Phase shift method of SSB generation

The phase shift (phase discrimination) method generates SSB by using two balanced modulators and $90^\circ$ phase shifters, so that one sideband cancels and the other adds. No sharp sideband filter is needed.

Block diagram

             +-------> Balanced  ---- v1 ----+
             |         modulator 1           |
             |            ^                  v
 m(t) -------+            | Ac cos wc t    (+/-) --> SSB
             |       Carrier osc             ^
             |            |                  |
          [-90 deg]   [-90 deg]              |
             |            | Ac sin wc t      |
             | m^(t)      v                  |
             +-------> Balanced  ---- v2 ----+
                       modulator 2

• Upper path: $m(t)$ and carrier $A_c\cos\omega_ct$.
• Lower path: message shifted by $-90^\circ$ (Hilbert transform $\hat m(t)$) and carrier shifted by $-90^\circ$ ($A_c\sin\omega_ct$).

Expressions

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

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

For a general message:

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

(minus → USB, plus → LSB).

Merits: no sharp filter, any carrier frequency, low audio frequencies preserved, easy sideband switching. Demerit: the wideband $90^\circ$ phase shifter must be accurate over the whole audio band (e.g. 300–3400 Hz); phase or amplitude errors leave part of the unwanted sideband.