Double-tone AM; DSB vs SSB

Double tone AM

In double tone (two-tone) AM the carrier is modulated by a message that contains two sinusoids of different frequencies.

Let the message and carrier be:

$$m(t) = A_{m1}\cos\omega_{m1}t + A_{m2}\cos\omega_{m2}t, \qquad c(t) = A_c\cos\omega_c t$$

The conventional (DSB-FC) AM wave is

$$\begin{aligned} s(t) &= [A_c + m(t)]\cos\omega_c t \\ &= A_c\left[1 + \frac{A_{m1}}{A_c}\cos\omega_{m1}t + \frac{A_{m2}}{A_c}\cos\omega_{m2}t\right]\cos\omega_c t \\ &= A_c\left[1 + \mu_1\cos\omega_{m1}t + \mu_2\cos\omega_{m2}t\right]\cos\omega_c t \end{aligned}$$

where $\mu_1 = A_{m1}/A_c$ and $\mu_2 = A_{m2}/A_c$ are the individual modulation indices. Expanding with $\cos A\cos B = \tfrac12[\cos(A+B)+\cos(A-B)]$:

$$\begin{aligned} s(t) = {} & A_c\cos\omega_c t \\ & + \frac{\mu_1A_c}{2}\cos(\omega_c+\omega_{m1})t + \frac{\mu_1A_c}{2}\cos(\omega_c-\omega_{m1})t \\ & + \frac{\mu_2A_c}{2}\cos(\omega_c+\omega_{m2})t + \frac{\mu_2A_c}{2}\cos(\omega_c-\omega_{m2})t \end{aligned}$$

So the spectrum has the carrier plus two upper and two lower side frequencies.

 amplitude
    |                       Ac
    |                       |
    |                       |
    |  u2Ac/2    u1Ac/2     |     u1Ac/2    u2Ac/2
    |     |         |       |        |         |
 ---+-----+---------+-------+--------+---------+----> f
        fc-fm2   fc-fm1     fc    fc+fm1    fc+fm2
                                         (fm1 < fm2)

Power (load $R$):

$$P_t = \frac{A_c^2}{2R}\left[1 + \frac{\mu_1^2}{2} + \frac{\mu_2^2}{2}\right] = P_c\left(1 + \frac{\mu_t^2}{2}\right)$$

with the net (total) modulation index

$$\mu_t = \sqrt{\mu_1^2 + \mu_2^2}$$

For no over-modulation, $\mu_t \le 1$. Bandwidth $= 2f_{m,\max}$ (twice the higher message frequency).

DSB vs SSB