DSB-AM, DSB-SC and SSB spectra; synchronous detection

Comparison of DSB-AM, DSB-SC and SSB

For message spectrum $M(f)$ band-limited to $W$:

Spectra

 DSB-AM:          carrier
          LSB      |      USB
         /////     |     \\\\\
 -------+-----+----+----+-----+------> f
      fc-W        fc         fc+W

 DSB-SC:  LSB    (no carrier)   USB
         /////                 \\\\\
 -------+-----+----+----+-----+------> f
      fc-W        fc         fc+W

 SSB (USB):                    USB
                              \\\\\
 ------------------+----------+------> f
                  fc         fc+W

Why DSB-SC detection is synchronous

DSB-SC has no carrier, and its envelope is $|m(t)|$, not $m(t)$; the carrier phase reverses each time $m(t)$ crosses zero. So an envelope detector would give $|m(t)|$ (distorted). The message can be recovered only by multiplying with a local carrier that is synchronised in frequency and phase with the transmitter carrier, hence "synchronous detection".

 s(t) --->( X )---> LPF ---> vo(t)
            ^
     cos(wc t + phi)   (from carrier recovery)

$$v_o(t) = \text{LPF}\{A_cm(t)\cos\omega_ct\cos(\omega_ct + \phi)\} = \frac{A_c}{2}m(t)\cos\phi$$

• $\phi = 0$: output $\tfrac{A_c}{2}m(t)$, ideal.
• $\phi \ne 0$: attenuation by $\cos\phi$; at $\phi = 90^\circ$ the output vanishes (quadrature null).
• Frequency error $\Delta\omega$: output $\propto m(t)\cos\Delta\omega t$, a beating distortion.

Hence exact synchronisation is required, provided by a pilot carrier, squaring loop or Costas loop.