---
title: "Phase error in coherent DSB-SC detection; AM demodulation by PLL"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:phase-error-dsbsc-am-pll"
source: https://www.bctengineer.com/q/semester_6/communication_systems/phase-error-dsbsc-am-pll
site: BCT Engineering
---

# Phase error in coherent DSB-SC detection; AM demodulation by PLL

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2071 Magh (CS I), 4+4 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/phase-error-dsbsc-am-pll
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Show the effect of phase error in coherent detection of DSB-SC AM. Explain the demodulation of AM using PLL.

## Short answer

In coherent DSB-SC detection, a constant phase error φ attenuates output by cosφ without distortion; at 90° (quadrature null) output vanishes, and time-varying φ causes distortion. A PLL demodulates AM by locking a VCO to the carrier; the recovered carrier drives a product detector, giving m(t) after lowpass filtering.

## Answer

#### Effect of phase error in coherent detection of DSB-SC

Received: $s(t) = A_cm(t)\cos\omega_ct$. Local carrier with phase error $\phi$: $\cos(\omega_ct + \phi)$.

```text
 s(t) --->( X )--- v(t) ---> LPF ---> vo(t)
            ^
     cos(wc t + phi)
```

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

LPF output:

$$
v_o(t) = \frac{A_c}{2}m(t)\cos\phi
$$

- $\phi = 0$: maximum output, no error.
- Constant $\phi$: output attenuated by $\cos\phi$ but not distorted.
- $\phi = \pm90^\circ$: $v_o = 0$, the **quadrature null effect**.
- Time-varying $\phi(t)$: output amplitude fluctuates (fading/distortion).

So the local carrier must be phase locked.

#### Demodulation of AM using PLL

A **phase-locked loop** locks a VCO to the carrier of the incoming AM signal; the locked carrier is then used in a product (coherent) detector.

```text
                +---> Phase   --> Loop  --> VCO --+
                |     detector    filter          |
 s(t) ----------+        ^                        |
 (AM)           |        +---------[-90 deg]<-----+
                |                       cos(wc t) |
                +--------------->( X )<-----------+
                                   |
                                  LPF ---> m(t)
```

**Working:**

1. The PLL phase detector compares the AM input with the VCO output (shifted by $90^\circ$). Its average output is proportional to $\sin\theta_e$, where $\theta_e$ is the phase error.
2. The loop filter removes high-frequency parts; the control voltage tunes the VCO until it is locked in frequency and in phase quadrature with the phase detector input. The $90^\circ$ shift then makes the carrier fed to the product detector **in phase** with the received carrier.
3. Product detector: for $s(t) = A_c[1 + k_am(t)]\cos\omega_ct$,

$$
s(t)\cos\omega_ct = \frac{A_c}{2}[1 + k_am(t)] + \frac{A_c}{2}[1 + k_am(t)]\cos2\omega_ct
$$

4. LPF and DC block give $\dfrac{A_ck_a}{2}m(t)$.

**Advantages:** linear detection, no diagonal clipping, works with over-modulation and low SNR better than an envelope detector; integrated PLL ICs (e.g. 565) make it practical.

## Check yourself

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

1. With local carrier cos(wc t + phi), what is the LPF output for DSB-SC Ac m(t) cos wc t?
   A. (Ac/2) m(t) cos(2wc t + phi)
   B. Ac m(t) cos(2 phi)
   C. (Ac/2) m(t) cos(phi)
   D. (Ac/2) m(t) sin(phi)
   Answer: C. The product contains (Ac/2)m(t)cos(phi) at baseband plus a 2wc term which the LPF removes. (Key point: Output with phase error)

2. What happens when the phase error is 90 degrees?
   A. The output is inverted
   B. The output is zero (quadrature null effect)
   C. The output is distorted but nonzero
   D. The output is maximum
   Answer: B. cos(90 deg) = 0, so the demodulated output vanishes. (Key point: Quadrature null)

3. What is the effect of a constant (non-zero) phase error below 90 degrees?
   A. Output frequency is shifted
   B. Output is distorted by harmonics
   C. Output is attenuated by cos(phi) without distortion
   D. Output is unaffected
   Answer: C. A constant phi only scales the message by cos(phi); only time-varying phi causes fading. (Key point: Constant vs varying phase)

4. In PLL-based AM demodulation, why is a 90 degree shift used between the VCO and the product detector?
   A. To double the carrier frequency
   B. To make the VCO run at 2fc
   C. The PLL locks in quadrature, so the shift puts the carrier in phase with the received carrier
   D. To remove the DC component
   Answer: C. The VCO locks 90 degrees out of phase with the input, so a further 90 degree shift gives an in-phase carrier for the product detector. (Key point: PLL demodulation)

## 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
- [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
- [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×
- [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