---
title: "Noise performance of DSB-AM, DSB-SC and SSB-SC"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:noise-dsbam-dsbsc-ssbsc"
source: https://www.bctengineer.com/q/semester_6/communication_systems/noise-dsbam-dsbsc-ssbsc
site: BCT Engineering
---

# Noise performance of DSB-AM, DSB-SC and SSB-SC

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2072 Chaitra (CS II), 8 marks
- Video: [NOISE CALCULATION IN DSBSC AND SSB SC](https://www.youtube.com/watch?v=CyvuC51e7Cw) — Thriveni Chowdary
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/noise-dsbam-dsbsc-ssbsc
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

With necessary derivation, compare noise performance of DSB-AM, DSB-SC, SSB-SC.

## Short answer

DSB-SC (FOM 1, gain 2) and SSB-SC (FOM 1, gain 1) give identical output SNR for equal transmitted power; SSB uses half the bandwidth. DSB-FC (AM) wastes power in the carrier, resulting in a figure of merit less than 1, e.g., 1/3 for 100% tone modulation.

## Answer

**Receiver model.** The received signal $s(t)$ plus white noise ($N_0/2$) passes through a band-pass (IF) filter of bandwidth $B_T$, then a demodulator and a low-pass filter of bandwidth $W$ (message bandwidth).

```text
 s(t)  +   +-------+    +-------+   +-----+
 ---->(+)-->|  BPF  |--->| demod |-->| LPF |--> y(t)
       ^    |  B_T  |    |       |   |  W  |
      w(t)  +-------+    +-------+   +-----+
```

The filtered noise is narrowband and is written in in-phase/quadrature form

$$
n(t)=n_c(t)\cos\omega_ct-n_s(t)\sin\omega_ct
$$

where $n_c$ and $n_s$ are low-pass, each with the same power as $n(t)$: $\overline{n_c^2}=\overline{n_s^2}=\overline{n^2}=N_0B_T$.

**Definitions.** Message power $P=\overline{m^2(t)}$.
- Input SNR $\text{SNR}_i=\dfrac{\text{received signal power}}{\text{noise power in } B_T}$
- Output SNR $\text{SNR}_o=\dfrac{\text{message power at output}}{\text{noise power at output}}$
- **Detection gain** $\gamma_d=\dfrac{\text{SNR}_o}{\text{SNR}_i}$; **figure of merit** $=\dfrac{\text{SNR}_o}{\text{SNR}_{ref}}$, where $\text{SNR}_{ref}$ is the channel SNR measured in the message band $W$ (baseband reference). Both show how much the demodulator improves or worsens SNR.

**DSB-FC (conventional AM).** $s(t)=A_c[1+k_am(t)]\cos\omega_ct$, $B_T=2W$. (Coherent and envelope detection give the same result at high SNR.)

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

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

**DSB-SC (coherent detection).** $s(t)=A_cm(t)\cos\omega_ct$, $B_T=2W$.

- Signal power $=\frac{A_c^2P}{2}$; noise in $2W$ $=2N_0W$, so $\text{SNR}_i=\dfrac{A_c^2P}{4N_0W}$.
- Multiply $x(t)=s(t)+n(t)$ by $\cos\omega_ct$ and low-pass filter:

$$
y(t)=\frac{A_c}{2}m(t)+\frac12n_c(t)
$$

- $\text{SNR}_o=\dfrac{A_c^2P/4}{2N_0W/4}=\dfrac{A_c^2P}{2N_0W}$

$$
\gamma_{DSB\text{-}SC}=\frac{\text{SNR}_o}{\text{SNR}_i}=2
$$

The quadrature noise $n_s$ is rejected by coherent detection, giving a 3 dB gain. With baseband reference $\text{SNR}_{ref}=\frac{A_c^2P}{2N_0W}$, figure of merit $=1$.

**SSB-SC (coherent detection).** $s(t)=\frac{A_c}{2}[m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct]$, $B_T=W$.

- Signal power $=\frac{A_c^2}{4}\cdot\frac{P+P}{2}=\frac{A_c^2P}{4}$ (since $\overline{\hat m^2}=P$); noise in $W$ $=N_0W$, so $\text{SNR}_i=\dfrac{A_c^2P}{4N_0W}$.
- Multiply by $\cos\omega_ct$ and low-pass filter:

$$
y(t)=\frac{A_c}{4}m(t)+\frac12n_c(t)
$$

- Signal power $\frac{A_c^2P}{16}$, noise power $\frac{N_0W}{4}$, so $\text{SNR}_o=\dfrac{A_c^2P}{4N_0W}$

$$
\gamma_{SSB}=\frac{\text{SNR}_o}{\text{SNR}_i}=1
$$

With baseband reference ($\text{SNR}_{ref}=\frac{A_c^2P}{4N_0W}$, same transmitted power), figure of merit $=1$.

**Comparison:**

| Parameter | DSB-FC (AM) | DSB-SC | SSB-SC |
|---|---|---|---|
| Transmission bandwidth $B_T$ | $2W$ | $2W$ | $W$ |
| Detector | Envelope or coherent | Coherent | Coherent |
| Detection gain $\text{SNR}_o/\text{SNR}_i$ | $\frac{2k_a^2P}{1+k_a^2P}$ ($=\frac23$ at $\mu=1$) | 2 | 1 |
| Figure of merit (baseband ref.) | $\frac{k_a^2P}{1+k_a^2P}$ ($=\frac13$ at $\mu=1$) | 1 | 1 |
| Threshold effect | Yes (envelope detector) | No | No |
| Power wasted in carrier | Yes (at least 1/2; 2/3 for a 100% tone) | No | No |

**Conclusions:** for the same transmitted power and noise PSD, DSB-SC and SSB give the **same output SNR** (equal to baseband); DSB-SC's detection gain of 2 is offset by its double noise bandwidth. SSB achieves this in **half the bandwidth**. DSB-FC is at least **3 dB worse** (since $k_a^2P\le1$ gives FOM $\le\frac12$, with equality only for a 100% square wave) and **4.77 dB worse** (a factor of 3) for 100% single-tone modulation, because the carrier carries no information; its advantage is that it allows a cheap envelope detector.

## Check yourself

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

1. In the receiver model, the narrowband noise after the IF filter is written as n_c cos ωct - n_s sin ωct. What are the powers of n_c and n_s?
   A. Each equals the power of n(t), N0 B_T
   B. Each is half of n(t)'s power
   C. n_c has all the power, n_s has none
   D. Each is twice n(t)'s power
   Answer: A. Both quadrature components have the same power as n(t), namely N0 B_T. (Key point: Noise model)

2. For DSB-SC, after multiplying by cos ωct and low-pass filtering, y(t) is:
   A. A_c m(t) + n_c(t)
   B. (A_c/2) m(t) + (1/2) n_c(t), rejecting n_s
   C. (A_c/4) m(t) + n_s(t)
   D. (A_c/2) m(t) + (1/2) n_s(t)
   Answer: B. Coherent detection keeps the in-phase noise n_c and rejects the quadrature n_s, giving detection gain 2. (Key point: DSB-SC coherent detection result)

3. With baseband reference, the figures of merit of DSB-FC (single tone, μ = 1), DSB-SC and SSB-SC are:
   A. 1/3, 1, 1
   B. 1, 1, 2
   C. 2/3, 2, 1
   D. 1/3, 2, 1
   Answer: A. FOM = k_a^2P/(1 + k_a^2P) = 1/3 for DSB-FC at μ = 1; DSB-SC and SSB both equal baseband, FOM = 1. (Key point: Figures of merit)

4. Which conclusion of the noise comparison is right?
   A. DSB-FC is best because it uses an envelope detector
   B. DSB-SC and SSB have equal output SNR, SSB in half the bandwidth; DSB-FC is at least 4.77 dB worse
   C. SSB is 3 dB better than DSB-SC
   D. All three give identical output SNR
   Answer: B. The carrier in DSB-FC wastes power (factor 3 at μ = 1, 4.77 dB), while SSB matches DSB-SC SNR with bandwidth W instead of 2W. (Key point: Overall conclusion)

## Related in Communication Systems

- [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
- [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
- [Compare DSB-AM, DSB-SC and SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Acompare-dsbam-dsbsc-ssb) — Communication Systems, chapter 2
- [DSB-AM vs SSB; ring modulator for DSB-SC](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adsbam-ssb-compare-ring-modulator) — 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
- [SSB vs AM and DSB-SC; balanced modulator](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-vs-am-balanced-modulator) — Communication Systems, chapter 2