Envelope detector; why not for DSB-SC and SSB

An envelope detector is a diode–RC circuit that tracks the peaks (envelope) of a standard AM wave $A_c[1+\mu m(t)]\cos\omega_ct$, giving $A_c[1+\mu m(t)]$ and hence $m(t)$ after removing DC.

Circuit

        D
 o----->|-----+--------+-----o
 AM           |        |
 input   C  ===       > RL   v_o
              |        >
 o------------+--------+-----o

Operation

1. Charging: in each positive carrier half cycle, when the input is above the capacitor voltage, the diode conducts and $C$ charges through the small source resistance $R_s$ almost to the peak.
2. Discharging: once the input falls below the capacitor voltage, the diode is cut off and $C$ discharges slowly through $R_L$.
3. Repeating every carrier cycle, the capacitor voltage follows the envelope with a small saw-tooth ripple at $f_c$.
4. A low-pass RC filter smooths the ripple and a blocking capacitor removes the DC term $A_c$, leaving $\mu A_c m(t)$.

 AM in:   /\/\/\  /\/\/\
         /\/\/\/\/\/\/\/\  (peaks follow m)
 Out:      ___      ___
         _/   \____/   \_  (envelope)

Design conditions

• $R_sC \ll 1/f_c$ (fast charge)
• $1/f_c \ll R_LC \ll 1/W$ (small ripple, yet follows the message)
• For a tone, to avoid diagonal clipping: $R_LC \le \dfrac{\sqrt{1-\mu^2}}{\mu\omega_m}$
• $\mu \le 1$ (no over-modulation)

Why DSB-SC cannot be envelope detected

$s_{DSB}(t) = A_cm(t)\cos\omega_ct$. Its envelope is $A_c|m(t)|$, not $m(t)$. When $m(t)$ goes negative, the carrier phase reverses by 180°, but the peaks stay positive. The envelope detector outputs $|m(t)|$, a rectified (full-wave) copy of the message, which is badly distorted.

 m(t):    /\      /\         |m(t)|:  /\  /\  /\
        _/  \    /  \_    ->         /  \/  \/  \
             \__/

Example: for $m(t) = \cos\omega_mt$ the output contains $|\cos\omega_mt|$, which has components at $0, 2f_m, 4f_m, \dots$ and none at $f_m$.

Why SSB cannot be envelope detected

$s_{SSB}(t) = \frac{A_c}{2}[m(t)\cos\omega_ct \mp \hat m(t)\sin\omega_ct]$. Its envelope is

$$a(t) = \frac{A_c}{2}\sqrt{m^2(t) + \hat m^2(t)}$$

which is not proportional to $m(t)$. For a single tone $m = A_m\cos\omega_mt$, $\hat m = A_m\sin\omega_mt$, so the envelope is constant, $A_cA_m/2$: the detector gives only DC and the message is lost completely.

Both DSB-SC and SSB therefore need coherent (synchronous) detection, or a large carrier must be added at the receiver so that the envelope again follows $m(t)$.