---
title: "Phase and frequency error in local oscillator (DSB and SSB)"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:phase-freq-error-lo-dsb-ssb"
source: https://www.bctengineer.com/q/semester_6/communication_systems/phase-freq-error-lo-dsb-ssb
site: BCT Engineering
---

# Phase and frequency error in local oscillator (DSB and SSB)

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2080 Chaitra (CS I), 8 marks; 2068 Bhadra (CS I), 8 marks (asked 2 times)
- Video: [Synchronous Detector : Amplitude Modulation I Phase & Frequency errors I Lect 32 I Comm. Engg I GATE](https://www.youtube.com/watch?v=qrOdw2gd4yQ) — OM learning classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/phase-freq-error-lo-dsb-ssb
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Explain the effect of phase and frequency error in local oscillator in demodulating DSB and SSB using PLL.

## Short answer

In DSB-SC, a phase error attenuates output by cos φ; a frequency error causes beating distortion. In SSB, a phase error introduces phase distortion (acceptable for speech), while a frequency error shifts all frequency components, making speech sound unnatural.

## Answer

In coherent (synchronous) demodulation, the receiver multiplies the received signal by a locally generated carrier and low-pass filters. A PLL (or Costas loop) is used to generate this carrier. If the local oscillator is not exactly matched, i.e. it has a **frequency error** $\Delta\omega$ and **phase error** $\phi$, the output is degraded.

Let the local carrier be $c(t)=2\cos[(\omega_c+\Delta\omega)t+\phi]$, and define $\theta(t)=\Delta\omega t+\phi$.

```text
 s(t) -->(X)-->[ LPF ]--> e0(t)
          ^
          | 2cos[(wc+dw)t + phi]
     [ PLL / local osc ]
```

#### Effect on DSB-SC
Received $s(t)=m(t)\cos\omega_ct$:

$$
\begin{aligned}
s(t)c(t)&=2m(t)\cos\omega_ct\cos(\omega_ct+\theta)\\
&=m(t)\cos\theta+m(t)\cos(2\omega_ct+\theta)\\
e_0(t)&=m(t)\cos(\Delta\omega t+\phi)\quad(\text{after LPF})
\end{aligned}
$$

- **Phase error only** ($\Delta\omega=0$): $e_0=m(t)\cos\phi$. The output is undistorted but **attenuated** by $\cos\phi$. If $\phi$ varies randomly, the output fades; at $\phi=\pm90°$ the output is zero (**quadrature null effect**).
- **Frequency error only** ($\phi=0$): $e_0=m(t)\cos(\Delta\omega t)$. The message is multiplied by a slowly varying sinusoid, so the output **beats** (rises and falls periodically at rate $\Delta f$), causing serious distortion even for small $\Delta f$.

So DSB-SC demodulation needs both frequency and phase to be correct; a Costas loop or squaring PLL is used.

#### Effect on SSB
Received USB: $s(t)=m(t)\cos\omega_ct-\hat m(t)\sin\omega_ct$. Multiplying and filtering:

$$
\begin{aligned}
e_0(t)&=m(t)\cos\theta+\hat m(t)\sin\theta\\
&=m(t)\cos(\Delta\omega t+\phi)+\hat m(t)\sin(\Delta\omega t+\phi)
\end{aligned}
$$

(For LSB the sign of the $\hat m$ term changes.)

- **Phase error only:** $e_0=m(t)\cos\phi+\hat m(t)\sin\phi$. Every frequency component of the message gets a constant phase shift $\phi$ (for a tone $A_m\cos\omega_mt$, $e_0=A_m\cos(\omega_mt-\phi)$). Amplitude is not reduced, but there is **phase distortion**. The human ear is insensitive to phase, so speech is acceptable, but data and video are affected.
- **Frequency error only:** for a tone, $e_0=A_m\cos(\omega_m-\Delta\omega)t$. Every component is **shifted in frequency by $\Delta f$**, so harmonic relationships are destroyed. Speech sounds unnatural ("Donald Duck" effect); a shift of a few Hz is tolerable for speech, but music becomes badly distorted.

#### Comparison

| Error | DSB-SC output | SSB output |
|---|---|---|
| Phase $\phi$ | $m(t)\cos\phi$: attenuation, null at 90° | Phase shift of all components (phase distortion) |
| Frequency $\Delta f$ | $m(t)\cos2\pi\Delta ft$: beating | All components shifted by $\Delta f$ |

**Remedy:** a PLL locks the local oscillator to a transmitted **pilot carrier** (SSB, DSB) or recovers the carrier from the signal itself (Costas loop for DSB-SC), reducing both errors close to zero.

## Check yourself

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

1. For DSB-SC demodulation with phase error phi only, the output is:
   A. m(t) cos(phi): attenuated, zero at 90 degrees
   B. m(t) + phi
   C. m(t) with beating
   D. m(t) shifted in frequency
   Answer: A. Multiplying by 2cos(wct+phi) and filtering gives m(t)cos(phi), which fades and nulls at quadrature. (Key point: DSB phase error)

2. What is the effect of a frequency error in DSB-SC demodulation?
   A. Constant attenuation
   B. Output m(t)cos(delta w t): the message beats up and down
   C. A fixed phase shift of the message
   D. No effect
   Answer: B. The message is multiplied by a slowly varying cosine, causing beating and serious distortion. (Key point: DSB frequency error)

3. In SSB demodulation with phase error only, what happens?
   A. Amplitude is nulled at 90 degrees
   B. Every component is phase-shifted by phi (phase distortion), amplitude unchanged
   C. Each component shifts in frequency
   D. Both sidebands reappear
   Answer: B. e0 = m cos(phi) + m^ sin(phi), a constant phase shift of every component; the ear tolerates it for speech. (Key point: SSB phase error)

4. In SSB demodulation with frequency error delta f, the recovered tone is:
   A. Unchanged
   B. Shifted by delta f so harmonic relations are destroyed (Donald Duck effect)
   C. Attenuated by cos(delta f)
   D. Rectified
   Answer: B. A tone Am cos(wm t) becomes Am cos(wm - delta w)t, shifting every component and making speech sound unnatural. (Key point: SSB frequency 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 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
- [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
- [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
- [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×

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