---
title: "Synchronous detection of DSB-SC; phase and frequency error"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:synchronous-dsbsc-phase-freq-error"
source: https://www.bctengineer.com/q/semester_6/communication_systems/synchronous-dsbsc-phase-freq-error
site: BCT Engineering
---

# Synchronous detection of DSB-SC; phase and frequency error

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2070 Bhadra (CS I), 3+3 marks
- Video: [Detection of DSBSC Signal| Synchronous Detection| Phase Error| Frequency Error| Amplitude Modulation](https://www.youtube.com/watch?v=8y1oDw4w6Ks) — Ashutosh Rasttogi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/synchronous-dsbsc-phase-freq-error
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

How can synchronous demodulator be used to detect DSB-SC wave? Explain mathematically, the effects of phase error and frequency error in local oscillator while demodulating DSB-SC.

## Short answer

Synchronous demodulator multiplies DSB-SC with local oscillator and low-pass filters. Phase error attenuates output by cosφ, zero when φ=90°. Frequency error multiplies output by cos(Δωt), causing periodic fading and distortion. Exact carrier synchronization is required.

## Answer

#### Synchronous (coherent) detection of DSB-SC

A DSB-SC wave $s(t) = A_c m(t)\cos\omega_ct$ has no carrier, so its envelope is $|m(t)|$, not $m(t)$. It is detected by multiplying with a locally generated carrier of the **same frequency and phase**, then low-pass filtering.

```text
 s(t) --->( X )--->[ LPF, BW = W ]---> v_o(t)
            ^
            |
   [ Local oscillator cos(wc t) ]
```

With an ideal local carrier $\cos\omega_ct$:

$$
v(t) = A_cm(t)\cos^2\omega_ct = \frac{A_c}{2}m(t) + \frac{A_c}{2}m(t)\cos 2\omega_ct
$$

The LPF removes the $2\omega_c$ term, so $v_o(t) = \frac{A_c}{2}m(t)$, an exact copy of the message.

#### Effect of phase error

Let the local carrier be $\cos(\omega_ct + \phi)$:

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

- The output is the message scaled by $\cos\phi$: an attenuation, not a distortion, if $\phi$ is constant.
- At $\phi = \pm 90^\circ$ the output is zero (**quadrature null effect**).
- If $\phi$ drifts randomly with time, the gain varies and the output fades.

#### Effect of frequency error

Let the local carrier be $\cos(\omega_c + \Delta\omega)t$:

$$
\begin{aligned}
v(t) &= \frac{A_c}{2}m(t)[\cos\Delta\omega t + \cos(2\omega_c + \Delta\omega)t] \\
v_o(t) &= \frac{A_c}{2}m(t)\cos(\Delta\omega\,t)
\end{aligned}
$$

- The message is multiplied by a slowly varying cosine at $\Delta f$. The output rises and falls periodically and passes through zero twice every $1/\Delta f$ seconds: a "beating" effect.
- In spectrum terms, $m(t)$ is shifted to $\pm\Delta f$, which badly distorts speech and music.

So DSB-SC needs exact carrier synchronization in both frequency and phase, using a pilot carrier or a Costas loop / squaring loop.

## Check yourself

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

1. Why can DSB-SC not be detected with an envelope detector?
   A. Its envelope is |m(t)|, not m(t)
   B. Its carrier is too large
   C. It has too little bandwidth
   D. It has too much power
   Answer: A. The carrier phase reverses when m(t) goes negative, but peaks stay positive. (Key point: Why not envelope)

2. With a frequency error, the output is
   A. (Ac/2) m(t) cos(phi)
   B. (Ac/2) m(t) cos(delta w t)
   C. Ac m(t)
   D. (Ac/2) m(t) cos(2 wc t)
   Answer: B. The message is multiplied by a slow beating cosine. (Key point: Frequency error)

3. A constant phase error causes
   A. attenuation by cos(phi), no distortion
   B. clipping
   C. zero output always
   D. frequency shift
   Answer: A. Output (Ac/2) m(t) cos phi. (Key point: Phase error)

4. How is the carrier recovered?
   A. Diode clipper
   B. Pilot carrier, Costas loop or squaring loop
   C. Envelope detector
   D. Ring modulator
   Answer: B. Exact frequency and phase synchronisation is needed. (Key point: Synchronisation)

## 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
- [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×
- [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
- [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
- [Detection gain of SSB-SC vs DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adetection-gain-ssbsc-dsbsc) — Communication Systems, chapter 9, asked 2×