Compare AM systems; one method of DSB-FC generation

AM systems differ in which parts of the spectrum (carrier, USB, LSB) are sent. This changes the power that must be transmitted and the bandwidth occupied.

Comparison (single tone, modulation index $\mu$, message bandwidth $W$)

Here $P_c = A_c^2/2$ (normalized power). For $\mu = 1$, DSB-FC needs $1.5P_c$, DSB-SC needs $0.5P_c$ (66.7% saving) and SSB needs $0.25P_c$ (83.3% saving).

Square-law modulator for DSB-FC

A square-law modulator uses a non-linear device (diode or transistor biased in its non-linear region) whose output is

$$v_2(t) = a_1 v_1(t) + a_2 v_1^2(t)$$

 m(t) --->(+)--->[ Non-linear ]--->[ BPF at fc ]---> s(t)
           ^      [  device   ]     [ BW = 2W  ]
 Accos wct-|

The input is $v_1(t) = A_c\cos\omega_c t + m(t)$. Then

$$\begin{aligned} v_2(t) &= a_1 A_c\cos\omega_c t + a_1 m(t) + a_2 m^2(t) \\ &\quad + a_2A_c^2\cos^2\omega_c t + 2a_2 A_c\, m(t)\cos\omega_c t \end{aligned}$$

A band-pass filter centred at $f_c$ with bandwidth $2W$ keeps only the terms near $f_c$:

$$s(t) = a_1A_c\left[1 + \frac{2a_2}{a_1}m(t)\right]\cos\omega_c t$$

This is DSB-FC AM with amplitude sensitivity $k_a = 2a_2/a_1$. The terms $m(t)$, $m^2(t)$ (around 0 to $2W$) and $\cos 2\omega_c t$ (around $2f_c$) are rejected. This needs $f_c > 3W$ so the spectra do not overlap.