---
title: "Costas loop as synchronous receiver for DSB-SC"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:costas-loop-dsbsc"
source: https://www.bctengineer.com/q/semester_6/communication_systems/costas-loop-dsbsc
site: BCT Engineering
---

# Costas loop as synchronous receiver for DSB-SC

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2073 Magh (CS I), 10 marks
- Video: [Costas Loop Receiver (Basics, Block Diagram, Working, Derivation & Advantages) Explained](https://www.youtube.com/watch?v=1y8goMRqYWs) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/costas-loop-dsbsc
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

With block diagram and necessary mathematics, show that the Costas loop can be used as a practical synchronous receiving system suitable for use with the DSB-SC modulated wave.

## Short answer

Costas loop uses I/Q coherent detectors with a VCO and phase discriminator. I-channel output (Ac/2)m(t)cosφ, Q-channel (Ac/2)m(t)sinφ. Their product gives error signal ∝ sin2φ, which after loop filter drives VCO to lock φ≈0. At lock, I-channel yields demodulated message (Ac/2)m(t).

## Answer

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

```text
                    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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. With VCO phase error phi, the LPF outputs of the I and Q channels are:
   A. (Ac/2) m(t) sin(phi) and (Ac/2) m(t) cos(phi)
   B. (Ac/2) m(t) cos(phi) and (Ac/2) m(t) sin(phi)
   C. Ac m(t) and zero
   D. (Ac/2) cos(phi) and (Ac/2) sin(phi)
   Answer: B. I channel multiplies by cos(wc t + phi) giving cos(phi); Q channel uses the -90 degree shifted VCO giving sin(phi). (Key point: I and Q channel outputs)

2. The phase discriminator multiplies yI and yQ to give e(t) equal to:
   A. (Ac^2/8) m^2(t) sin(2 phi)
   B. (Ac/2) m(t) cos(2 phi)
   C. (Ac^2/4) m(t) cos(phi)
   D. (Ac^2/8) m(t) cos(2 phi)
   Answer: A. yI yQ = (Ac^2/4) m^2 sin(phi)cos(phi) = (Ac^2/8) m^2 sin(2 phi). (Key point: Error signal)

3. Why does the control voltage direction not depend on the sign of m(t)?
   A. The LPF removes m(t)
   B. The VCO ignores the sign
   C. The I channel cancels it
   D. The error contains m^2(t), which is never negative
   Answer: D. Since m^2(t) >= 0, the sign of vc depends only on phi, so polarity reversals of DSB-SC do not disturb the loop. (Key point: Independence from message sign)

4. When the loop is locked (phi = 0), what do the channels give?
   A. I output is zero and Q is the message
   B. Both outputs are the message
   C. I output is (Ac/2) m(t), the message, and Q output is zero
   D. Both outputs are zero
   Answer: C. At phi = 0, cos = 1 and sin = 0, so I demodulates the message while Q acts only as error detector. (Key point: Locked operation)

## Related in Communication Systems

- [Phase error in synchronous DSB-SC detection; synchronized receiver](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-costas) — Communication Systems, chapter 2
- [Synchronous detection of DSB-SC; phase and frequency error](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asynchronous-dsbsc-phase-freq-error) — Communication Systems, chapter 2
- [Costas loop detector and phase error correction](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acostas-loop-phase-error) — Communication Systems, chapter 2
- [DSB-SC vs SSB; synchronous demodulation of DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbsc-vs-ssb-synchronous-demod) — Communication Systems, chapter 2
- [DSB-AM, DSB-SC and SSB spectra; synchronous detection](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbam-dsbsc-ssb-spectrum-synchronous) — Communication Systems, chapter 2
- [DSB-FC vs DSB-SC; phase shift method for SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-dsbsc-ssb-phase-shift) — Communication Systems, chapter 2

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