# Squaring signal for amplitude demodulation

I read the following here. Can someone explain/elaborate on this?

Squaring the signal effectively demodulates the input by using itself as its own carrier wave. This means that half the energy of the signal is pushed up to higher frequencies and half is shifted down toward DC

I didn't find any good source giving more information or examples of this.

• Please see this link. It's a standard technique for amp.Demod. – A_A Jul 15 '16 at 9:08

If $m(t)$ is the message signal with $|m(t)|<1$, 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:

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