Why AM wastes power and bandwidth; switching modulator

Why conventional AM wastes power and bandwidth

Power: In DSB-FC AM, $P_t = P_c(1 + \mu^2/2)$. The carrier carries no information, yet it takes most of the power. Even at $\mu = 1$, the carrier uses $\frac{1}{1.5} = 66.7\%$ of the total and the sidebands only 33.3%. In practice $\mu \approx 0.3$ on average, giving efficiency $\frac{0.09}{2.09} \approx 4.3\%$ only.

Bandwidth: Both sidebands carry the same information (each is a mirror image of the other about $f_c$). Sending both needs $2W$ bandwidth, while $W$ (one sideband) is enough. So half the bandwidth is wasted.

Switching modulator

A diode used as a switch, driven by a large carrier, generates DSB-FC AM.

 m(t) --->(+)--- v1 ----|>|----+------> v2 --> BPF --> s(t)
           ^             D     |              (fc, 2W)
           |                   R_L
 Ac cos wc t                   |
 (Ac >> |m(t)|)               ---

Assumption: carrier amplitude $A_c \gg |m(t)|$, so the diode is ON during the positive half-cycles of the carrier and OFF during the negative half-cycles, regardless of $m(t)$.

Input: $v_1(t) = A_c\cos\omega_ct + m(t)$.

Output:

$$v_2(t) = \begin{cases} v_1(t), & c(t) > 0 \\ 0, & c(t) < 0 \end{cases} \;=\; v_1(t)\,g_p(t)$$

where $g_p(t)$ is a unit-amplitude (0/1) pulse train at $f_c$ with 50% duty:

$$g_p(t) = \frac12 + \frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{2n-1}\cos[(2n-1)\omega_ct]$$ $$\begin{aligned} v_2(t) &= [A_c\cos\omega_ct + m(t)]\left[\frac12 + \frac{2}{\pi}\cos\omega_ct - \frac{2}{3\pi}\cos3\omega_ct + \dots\right] \\ &= \frac{A_c}{2}\cos\omega_ct + \frac{2}{\pi}m(t)\cos\omega_ct + \frac{m(t)}{2} + \frac{A_c}{\pi} + \frac{A_c}{\pi}\cos2\omega_ct + \dots \end{aligned}$$

A BPF at $f_c$ with bandwidth $2W$ keeps only the first two terms:

$$s(t) = \frac{A_c}{2}\left[1 + \frac{4}{\pi A_c}m(t)\right]\cos\omega_ct$$

This is DSB-FC AM with $k_a = \dfrac{4}{\pi A_c}$.

 v1:  carrier + m(t)      v2: positive half-cycles only
   /\  /\  /\               /\  /\  /\
  /  \/  \/  \/            /  \_/  \_/  \_
                         (then BPF -> AM wave)

Advantage: works at high power levels with good efficiency, does not depend on an exact square-law characteristic.