4}[\delta(f-f_c-f_m) + \delta(f+f_c+f_m)] \\ & + \frac{\mu A_c}{4}[\delta(f-f_c+f_m) + \delta(f+f_c-f_m)] \end{aligned}$$

Spectrum

                 S(f)
         Ac/2              Ac/2
          |                 |
   uAc/4  |  uAc/4   uAc/4  |  uAc/4
     |    |    |       |    |    |
 ----+----+----+---0---+----+----+---> f
 -fc-fm -fc -fc+fm   fc-fm  fc  fc+fm

• Carrier impulses of weight $A_c/2$ at $\pm f_c$.
• Side-frequency impulses of weight $\mu A_c/4$ at $\pm(f_c \pm f_m)$.
• Bandwidth: $B = (f_c + f_m) - (f_c - f_m) = 2f_m$.

Power (load $R$): $P_t = \dfrac{A_c^2}{2R}\left(1 + \dfrac{\mu^2}{2}\right)$, efficiency $\eta = \dfrac{\mu^2}{2 + \mu^2}$ (max 33.3% at $\mu = 1$).