DSB-FC generation using square law modulator

A square law modulator generates DSB-FC (conventional AM) by passing the sum of carrier and message through a non-linear device (diode or transistor biased in its non-linear region) whose output contains a square term, and then filtering out the AM band.

Block diagram

 m(t) -->(+)--> v1 --> Non-linear --> v2 --> BPF --> s(t)
          ^             device             (fc,2W)   (AM)
          |             (diode)
 c(t) ----+
 Ac cos wc t

Analysis

Input to the non-linear device:

$$v_1(t) = m(t) + A_c\cos\omega_c t$$

The device characteristic (square law approximation):

$$v_2(t) = a_1v_1(t) + a_2v_1^2(t)$$

Substituting:

$$\begin{aligned} v_2(t) = {} & a_1m(t) + a_1A_c\cos\omega_c t \\ & + a_2m^2(t) + 2a_2A_c\,m(t)\cos\omega_c t + a_2A_c^2\cos^2\omega_c t \end{aligned}$$

The terms and their frequencies:

The BPF centred at $f_c$ with bandwidth $2W$ passes:

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

This is DSB-FC AM with amplitude sensitivity $k_a = 2a_2/a_1$. Condition for separation: $f_c > 3W$, so that the $m^2(t)$ band (up to $2W$) does not overlap the lower sideband ($f_c - W$).

Spectrum

 M(f)                      S(f)
   /\                   /\    |    /\     /\    |    /\
  /  \                 /  \   |   /  \   /  \   |   /  \
-W 0  W        ...  -fc-W -fc -fc+W ... fc-W  fc  fc+W
                    (LSB carr USB)     (LSB carr USB)

The transmitted spectrum has the carrier impulse at $\pm f_c$ with the USB and LSB on either side, total bandwidth $2W$.

Limitation: distortion if higher-order terms are present and the device is not ideally square law; output is low-level, so it is used in low-power transmitters.