DSB-FC single tone expression and modulation index

DSB-FC (standard AM) is the AM wave in which the carrier amplitude varies linearly with the message, and the carrier and both sidebands are transmitted.

Derivation

Let the message and carrier be

$$m(t) = A_m\cos\omega_mt, \qquad c(t) = A_c\cos\omega_ct, \qquad \omega_c \gg \omega_m$$

In AM the instantaneous amplitude of the carrier is

$$A(t) = A_c + k_am(t) = A_c + k_aA_m\cos\omega_mt$$

where $k_a$ is the amplitude sensitivity of the modulator. The AM wave is

$$\begin{aligned} s(t) &= [A_c + k_aA_m\cos\omega_mt]\cos\omega_ct \\ &= A_c\left[1 + \frac{k_aA_m}{A_c}\cos\omega_mt\right]\cos\omega_ct \\ &= A_c[1 + \mu\cos\omega_mt]\cos\omega_ct \end{aligned}$$

with $\mu = k_aA_m/A_c$. Using $\cos A\cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$:

$$s(t) = \underbrace{A_c\cos\omega_ct}_{\text{carrier}} + \underbrace{\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t}_{\text{USB}} + \underbrace{\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t}_{\text{LSB}}$$

So the wave has three components: the carrier at $f_c$ and two sidebands at $f_c \pm f_m$, each of amplitude $\mu A_c/2$. The bandwidth is $2f_m$.

 Amp
  Ac  |         |
      |         |
 uAc/2|    |    |    |
      +----+----+----+---> f
        fc-fm  fc  fc+fm

Power (1 Ω): $P_t = \dfrac{A_c^2}{2}\left(1 + \dfrac{\mu^2}{2}\right)$.

Modulation index

$$\mu = \frac{k_aA_m}{A_c} = \frac{\text{peak amplitude change}}{\text{carrier amplitude}}$$

From the waveform, $A_{max} = A_c(1+\mu)$ and $A_{min} = A_c(1-\mu)$. Solving:

$$\mu = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}$$

For distortion-free envelope detection, $0 < \mu \le 1$; $\mu > 1$ is over-modulation.