Multi-tone AM: spectrum, bandwidth, power efficiency

Compare with $s(t) = A_c[1 + \mu_1\cos\omega_1t + \mu_2\cos\omega_2t]\cos\omega_ct$:

i) Amplitude spectrum

Expanding:

$$\begin{aligned} s(t) = {} & 50\cos\omega_ct + 7.5\cos(\omega_c \pm 3141.6)t + 5\cos(\omega_c \pm 2513.28)t \end{aligned}$$

(Side amplitudes $\mu A_c/2$: $0.3 \times 50/2 = 7.5$ V; $0.2 \times 50/2 = 5$ V.)

  A(V)
  50 |                  |
     |                  |
 7.5 |   |              |              |
   5 |   |     |        |        |     |
     +---+-----+--------+--------+-----+---> f
       fc-500 fc-400    fc    fc+400 fc+500 (Hz)
                   fc = 159.155 kHz

ii) Bandwidth

$$B = 2f_{max} = 2 \times 500 = 1000\ \text{Hz}\ (= 6283.2\ \text{rad/s})$$

iii) Power efficiency

$$\mu_t^2 = \mu_1^2 + \mu_2^2 = 0.09 + 0.04 = 0.13$$ $$\eta = \frac{\mu_t^2}{2 + \mu_t^2} = \frac{0.13}{2.13} = 0.0610 = 6.10\%$$

Check with powers (1 Ω): $P_c = 1250$ W, $P_{SB} = 1250 \times 0.13/2 = 81.25$ W, $P_t = 1331.25$ W, $\eta = 81.25/1331.25 = 6.10\%$.

Answer: $B = 1$ kHz; $\eta = 6.10\%$.