Squaring signal for amplitude demodulation
If $m(t)$ is the message signal with $|m(t)|&lt;1$, the AM signal can be written as

$$s(t)=(1+m(t))\cos(\omega_ct)\tag{1}$$

where $\omega_c$ is the carrier frequency in radians per second. Squaring $s(t)$ gives

$$s^2(t)=(1+m(t))^2\cos^2(\omega_ct)=(1+m(t))^2\frac12(1+\cos(2\omega_ct))\tag{2}$$

Low pass filtering $(2)$ gives

$$x(t)=\frac12(1+m(t))^2\tag{3}$$

from which the message $m(t)$ can be retrieved:

$$m(t)=\sqrt{2x(t)}-1\tag{4}$$