Costas loop as synchronous receiver for DSB-SC

The Costas loop is a practical synchronous (coherent) receiver for DSB-SC. Since DSB-SC contains no carrier, the loop derives a phase-locked carrier from the sidebands and demodulates at the same time.

Block diagram

                    I-channel
           +---->( X )----> LPF ----+--------> demodulated
           |       ^                |          output
           |       | cos(wc t+phi)  |   (Ac/2)m(t)cos(phi)
           |       |                v
 s(t) -----+     [VCO] <-- Loop <-( X )  phase
 Ac m(t)   |       |       filter   ^    discriminator
 cos wc t  |    [-90 deg]           |
           |       | sin(wc t+phi)  |
           |       v                |
           +---->( X )----> LPF ----+
                    Q-channel   (Ac/2)m(t)sin(phi)

It has two coherent detectors (I and Q) fed by the same VCO, one through a $-90^\circ$ phase shifter, and a phase discriminator (multiplier + loop filter) that controls the VCO.

Mathematics

Input: $s(t) = A_cm(t)\cos\omega_ct$. VCO output: $\cos(\omega_ct + \phi)$, where $\phi$ is the phase error.

I-channel:

$$\begin{aligned} s(t)\cos(\omega_ct + \phi) &= \frac{A_c}{2}m(t)[\cos\phi + \cos(2\omega_ct + \phi)] \\ y_I(t) &= \frac{A_c}{2}m(t)\cos\phi \quad \text{(after LPF)} \end{aligned}$$

Q-channel:

$$\begin{aligned} s(t)\sin(\omega_ct + \phi) &= \frac{A_c}{2}m(t)[\sin\phi + \sin(2\omega_ct + \phi)] \\ y_Q(t) &= \frac{A_c}{2}m(t)\sin\phi \quad \text{(after LPF)} \end{aligned}$$

Phase discriminator:

$$\begin{aligned} e(t) = y_Iy_Q &= \frac{A_c^2}{4}m^2(t)\sin\phi\cos\phi \\ &= \frac{A_c^2}{8}m^2(t)\sin2\phi \end{aligned}$$

The loop filter (narrow LPF) averages $m^2(t)$ to its mean value $\overline{m^2}$:

$$v_c = \frac{A_c^2}{8}\overline{m^2}\sin2\phi \approx \frac{A_c^2}{4}\overline{m^2}\,\phi \quad (\text{small }\phi)$$

Operation

1. When $\phi = 0$ (locked): $y_Q = 0$ and $y_I = \tfrac{A_c}{2}m(t)$, the desired message.
2. If the VCO phase drifts so $\phi > 0$, the control voltage $v_c$ becomes positive and drives the VCO frequency/phase to reduce $\phi$. If $\phi < 0$, $v_c$ is negative and corrects in the other direction.
3. Thus the loop is a negative-feedback system that keeps $\phi \approx 0$, giving automatic carrier synchronisation. The Q channel acts as the error detector, the I channel as the demodulator.
4. The sign of $v_c$ does not depend on the sign of $m(t)$ because $m^2(t) \ge 0$, so modulation reversals in DSB-SC do not disturb the loop.

Remarks

• Phase ambiguity: $\sin2\phi$ is zero at $\phi = 0$ and $\phi = 180^\circ$, so the loop can lock with inverted polarity; for speech this is harmless.
• Message must not be zero for long periods, or the loop loses its error signal.
• Advantages: no pilot carrier required, tracks slow frequency and phase drift, demodulates and recovers the carrier in one circuit; also used for BPSK carrier recovery.