A receiver must reverse the modulation process to recover the baseband message.

For a suppressed-carrier signal

$$s(t)=m(t)\cos(2\pi f_ct),$$

coherent demodulation multiplies the received signal by a synchronized carrier. Using

$$\cos^2(2\pi f_ct)=\frac12[1+\cos(4\pi f_ct)],$$

the product contains one copy of the message near zero frequency and another translated near $2f_c$. A low-pass filter removes the high-frequency term and leaves a scaled version of $m(t)$.

Coherent detection requires sufficiently accurate carrier frequency and phase.

Conventional AM retains a carrier component, so its slowly varying envelope can instead be detected without reproducing carrier phase explicitly. A rectifying element followed by suitable smoothing can track that envelope when the modulation is not overdriven.

Retaining the carrier permits simpler envelope detection; suppressing it requires coherent synchronization but avoids transmitting that carrier component.