Square-law modulator for DSB-AM; characteristics of AM

A square-law modulator generates standard AM (DSB-FC) using a non-linear device (diode or transistor in its non-linear region) whose output contains a squared term, followed by a band-pass filter.

Block diagram

 m(t) -->(+)-v1->[Non-linear]-v2->[ BPF ]--> s(t)
          ^       [ device   ]     fc, 2W     AM
          |
 Ac cos wct

Analysis

The sum of message and carrier is applied to the device:

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

The device characteristic (up to second order) is

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

Substituting:

$$\begin{aligned} v_2(t)&=a_1A_c\cos\omega_ct+a_1m(t)+a_2\left[A_c^2\cos^2\omega_ct+2A_cm(t)\cos\omega_ct+m^2(t)\right]\\ &=\underbrace{a_1A_c\left[1+\frac{2a_2}{a_1}m(t)\right]\cos\omega_ct}_{\text{wanted AM}}+\underbrace{a_1m(t)+a_2m^2(t)+\frac{a_2A_c^2}{2}+\frac{a_2A_c^2}{2}\cos2\omega_ct}_{\text{unwanted}} \end{aligned}$$

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

$$s(t)=a_1A_c\left[1+k_am(t)\right]\cos\omega_ct,\qquad k_a=\frac{2a_2}{a_1}$$

This is exactly a DSB-AM (full carrier) signal with amplitude sensitivity $k_a$. Hence a square-law device plus a BPF works as an AM modulator.

Spectrum at the input of the BPF

For a message of bandwidth $W$, the components of $v_2(t)$ are:

 |V2(f)|      BPF passband
 |              +-------+
 |^             |   ^   |
 ||\_           |  /|\  |              ^
 ||  \__        | / | \ |              |
 +|-----\-------+/--+--\+--------------+--> f
  0  W  2W   fc-W  fc  fc+W           2fc
   m(t)  m^2(t)        AM             2fc

To separate the AM band from $m^2(t)$ without overlap, we need $f_c-W>2W$, i.e. $f_c>3W$.

Basic characteristics of AM

1. Carrier amplitude varies linearly with the message; envelope is $A_c[1+k_am(t)]$.
2. Spectrum has a carrier plus two sidebands (USB and LSB), each a copy of the message spectrum.
3. Bandwidth $B=2W$.
4. Modulation index $\mu=k_a\lvert m(t)\rvert_{max}\le1$ to avoid over-modulation.
5. Power: $P_t=P_c\left(1+\frac{\mu^2}{2}\right)$; maximum efficiency only 33.3% at $\mu=1$ (most power is in the carrier).
6. Can be demodulated by a simple envelope detector, making receivers cheap.