Double-tone AM expression, bandwidth and modulation index

Double-tone AM is amplitude modulation in which the message consists of two sinusoids:

$$m(t)=A_{m1}\cos\omega_1t+A_{m2}\cos\omega_2t,\qquad f_1<f_2\ll f_c$$

Expression

With carrier $A_c\cos\omega_ct$, the standard AM wave is

$$\begin{aligned} s(t)&=[A_c+A_{m1}\cos\omega_1t+A_{m2}\cos\omega_2t]\cos\omega_ct\\ &=A_c[1+\mu_1\cos\omega_1t+\mu_2\cos\omega_2t]\cos\omega_ct \end{aligned}$$

where $\mu_1=A_{m1}/A_c$ and $\mu_2=A_{m2}/A_c$ are the individual modulation indices. Expanding:

$$\begin{aligned} s(t)&=A_c\cos\omega_ct\\ &+\frac{\mu_1A_c}{2}\left[\cos(\omega_c+\omega_1)t+\cos(\omega_c-\omega_1)t\right]\\ &+\frac{\mu_2A_c}{2}\left[\cos(\omega_c+\omega_2)t+\cos(\omega_c-\omega_2)t\right] \end{aligned}$$

So the spectrum has five lines: the carrier at $f_c$, and side frequencies at $f_c\pm f_1$ and $f_c\pm f_2$.

                         Ac
                         |
                         |
   mu2Ac/2   mu1Ac/2     |     mu1Ac/2   mu2Ac/2
      |         |        |        |         |
 -----+---------+--------+--------+---------+-----> f
    fc-f2     fc-f1      fc     fc+f1     fc+f2

Bandwidth

The bandwidth is set by the highest modulating frequency:

$$B=(f_c+f_2)-(f_c-f_2)=2f_2=2f_{max}$$

Modulation indices

• Individual indices: $\mu_1=\frac{A_{m1}}{A_c}$, $\mu_2=\frac{A_{m2}}{A_c}$.
• Net (total, effective) index: the single-tone index that gives the same total sideband power:

$$P_t=P_c\left(1+\frac{\mu_1^2}{2}+\frac{\mu_2^2}{2}\right)=P_c\left(1+\frac{\mu_t^2}{2}\right)\ \Rightarrow\ \mu_t=\sqrt{\mu_1^2+\mu_2^2}$$

For no over-modulation, $\mu_t\le1$ (strictly, $\mu_1+\mu_2\le1$ guarantees that the envelope never goes negative).

Example: $\mu_1=0.3$, $\mu_2=0.4$ gives $\mu_t=0.5$.