---
title: "Threshold effect in envelope detection of DSB-FC"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:threshold-envelope-dsbfc"
source: https://www.bctengineer.com/q/semester_6/communication_systems/threshold-envelope-dsbfc
site: BCT Engineering
---

# Threshold effect in envelope detection of DSB-FC

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2072 Kartik (CS II), 7 marks
- Video: [Threshold Effect | Lec 15 । Noise Analysis |Communication System । NET ESE GATE](https://www.youtube.com/watch?v=0j2GB2Q8tJQ) — Dopamine
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/threshold-envelope-dsbfc
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

With necessary derivations, explain the threshold effect in envelope detector for DSB-FC modulation in analog communication system.

## Short answer

Below a carrier-to-noise ratio of about 10 dB, the output SNR of an envelope detector for DSB-FC collapses abruptly because the message is multiplied by a random noise term rather than being additive. This non-linear threshold effect does not occur in coherent detection.

## Answer

An **envelope detector** is a non-linear device whose output follows the magnitude of the resultant of signal plus noise. Its noise performance changes sharply when the carrier-to-noise ratio drops below a certain level; this is the **threshold effect**.

**Envelope detector input.** DSB-FC signal plus narrowband noise:

$$
x(t)=A_c[1+k_am(t)]\cos\omega_ct+n_c(t)\cos\omega_ct-n_s(t)\sin\omega_ct
$$

The envelope detector output is the magnitude of the resultant phasor:

$$
y(t)=\sqrt{\{A_c[1+k_am(t)]+n_c(t)\}^2+n_s^2(t)}
$$

```text
              n_s
          .---->           resultant length = y(t)
         /    ^
        /     | n_c
  ----------->+
  Ac[1+ka m(t)]
```

**Case 1: high carrier-to-noise ratio** ($A_c[1+k_am]\gg$ noise). The quadrature term is negligible:

$$
y(t)\approx A_c+A_ck_am(t)+n_c(t)
$$

The message appears **additively** with the noise. After removing DC, $\text{SNR}_o=\dfrac{A_c^2k_a^2P}{2N_0W}$, the same as coherent detection. Figure of merit $\frac{k_a^2P}{1+k_a^2P}$ ($=1/3$ for a single tone with $\mu=1$).

**Case 2: low carrier-to-noise ratio** (noise $\gg$ carrier). Write the noise as $n(t)=r(t)\cos[\omega_ct+\psi(t)]$, with Rayleigh envelope $r$ and uniform phase $\psi$. Then

$$
\begin{aligned}
y(t) &= \sqrt{r^2+2rA_c[1+k_am]\cos\psi+A_c^2[1+k_am]^2} \\
&\approx r(t)\sqrt{1+\frac{2A_c}{r(t)}[1+k_am(t)]\cos\psi(t)} \\
&\approx r(t)+A_c\cos\psi(t)+A_ck_am(t)\cos\psi(t)
\end{aligned}
$$

using $\sqrt{1+x}\approx1+x/2$. Now the message $m(t)$ is **multiplied by the random noise $\cos\psi(t)$**; there is no term proportional to $m(t)$ alone. The message is mutilated and cannot be recovered; output SNR collapses.

**Threshold effect.** The change from Case 1 to Case 2 is not gradual: below a certain carrier-to-noise ratio, the **threshold** (roughly 10 dB carrier-to-noise ratio), the output SNR falls much faster than the input SNR. The threshold is a property of **non-linear (envelope) detection**; coherent detection of DSB-FC, DSB-SC or SSB has **no threshold**, because the product detector is linear and the message always stays additive with noise.

```text
 SNRo (dB)                 coherent / envelope
     |                   . (above threshold)
     |                 .
     |               .
     |             .  <- threshold
     |          ..'
     |      ..''    envelope detector
     |  ..''        (below threshold)
     +-------------------------> CNR (dB)
```

## Check yourself

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

1. For the envelope detector, the output y(t) equals:
   A. The product of the input and a local carrier
   B. The magnitude of the resultant phasor of signal plus noise: sqrt({A_c[1+k_a m]+n_c}^2 + n_s^2)
   C. Only the noise component n_c(t)
   D. The phase of the signal plus noise
   Answer: B. The envelope is the length of the resultant of the signal and the in-phase and quadrature noise components. (Key point: Envelope detector output)

2. When the carrier is much larger than the noise, the output is approximately:
   A. r(t) + A_c cos ψ(t)
   B. A_c + A_c k_a m(t) + n_c(t), with the message additive to the noise
   C. k_a m(t) cos ψ(t)
   D. n_s(t) only
   Answer: B. The n_s^2 term is negligible, so the message appears additively with noise and SNR_o matches coherent detection. (Key point: High CNR case)

3. When noise is much larger than the carrier, why is the message lost?
   A. The message is multiplied by random noise cos ψ(t), so no term is proportional to m(t) alone
   B. The message is filtered out by the LPF
   C. The carrier phase flips to 180°
   D. The noise bandwidth becomes zero
   Answer: A. The output has the term A_c k_a m(t) cos ψ(t); the message is mutilated by random noise and SNR collapses. (Key point: Low CNR case)

4. Which statement about the threshold effect is right?
   A. It occurs in coherent detection of DSB-SC
   B. It occurs in the non-linear envelope detector below about 10 dB CNR, but not in coherent detection
   C. It occurs only in SSB
   D. It appears only above a high CNR
   Answer: B. Non-linear envelope detection makes output SNR drop faster than input SNR below the threshold; linear coherent detection has no threshold. (Key point: Threshold effect scope)

## Related in Communication Systems

- [Capture effect; gain parameter of DSB-FC envelope detection](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acapture-effect-dsbfc-envelope-gain) — Communication Systems, chapter 9
- [Noise performance of coherent DSB-FC detection](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-noise-coherent-100pct) — Communication Systems, chapter 9
- [Envelope detector; can it demodulate DSB-SC?](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-dsbsc) — Communication Systems, chapter 2, asked 2×
- [Envelope detector; why not for DSB-SC and SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aenvelope-detector-why-not-dsbsc-ssb) — Communication Systems, chapter 2
- [Detection gains of DSB-FC, DSB-SC and SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adetection-gain-dsbfc-dsbsc-ssb) — Communication Systems, chapter 9
- [Modulation; analog communication system block diagram](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amodulation-analog-block-diagram) — Communication Systems, chapter 1, asked 3×