Phase error in synchronous DSB-SC detection; synchronized receiver

Effect of phase error in synchronous detection

DSB-SC wave: $s(t) = A_cm(t)\cos\omega_ct$. Local oscillator with phase error $\phi$: $\cos(\omega_ct + \phi)$.

 s(t) --->( X )---> v(t) ---> LPF ---> vo(t)
            ^
     cos(wc t + phi)

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

After the LPF:

$$v_o(t) = \frac{A_c}{2}m(t)\cos\phi$$

• $\phi = 0$: maximum output $\tfrac{A_c}{2}m(t)$.
• Small $\phi$: $\cos\phi \approx 1 - \phi^2/2$, so output is only slightly reduced and undistorted.
• $\phi = \pm90^\circ$: output is zero: the quadrature null effect.
• If $\phi$ varies randomly with time, the output amplitude fluctuates (fading). So the receiver needs a carrier that is locked in phase.

Practical synchronous receiver: Costas loop

               +-->(X)-->LPF--> (Ac/2)m(t)cos(phi) -> out
               |    ^                       |
               |  cos(wc t+phi)             v
 s(t) ---------+   VCO <--- loop filter <--(X)
               |    |                       ^
               |  [-90 deg]                 |
               |  sin(wc t+phi)             |
               +-->(X)-->LPF--> (Ac/2)m(t)sin(phi)

• I-channel output: $\tfrac{A_c}{2}m(t)\cos\phi$ (message).
• Q-channel output: $\tfrac{A_c}{2}m(t)\sin\phi$ (error signal, zero when locked).
• Phase discriminator (multiplier + loop filter):

$$e = \overline{\frac{A_c^2}{4}m^2(t)\sin\phi\cos\phi} = \frac{A_c^2}{8}\overline{m^2(t)}\sin2\phi$$

• For small $\phi$, $e \propto \phi$. This voltage drives the VCO to reduce $\phi$ towards zero. When $\phi \to 0$, the Q output vanishes and the I output gives the message.

So the Costas loop gives automatic phase and frequency tracking and works even though DSB-SC has no carrier component.