Advantages of SSB over DSB-FC; SSB expression

Advantages of SSB over DSB-FC

• Half bandwidth: $W$ instead of $2W$, so twice the number of channels.
• Power saving: at $\mu = 1$, SSB needs only $0.25P_c$ against $1.5P_c$, a saving of 83.3%.
• Better S/N: narrower band admits less noise.
• Less selective fading: with one sideband and no carrier, phase shifts between components cannot cancel each other.
• Higher efficiency: all transmitted power carries information.

Derivation of SSB signal

Start with DSB-SC for a single tone $m(t) = A_m\cos\omega_mt$:

$$s_{DSB}(t) = A_cA_m\cos\omega_mt\cos\omega_ct = \frac{A_cA_m}{2}[\cos(\omega_c+\omega_m)t + \cos(\omega_c-\omega_m)t]$$

Keeping only the upper sideband:

$$\begin{aligned} s_{USB}(t) &= \frac{A_cA_m}{2}\cos(\omega_c + \omega_m)t \\ &= \frac{A_c}{2}[A_m\cos\omega_mt\cos\omega_ct - A_m\sin\omega_mt\sin\omega_ct] \end{aligned}$$

Similarly, the lower sideband:

$$s_{LSB}(t) = \frac{A_c}{2}[A_m\cos\omega_mt\cos\omega_ct + A_m\sin\omega_mt\sin\omega_ct]$$

Here $A_m\sin\omega_mt$ is the Hilbert transform of $A_m\cos\omega_mt$. Since any message is a sum of such tones, for a general $m(t)$:

$$s_{SSB}(t) = \frac{A_c}{2}\left[m(t)\cos\omega_ct \mp \hat m(t)\sin\omega_ct\right]$$

(minus sign: USB; plus sign: LSB), where $\hat m(t)$ is the Hilbert transform of $m(t)$.

Frequency domain check (USB): $\hat M(f) = -j\,\text{sgn}(f)M(f)$, so the in-phase and quadrature terms add for $|f| > f_c$ and cancel for $|f| < f_c$, leaving only the upper sideband.

Power (single tone, load $R$): $P_{SSB} = \dfrac{(A_cA_m/2)^2}{2R} = \dfrac{A_c^2A_m^2}{8R}$.