---
title: "Phase error in synchronous DSB-SC detection; synchronized receiver"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:phase-error-dsbsc-costas"
source: https://www.bctengineer.com/q/semester_6/communication_systems/phase-error-dsbsc-costas
site: BCT Engineering
---

# Phase error in synchronous DSB-SC detection; synchronized receiver

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2076 Bhadra (CS I), 3+5 marks
- Video: [Coherent Detection of DSB-SC or DSB-SC Detection (Basics, Working, Block Diagram & Impotence)](https://www.youtube.com/watch?v=uiTHa7g5ssg) — 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/phase-error-dsbsc-costas
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Evaluate the effect of small phase error in the local oscillator on synchronous detection of DSB-SC AM. Propose one of practically synchronized receiving system for DSB-SC wave?

## Short answer

A small phase error reduces output by cos φ without distortion, but time-varying φ causes fading. The Costas loop uses I and Q channels; their product gives a phase error signal driving a VCO to lock local oscillator phase to the suppressed carrier, enabling coherent detection.

## Answer

#### Effect of phase error in synchronous detection

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

```text
 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

```text
               +-->(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.

## Check yourself

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

1. With local carrier phase error phi, how does the output change?
   A. It is scaled by sin(phi)
   B. It is scaled by cos(phi)
   C. It is unaffected
   D. It is scaled by 2 cos(phi)
   Answer: B. vo = (Ac/2) m(t) cos phi. (Key point: Effect of phase error)

2. What happens for a small constant phi?
   A. Output is heavily distorted
   B. Output becomes zero
   C. Output becomes the 2wc term
   D. Output is slightly reduced but undistorted
   Answer: D. cos phi is about 1 - phi^2/2. (Key point: Small phase error)

3. At phi = 90°, what is the output?
   A. Zero
   B. Maximum
   C. Half of maximum
   D. Negative maximum
   Answer: A. cos 90° = 0. (Key point: Quadrature null)

4. In the Costas loop, what error signal drives the VCO?
   A. A constant voltage independent of phi
   B. The message itself
   C. About (Ac^2/8) m^2 sin 2phi, proportional to phi for small phi
   D. Ac cos phi only
   Answer: C. The product of I and Q outputs, averaged, adjusts the VCO until phi tends to 0. (Key point: Costas loop error)

## Related in Communication Systems

- [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
- [Phase error in coherent DSB-SC detection; AM demodulation by PLL](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-error-dsbsc-am-pll) — Communication Systems, chapter 2
- [Phase and frequency error in local oscillator (DSB and SSB)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aphase-freq-error-lo-dsb-ssb) — Communication Systems, chapter 2, asked 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
- [Costas loop as synchronous receiver for DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acostas-loop-dsbsc) — Communication Systems, chapter 2
- [Compare DSB-AM, DSB-SC, SSB, VSB; SSB by phase discrimination](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acompare-am-variants-ssb-phase) — Communication Systems, chapter 2