Noise performance of coherent DSB-FC detection

Noise performance of coherent DSB-FC detection

Noise performance is measured by the output SNR and the detection gain $\gamma=\text{SNR}_o/\text{SNR}_i$ (or figure of merit, with SNR referred to the message band $W$).

 s(t)+w(t) +-----+         +-----+   +-------+
 --------->| BPF |-->(x)-->| LPF |-->|  DC   |--> y(t)
           | 2W  |    ^    |  W  |   | block |
           +-----+    |    +-----+   +-------+
                  cos(wc t)

Receiver: BPF ($B_T=2W$) → product detector with $\cos\omega_ct$ → LPF ($W$) → DC block. Noise PSD $N_0/2$, narrowband noise $n(t)=n_c\cos\omega_ct-n_s\sin\omega_ct$ with $\overline{n_c^2}=2N_0W$.

• Signal power $=\frac{A_c^2}{2}(1+k_a^2P)$; noise in $2W$ $=2N_0W$:

$$\text{SNR}_i=\frac{A_c^2(1+k_a^2P)}{4N_0W}$$

• Coherent detection (or envelope detection at high SNR) and DC blocking:

$$y(t)=\frac{A_c}{2}k_am(t)+\frac12n_c(t)\ \Rightarrow\ \text{SNR}_o=\frac{A_c^2k_a^2P/4}{2N_0W/4}=\frac{A_c^2k_a^2P}{2N_0W}$$ $$\gamma_{DSB\text{-}FC}=\frac{\text{SNR}_o}{\text{SNR}_i}=\frac{2k_a^2P}{1+k_a^2P},\qquad \text{FOM}=\frac{k_a^2P}{1+k_a^2P}$$

The carrier uses power but carries no information, so the gain is reduced.

Single-tone DSB-FC with 100% modulation

Single tone, 100% modulation. $m(t)=A_m\cos\omega_mt$, $\mu=k_aA_m=1$, so $k_a^2P=\frac{\mu^2}{2}=\frac12$:

$$\begin{aligned} \gamma_{DSB\text{-}FC} &= \frac{2(\mu^2/2)}{1+\mu^2/2}=\frac{2\mu^2}{2+\mu^2}=\frac{2}{3}\approx0.667 \\ \text{FOM} &= \frac{\mu^2}{2+\mu^2}=\frac13 \end{aligned}$$

So with 100% single-tone modulation the output SNR is only one-third of that of DSB-SC or SSB for the same transmitted power ($10\log_{10}3=4.77$ dB worse), and the detection gain is less than 1, because two-thirds of the power is in the carrier.