Detection gain of SSB-SC vs DSB-SC

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.

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 $=1$.

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 $=1$.

Comparison:

The DSB-SC detection gain is twice that of SSB because the two sidebands add coherently (voltage) while the noise in them adds in power. But SSB lets in only half the noise bandwidth, so for the same transmitted power and same noise PSD, both give the same output SNR (figure of merit 1). SSB is preferred because it needs half the bandwidth.