Capture effect; gain parameter of DSB-FC envelope detection

Capture effect

Capture effect is the property of an FM receiver whereby, when two FM signals on (nearly) the same frequency are received, the stronger one is reproduced and the weaker one is almost completely suppressed. The limiter and discriminator respond to the dominant signal's phase, so the weaker signal appears only as small noise. If the two strengths are nearly equal (within about 1 dB), the receiver may switch between them. AM receivers do not show this; both signals are heard together.

Gain parameter (figure of merit) of DSB-FC with envelope detection

Assumptions: message $m(t)$ of bandwidth $W$ and power $P = \overline{m^2(t)}$; white noise of PSD $N_0/2$; IF filter bandwidth $2W$; high carrier-to-noise ratio.

Transmitted signal: $s(t) = A_c[1 + k_a m(t)]\cos\omega_c t$

Channel (input) SNR. Signal power $= \dfrac{A_c^2(1+k_a^2P)}{2}$; noise power in bandwidth $W$ (reference baseband) $= WN_0$:

$$(SNR)_C = \frac{A_c^2(1+k_a^2P)}{2WN_0}$$

Received signal with narrowband noise

$$\begin{aligned} x(t) &= [A_c + A_ck_am(t) + n_I(t)]\cos\omega_c t - n_Q(t)\sin\omega_c t \end{aligned}$$

where $n_I, n_Q$ each have power $2WN_0$. The envelope is

$$y(t) = \sqrt{[A_c + A_ck_am(t) + n_I(t)]^2 + n_Q^2(t)}$$

For high SNR, $A_c[1+k_am(t)] \gg |n_Q(t)|$, so

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

The DC term $A_c$ is removed by a blocking capacitor.

Output SNR

$$(SNR)_O = \frac{A_c^2k_a^2P}{2WN_0}$$

Figure of merit (gain parameter)

$$\gamma = \frac{(SNR)_O}{(SNR)_C} = \frac{k_a^2P}{1+k_a^2P}$$

which is always less than 1.

Single tone $m(t) = A_m\cos\omega_m t$, $\mu = k_aA_m$: $k_a^2P = \mu^2/2$, so

$$\gamma = \frac{\mu^2}{2+\mu^2}$$

For 100 % modulation ($\mu = 1$): $\gamma = 1/3$. So DSB-FC with envelope detection needs three times (4.77 dB) more transmitted power than DSB-SC/SSB with coherent detection ($\gamma = 1$) for the same output SNR, because much power goes into the carrier.

Low SNR: when noise dominates the carrier, the message is no longer separable in the envelope and output SNR falls sharply: this is the threshold effect of the envelope detector.