Envelope detector
= Envelope detector
{wiki}
An envelope detector extracts the slowly varying nonnegative <wave envelope> of a rapid <carrier wave>. In a simple electrical version a <diode> charges a <capacitor> near positive carrier peaks and an attached <resistor> discharges it between peaks. When $0\leq m<1$, ideal envelope extraction gives $E(t)=A[1+m\cos(\Omega t)]$; removing the constant $A$ leaves $Am\cos(\Omega t)$. This permits recovery without a synchronized carrier reference. Carrier injection thus trades transmitted <power> for simplicity.<Continuous-Time Modulation by Alan V. Oppenheim>[Signals and Systems, Lecture 13: Continuous-Time Modulation, printed p. 13-2 and transparencies 13.17–13.19 on pp. 13-11–13-12: asynchronous recovery and injected carrier]{ref}
A working peak detector needs margins as well as $m<1$: $E_{\min}=A(1-m)$ should comfortably exceed the <diode>'s forward-voltage scale and <noise>, and the <time constant> $\tau=RC$ must suppress carrier ripple while following a falling <wave envelope>. The coarse scale window is
$$
\frac{2\pi}{\omega_c}\ll\tau\ll\frac1\Omega.
$$
A <time constant> that is too short leaves carrier ripple; one that is too long makes the discharging <capacitor> fall more slowly than the envelope. In the ideal zero-drop, continuous-tracking approximation, following a falling $E$ requires $E/\tau\geq-E'$. For $0<m<1$,
$$
\max_{0<\theta<\pi}\frac{m\Omega\sin\theta}{1+m\cos\theta}
=\frac{m\Omega}{\sqrt{1-m^2}},\qquad
\tau\leq\frac{\sqrt{1-m^2}}{m\Omega}.
$$
Differentiating the ratio locates its maximum at $\cos\theta=-m$. This original slope check explains why operation near $m=1$ is demanding even before <overmodulation>: the upper bound tends to zero while the need to smooth carrier cycles persists. It is an approximate circuit criterion, not a guarantee in the presence of <diode> drops, loading or <noise>. At $m=0$ there is no varying message to track; at $m=1$ ideal envelope extraction remains algebraically possible, but the simple detector has no positive trough margin.
For $m>1$, even an ideal noiseless envelope detector returns $A|q(t)|$. Removing its mean or its original offset cannot restore the discarded sign. A more elaborate receiver can use the retained <wave phase> through <coherent demodulation>.