Noise performance of DSB-SC and SSB-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.

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:

DSB-SC has twice the detection gain because its two sidebands add coherently, but it collects twice the noise. For the same transmitted power both give identical output SNR; SSB is preferred because it needs half the bandwidth.