DSB-FC vs DSB-SC; phase shift method for SSB

DSB-FC vs DSB-SC

Phase shift method of SSB generation

The method uses the Hilbert transform (90° phase shift) to cancel one sideband instead of filtering it.

           +--->( X )<--- Ac cos wc t
           |      |              ^
 m(t) -----+      |              |   Carrier
           |      v              |   oscillator
           |     (+/-)---> s_SSB(t)
           |      ^              |
           v      |              v
       [-90 deg]  |          [-90 deg]
        (HT)      |              |
           |      |              v
           +--->( X )<--- Ac sin wc t
          m^(t)

Two balanced modulators are used:

• Modulator 1: $m(t)$ multiplied by $A_c\cos\omega_ct$.
• Modulator 2: $\hat m(t)$ (message shifted by $-90^\circ$) multiplied by $A_c\sin\omega_ct$ (carrier shifted by $-90^\circ$).

Derivation (single tone)

Let $m(t) = A_m\cos\omega_mt$, so $\hat m(t) = A_m\sin\omega_mt$.

$$\begin{aligned} v_1(t) &= A_mA_c\cos\omega_mt\cos\omega_ct = \frac{A_mA_c}{2}[\cos(\omega_c-\omega_m)t + \cos(\omega_c+\omega_m)t] \\ v_2(t) &= A_mA_c\sin\omega_mt\sin\omega_ct = \frac{A_mA_c}{2}[\cos(\omega_c-\omega_m)t - \cos(\omega_c+\omega_m)t] \end{aligned}$$

Subtracting gives USB, adding gives LSB:

$$\begin{aligned} v_1 - v_2 &= A_mA_c\cos(\omega_c+\omega_m)t \quad (\text{USB}) \\ v_1 + v_2 &= A_mA_c\cos(\omega_c-\omega_m)t \quad (\text{LSB}) \end{aligned}$$

In general, $s_{SSB}(t) = A_c[m(t)\cos\omega_ct \mp \hat m(t)\sin\omega_ct]$ (minus: USB, plus: LSB).

Waveforms / spectra

 v1 spectrum:     LSB  USB      (both, in phase)
 v2 spectrum:     LSB -USB      (USB inverted)
 v1 - v2:              2*USB    -> USB only
 v1 + v2:         2*LSB         -> LSB only

Advantages: no sharp sideband filter, works for messages with low-frequency content, sideband can be switched easily. Limitation: the wideband $90^\circ$ phase shifter must be accurate over the whole message band; any error leaves a residue of the unwanted sideband.