Detection gains of DSB-FC, DSB-SC and SSB

Receiver model. The received signal $s(t)$ plus white noise ($N_0/2$) passes through a band-pass (IF) filter of bandwidth $B_T$, then a demodulator and a low-pass filter of bandwidth $W$ (message bandwidth).

 s(t)  +   +-------+    +-------+   +-----+
 ---->(+)-->|  BPF  |--->| demod |-->| LPF |--> y(t)
       ^    |  B_T  |    |       |   |  W  |
      w(t)  +-------+    +-------+   +-----+

The filtered noise is narrowband and is written in in-phase/quadrature form

$$n(t)=n_c(t)\cos\omega_ct-n_s(t)\sin\omega_ct$$

where $n_c$ and $n_s$ are low-pass, each with the same power as $n(t)$: $\overline{n_c^2}=\overline{n_s^2}=\overline{n^2}=N_0B_T$.

Definitions. Message power $P=\overline{m^2(t)}$.

• Input SNR $\text{SNR}_i=\dfrac{\text{received signal power}}{\text{noise power in } B_T}$
• Output SNR $\text{SNR}_o=\dfrac{\text{message power at output}}{\text{noise power at output}}$
• Detection gain $\gamma_d=\dfrac{\text{SNR}_o}{\text{SNR}_i}$; figure of merit $=\dfrac{\text{SNR}_o}{\text{SNR}_{ref}}$, where $\text{SNR}_{ref}$ is the channel SNR measured in the message band $W$ (baseband reference). Both show how much the demodulator improves or worsens SNR.

DSB-FC (conventional AM). $s(t)=A_c[1+k_am(t)]\cos\omega_ct$, $B_T=2W$. (Coherent and envelope detection give the same result at high SNR.)

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

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

DSB-SC (coherent detection). $s(t)=A_cm(t)\cos\omega_ct$, $B_T=2W$.

• Signal power $=\frac{A_c^2P}{2}$; noise in $2W$ $=2N_0W$, so $\text{SNR}_i=\dfrac{A_c^2P}{4N_0W}$.
• Multiply $x(t)=s(t)+n(t)$ by $\cos\omega_ct$ and low-pass filter:

$$y(t)=\frac{A_c}{2}m(t)+\frac12n_c(t)$$

• $\text{SNR}_o=\dfrac{A_c^2P/4}{2N_0W/4}=\dfrac{A_c^2P}{2N_0W}$

$$\gamma_{DSB\text{-}SC}=\frac{\text{SNR}_o}{\text{SNR}_i}=2$$

The quadrature noise $n_s$ is rejected by coherent detection, giving a 3 dB gain. With baseband reference $\text{SNR}_{ref}=\frac{A_c^2P}{2N_0W}$, figure of merit $=1$.

SSB-SC (coherent detection). $s(t)=\frac{A_c}{2}[m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct]$, $B_T=W$.

• Signal power $=\frac{A_c^2}{4}\cdot\frac{P+P}{2}=\frac{A_c^2P}{4}$ (since $\overline{\hat m^2}=P$); noise in $W$ $=N_0W$, so $\text{SNR}_i=\dfrac{A_c^2P}{4N_0W}$.
• Multiply by $\cos\omega_ct$ and low-pass filter:

$$y(t)=\frac{A_c}{4}m(t)+\frac12n_c(t)$$

• Signal power $\frac{A_c^2P}{16}$, noise power $\frac{N_0W}{4}$, so $\text{SNR}_o=\dfrac{A_c^2P}{4N_0W}$

$$\gamma_{SSB}=\frac{\text{SNR}_o}{\text{SNR}_i}=1$$

With baseband reference ($\text{SNR}_{ref}=\frac{A_c^2P}{4N_0W}$, same transmitted power), figure of merit $=1$.

Comparison:

Conclusions: for the same transmitted power and noise PSD, DSB-SC and SSB give the same output SNR (equal to baseband); DSB-SC's detection gain of 2 is offset by its double noise bandwidth. SSB achieves this in half the bandwidth. DSB-FC is at least 3 dB worse (since $k_a^2P\le1$ gives FOM $\le\frac12$, with equality only for a 100% square wave) and 4.77 dB worse (a factor of 3) for 100% single-tone modulation, because the carrier carries no information; its advantage is that it allows a cheap envelope detector.