Envelope detector; can it demodulate DSB-SC?

The envelope detector is a simple non-coherent demodulator whose output follows the envelope of the AM wave. Since standard AM has envelope $A_c[1+k_am(t)]$, which is a shifted copy of $m(t)$, it recovers the message without any local carrier.

Circuit

        D
 o---->|---+------+--------o
           |      |
 AM in     C      R      vo(t) ~ envelope
           |      |
 o---------+------+--------o

Working

1. Positive half cycle: when the input exceeds the capacitor voltage, the diode conducts and $C$ charges quickly to the peak of the carrier through the small source resistance ($R_sC\ll1/f_c$).
2. Between peaks: when the input falls below the capacitor voltage, the diode becomes reverse-biased and $C$ discharges slowly through $R$.
3. On the next peak the diode conducts again and recharges $C$. Thus the capacitor voltage follows the peaks of the carrier, i.e. the envelope, with a small ripple at $f_c$.
4. A DC-blocking capacitor removes the DC ($A_c$) and a small RC low-pass filter removes ripple, giving $m(t)$.

Time constant condition:

$$\frac{1}{f_c}\ll RC\ll\frac{1}{W}$$

• If $RC$ is too small, the output follows the carrier cycles (large ripple).
• If $RC$ is too large, the capacitor cannot follow a falling envelope (diagonal clipping); for a tone with index $\mu$ we need $RC\le\frac{\sqrt{1-\mu^2}}{\mu\,\omega_m}$.

 AM wave       envelope      detector output
  /\  /\/\       ___           ___
 /  \/    \/\   /   \  __     /   \  __ (small ripple)

Requirements: $\mu\le1$ (no over-modulation) and $f_c\gg W$.

Can an envelope detector demodulate DSB-SC?

No. A DSB-SC wave is $s(t)=A_cm(t)\cos\omega_ct$. Its envelope is $A_c\lvert m(t)\rvert$, not $m(t)$. Whenever $m(t)$ becomes negative, the carrier phase reverses by $180°$ but the envelope stays positive, so the detector output is the rectified message $\lvert m(t)\rvert$, which is badly distorted. For example, a tone $\cos\omega_mt$ comes out as $\lvert\cos\omega_mt\rvert$, which contains $2f_m$ and higher harmonics. DSB-SC therefore needs coherent (synchronous) detection, using a carrier recovered by a Costas loop or squaring loop. (Envelope detection would only work if a large carrier were re-inserted, which converts it back to standard AM.)