Phase error in coherent DSB-SC detection; AM demodulation by PLL

Effect of phase error in coherent detection of DSB-SC

Received: $s(t) = A_cm(t)\cos\omega_ct$. Local carrier with phase error $\phi$: $\cos(\omega_ct + \phi)$.

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

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

LPF output:

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

• $\phi = 0$: maximum output, no error.
• Constant $\phi$: output attenuated by $\cos\phi$ but not distorted.
• $\phi = \pm90^\circ$: $v_o = 0$, the quadrature null effect.
• Time-varying $\phi(t)$: output amplitude fluctuates (fading/distortion).

So the local carrier must be phase locked.

Demodulation of AM using PLL

A phase-locked loop locks a VCO to the carrier of the incoming AM signal; the locked carrier is then used in a product (coherent) detector.

                +---> Phase   --> Loop  --> VCO --+
                |     detector    filter          |
 s(t) ----------+        ^                        |
 (AM)           |        +---------[-90 deg]<-----+
                |                       cos(wc t) |
                +--------------->( X )<-----------+
                                   |
                                  LPF ---> m(t)

Working:

1. The PLL phase detector compares the AM input with the VCO output (shifted by $90^\circ$). Its average output is proportional to $\sin\theta_e$, where $\theta_e$ is the phase error.
2. The loop filter removes high-frequency parts; the control voltage tunes the VCO until it is locked in frequency and in phase quadrature with the phase detector input. The $90^\circ$ shift then makes the carrier fed to the product detector in phase with the received carrier.
3. Product detector: for $s(t) = A_c[1 + k_am(t)]\cos\omega_ct$,

$$s(t)\cos\omega_ct = \frac{A_c}{2}[1 + k_am(t)] + \frac{A_c}{2}[1 + k_am(t)]\cos2\omega_ct$$

4. LPF and DC block give $\dfrac{A_ck_a}{2}m(t)$.

Advantages: linear detection, no diagonal clipping, works with over-modulation and low SNR better than an envelope detector; integrated PLL ICs (e.g. 565) make it practical.