Threshold effect in envelope detection of DSB-FC

An envelope detector is a non-linear device whose output follows the magnitude of the resultant of signal plus noise. Its noise performance changes sharply when the carrier-to-noise ratio drops below a certain level; this is the threshold effect.

Envelope detector input. DSB-FC signal plus narrowband noise:

$$x(t)=A_c[1+k_am(t)]\cos\omega_ct+n_c(t)\cos\omega_ct-n_s(t)\sin\omega_ct$$

The envelope detector output is the magnitude of the resultant phasor:

$$y(t)=\sqrt{\{A_c[1+k_am(t)]+n_c(t)\}^2+n_s^2(t)}$$

              n_s
          .---->           resultant length = y(t)
         /    ^
        /     | n_c
  ----------->+
  Ac[1+ka m(t)]

Case 1: high carrier-to-noise ratio ($A_c[1+k_am]\gg$ noise). The quadrature term is negligible:

$$y(t)\approx A_c+A_ck_am(t)+n_c(t)$$

The message appears additively with the noise. After removing DC, $\text{SNR}_o=\dfrac{A_c^2k_a^2P}{2N_0W}$, the same as coherent detection. Figure of merit $\frac{k_a^2P}{1+k_a^2P}$ ($=1/3$ for a single tone with $\mu=1$).

Case 2: low carrier-to-noise ratio (noise $\gg$ carrier). Write the noise as $n(t)=r(t)\cos[\omega_ct+\psi(t)]$, with Rayleigh envelope $r$ and uniform phase $\psi$. Then

$$\begin{aligned} y(t) &= \sqrt{r^2+2rA_c[1+k_am]\cos\psi+A_c^2[1+k_am]^2} \\ &\approx r(t)\sqrt{1+\frac{2A_c}{r(t)}[1+k_am(t)]\cos\psi(t)} \\ &\approx r(t)+A_c\cos\psi(t)+A_ck_am(t)\cos\psi(t) \end{aligned}$$

using $\sqrt{1+x}\approx1+x/2$. Now the message $m(t)$ is multiplied by the random noise $\cos\psi(t)$; there is no term proportional to $m(t)$ alone. The message is mutilated and cannot be recovered; output SNR collapses.

Threshold effect. The change from Case 1 to Case 2 is not gradual: below a certain carrier-to-noise ratio, the threshold (roughly 10 dB carrier-to-noise ratio), the output SNR falls much faster than the input SNR. The threshold is a property of non-linear (envelope) detection; coherent detection of DSB-FC, DSB-SC or SSB has no threshold, because the product detector is linear and the message always stays additive with noise.

 SNRo (dB)                 coherent / envelope
     |                   . (above threshold)
     |                 .
     |               .
     |             .  <- threshold
     |          ..'
     |      ..''    envelope detector
     |  ..''        (below threshold)
     +-------------------------> CNR (dB)