Short note: Coherent detector

A coherent (synchronous) detector recovers the message by multiplying the received modulated signal with a locally generated carrier of the same frequency and phase as the transmitted carrier, followed by a low-pass filter. It is needed for DSB-SC, SSB and VSB, and can also detect DSB-FC.

 s(t) ---->( X )----> LPF ----> vo(t)
             ^       (0 to W)
             |
     cos(wc t + phi)  <-- carrier recovery
                          (Costas loop / PLL)

For DSB-SC, $s(t) = A_cm(t)\cos\omega_ct$:

$$\begin{aligned} v(t) &= A_cm(t)\cos\omega_ct\cos(\omega_ct + \phi) \\ &= \frac{A_c}{2}m(t)\cos\phi + \frac{A_c}{2}m(t)\cos(2\omega_ct + \phi) \end{aligned}$$

After LPF: $v_o(t) = \tfrac{A_c}{2}m(t)\cos\phi$.

Effect of errors

• Phase error $\phi$: output reduced by $\cos\phi$; at $\phi = 90^\circ$ output is zero (quadrature null).
• Frequency error $\Delta\omega$: output $\propto m(t)\cos\Delta\omega t$ (beating); for SSB it shifts the audio pitch.

Carrier recovery: pilot carrier, squaring loop, or Costas loop.

Merits: works for all AM types, linear, better noise performance than envelope detection at low SNR. Demerits: complex and costly because of the synchronisation circuit.