Costas loop detector and phase error correction

Block diagram of Costas loop

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

Parts: two product modulators (I and Q), two LPFs, a VCO with a $90^\circ$ phase shifter, and a phase discriminator (multiplier + loop filter).

How it demodulates DSB-SC

Input $s(t) = A_cm(t)\cos\omega_ct$; VCO output $\cos(\omega_ct + \phi)$.

$$\begin{aligned} y_I(t) &= \text{LPF}\{A_cm(t)\cos\omega_ct\cos(\omega_ct+\phi)\} = \frac{A_c}{2}m(t)\cos\phi \\ y_Q(t) &= \text{LPF}\{A_cm(t)\cos\omega_ct\sin(\omega_ct+\phi)\} = \frac{A_c}{2}m(t)\sin\phi \end{aligned}$$

When the VCO is locked ($\phi = 0$), $y_I = \tfrac{A_c}{2}m(t)$: the I channel output is the demodulated message, and $y_Q = 0$.

How it corrects phase error

The phase discriminator multiplies the I and Q outputs:

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

The loop filter averages this to a slowly varying control voltage $v_c \propto \overline{m^2}\sin2\phi \approx 2\overline{m^2}\phi$ for small $\phi$.

• If $\phi > 0$, $v_c > 0$: the VCO is pulled back to reduce $\phi$.
• If $\phi < 0$, $v_c < 0$: the VCO is pushed the other way.
• At $\phi = 0$: $v_c = 0$ and the loop is in lock.

Because $m^2(t) \ge 0$, the correction direction does not depend on the polarity of the message. So the Q channel acts as an error detector, and the loop keeps the local carrier coherent with the received one, avoiding the $\cos\phi$ loss and quadrature null.

Key points

• Automatic phase and frequency tracking without a pilot carrier.
• 180° phase ambiguity (lock at $\phi = \pi$ gives $-m(t)$), harmless for audio.
• Needs $m(t)$ to be non-zero most of the time.