---
title: "Noise performance of coherent DSB-FC detection"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:dsbfc-noise-coherent-100pct"
source: https://www.bctengineer.com/q/semester_6/communication_systems/dsbfc-noise-coherent-100pct
site: BCT Engineering
---

# Noise performance of coherent DSB-FC detection

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2081 Baisakh (CS II), 5+2 marks
- Video: [Noise in AM(DSBFC) Receiver](https://www.youtube.com/watch?v=q7KMmvy1c-E) — Communication Systems
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/dsbfc-noise-coherent-100pct
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

Evaluate the noise performance of the coherent detection of DSB-FC using necessary derivations. Use the result to evaluate the noise performance of the single-tone DSB-FC modulation with 100% modulation.

## Short answer

Coherent DSB-FC detection yields input SNR_i = A_c²(1+k_a²P)/(4N_0W) and output SNR_o = A_c²k_a²P/(2N_0W). Detection gain γ = 2k_a²P/(1+k_a²P). For 100% single-tone modulation (μ=1, k_a²P=1/2), γ = 2/3 and FOM = 1/3, making output SNR 4.77 dB worse than DSB-SC/SSB for same transmitted power.

## Answer

#### Noise performance of coherent DSB-FC detection

Noise performance is measured by the output SNR and the **detection gain** $\gamma=\text{SNR}_o/\text{SNR}_i$ (or figure of merit, with SNR referred to the message band $W$).

```text
 s(t)+w(t) +-----+         +-----+   +-------+
 --------->| BPF |-->(x)-->| LPF |-->|  DC   |--> y(t)
           | 2W  |    ^    |  W  |   | block |
           +-----+    |    +-----+   +-------+
                  cos(wc t)
```

Receiver: BPF ($B_T=2W$) → product detector with $\cos\omega_ct$ → LPF ($W$) → DC block. Noise PSD $N_0/2$, narrowband noise $n(t)=n_c\cos\omega_ct-n_s\sin\omega_ct$ with $\overline{n_c^2}=2N_0W$.

- Signal power $=\frac{A_c^2}{2}(1+k_a^2P)$; noise in $2W$ $=2N_0W$:

$$
\text{SNR}_i=\frac{A_c^2(1+k_a^2P)}{4N_0W}
$$

- Coherent detection (or envelope detection at high SNR) and DC blocking:

$$
y(t)=\frac{A_c}{2}k_am(t)+\frac12n_c(t)\ \Rightarrow\ \text{SNR}_o=\frac{A_c^2k_a^2P/4}{2N_0W/4}=\frac{A_c^2k_a^2P}{2N_0W}
$$

$$
\gamma_{DSB\text{-}FC}=\frac{\text{SNR}_o}{\text{SNR}_i}=\frac{2k_a^2P}{1+k_a^2P},\qquad \text{FOM}=\frac{k_a^2P}{1+k_a^2P}
$$

The carrier uses power but carries no information, so the gain is reduced.

#### Single-tone DSB-FC with 100% modulation

**Single tone, 100% modulation.** $m(t)=A_m\cos\omega_mt$, $\mu=k_aA_m=1$, so $k_a^2P=\frac{\mu^2}{2}=\frac12$:

$$
\begin{aligned}
\gamma_{DSB\text{-}FC} &= \frac{2(\mu^2/2)}{1+\mu^2/2}=\frac{2\mu^2}{2+\mu^2}=\frac{2}{3}\approx0.667 \\
\text{FOM} &= \frac{\mu^2}{2+\mu^2}=\frac13
\end{aligned}
$$

So with 100% single-tone modulation the output SNR is only one-third of that of DSB-SC or SSB for the same transmitted power ($10\log_{10}3=4.77$ dB worse), and the detection gain is less than 1, because two-thirds of the power is in the carrier.

## Check yourself

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

1. What is the detection gain of coherent DSB-FC in terms of k_a^2 P?
   A. 2 k_a^2 P / (1 + k_a^2 P)
   B. k_a^2 P / (1 + k_a^2 P)
   C. (1 + k_a^2 P) / (2 k_a^2 P)
   D. 2 k_a^2 P
   Answer: A. SNR_o / SNR_i = [A^2 k_a^2 P / 2N0W] / [A^2(1+k_a^2P)/4N0W] = 2k_a^2P/(1+k_a^2P). (Key point: Detection gain of DSB-FC)

2. Why is the noise performance of DSB-FC worse than DSB-SC?
   A. The BPF bandwidth is doubled
   B. Noise is added at the transmitter
   C. Coherent detection cannot be used for DSB-FC
   D. The carrier consumes power but carries no information
   Answer: D. Carrier power counts in SNR_i but contributes nothing to the recovered message after the DC block. (Key point: Why gain is reduced)

3. For single-tone modulation with mu = 1, what are the detection gain and FOM?
   A. Gain 1, FOM 1
   B. Gain 2/3, FOM 1/3
   C. Gain 1/3, FOM 2/3
   D. Gain 3/2, FOM 1/2
   Answer: B. k_a^2 P = mu^2/2 = 1/2, so gain = 2(1/2)/(1+1/2) = 2/3 and FOM = (1/2)/(3/2) = 1/3. (Key point: Single-tone 100% result)

4. After coherent detection and DC blocking, what does the output y(t) contain?
   A. A_c m(t) + n_s(t)
   B. (A_c/2) k_a m(t) + (1/2) n_c(t)
   C. A_c + k_a m(t) + n(t)
   D. (A_c/2) k_a m(t) + n_s(t)
   Answer: B. The product detector and LPF give half amplitude: (A_c/2)k_a m + n_c/2; the quadrature noise n_s is rejected. (Key point: Detector output)

## Related in Communication Systems

- [DSB-FC single tone expression and modulation index](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbfc-single-tone-index) — Communication Systems, chapter 2
- [Single tone DSB-FC AM in time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asingle-tone-dsbfc-time-freq) — Communication Systems, chapter 2
- [Threshold effect in envelope detection of DSB-FC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Athreshold-envelope-dsbfc) — Communication Systems, chapter 9
- [Noise performance of DSB-SC and SSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anoise-dsbsc-ssbsc) — Communication Systems, chapter 9
- [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
- [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

## Related in other subjects

- [Single tone FM parameters](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afm-single-tone-numeric) — Data Communication, chapter 5