---
title: "Double-tone AM"
subject: "Data Communication"
chapter: 5
key: "semester_5/data_communication:double-tone-am"
source: https://www.bctengineer.com/q/semester_5/data_communication/double-tone-am
site: BCT Engineering
---

# Double-tone AM

Data Communication · Chapter 5: Data Encoding and Modulation

- Asked in: 2078 Bhadra, 4 marks; 2075 Chaitra, 8 marks (asked 2 times)
- Video: [L 7 |Multi-tone Amplitude Modulation | Double Side Band Full Carrier | DSB-FC | Analog Communication](https://www.youtube.com/watch?v=1KdQkDJUbL0) — Dopamine
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/double-tone-am
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-5

## Question

Write a short note on double-tone AM.

## Short answer

Double-tone AM uses two modulating sinusoids. Total modulation index μt=√(μ₁²+μ₂²), peak condition μ₁+μ₂≤1. Sideband power is Pc μt²/2, bandwidth 2×max(f₁,f₂). For Ac=10V, μ₁=0.4, μ₂=0.3, fc=100kHz, f₁=1kHz, f₂=2kHz, R=50Ω: μt=0.5, Pc=1W, Pt=1.125W, BW=4kHz, η=11.1%.

## Answer

In **double-tone (two-tone) AM**, the carrier is amplitude-modulated by a message that contains two sinusoids of different frequencies f₁ and f₂ at the same time (for example a two-tone test signal).

#### Expression
m(t) = Am₁ cos(2πf₁t) + Am₂ cos(2πf₂t)

s(t) = Ac [1 + μ₁ cos(2πf₁t) + μ₂ cos(2πf₂t)] cos(2πfct)

where μ₁ = Am₁/Ac and μ₂ = Am₂/Ac. Expanding:

s(t) = Ac cos ωct + (Acμ₁/2)[cos(ωc+ω₁)t + cos(ωc-ω₁)t] + (Acμ₂/2)[cos(ωc+ω₂)t + cos(ωc-ω₂)t]

#### Spectrum
```text
 amplitude
       |
 Ac    |           |
       |           |
 Acu1/2|    |      |      |
 Acu2/2| |  |      |      |  |
       +-+--+------+------+--+----> f
       fc-f2 fc-f1 fc  fc+f1 fc+f2
```
The signal has the carrier and four side frequencies (two for each tone).

#### Total modulation index
μt = √(μ₁² + μ₂²). To avoid over-modulation and allow distortion-free envelope detection, the peak condition μ₁ + μ₂ ≤ 1 must hold. This is stricter than μt ≤ 1.

#### Power
Pc = Ac²/2R and Pt = Pc (1 + μ₁²/2 + μ₂²/2) = Pc (1 + μt²/2)

Sideband power Ps = Pc μt²/2 and efficiency η = μt² / (2 + μt²).

#### Bandwidth
BW = 2·f_max = 2·max(f₁, f₂).

#### Example
Ac = 10 V, μ₁ = 0.4, μ₂ = 0.3, fc = 100 kHz, f₁ = 1 kHz, f₂ = 2 kHz, R = 50 Ω.
- μt = √(0.16 + 0.09) = 0.5
- Pc = 100/(2×50) = 1 W
- Pt = 1 (1 + 0.25/2) = 1.125 W
- BW = 2 × 2 kHz = 4 kHz
- η = 0.25/2.25 = 11.1%

## Check yourself

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

1. Double-tone AM spectrum contains:
   A. Carrier and four side frequencies
   B. Carrier and two sidebands
   C. Only four side frequencies
   D. Two carriers
   Answer: A. Two per tone. (Key point: Spectrum)

2. Total modulation index:
   A. mu1 + mu2
   B. sqrt(mu1^2 + mu2^2)
   C. mu1 x mu2
   D. (mu1 + mu2)/2
   Answer: B. Root sum of squares. (Key point: Total index)

3. Bandwidth is:
   A. f1 + f2
   B. 2 x lowest frequency
   C. 2 x highest message frequency
   D. f2 - f1
   Answer: C. BW = 2fmax. (Key point: Bandwidth)

4. With mu1=0.4, mu2=0.3, Pc=1 W, Pt is:
   A. 1.25 W
   B. 1.5 W
   C. 1.0 W
   D. 1.125 W
   Answer: D. mu_t^2 = 0.25; 1(1+0.125). (Key point: Example)

## Related in Data Communication

- [AM numerical: 10 sin1000πt on 50 sin 2π×10⁵t](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-10sin1000pi) — Data Communication, chapter 5

## Related in other subjects

- [Double-tone AM expression, bandwidth and modulation index](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adouble-tone-am-bw-index) — Communication Systems, chapter 2
- [Double-tone AM; DSB vs SSB](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adouble-tone-am-dsb-vs-ssb) — Communication Systems, chapter 2
- [Need for modulation; single tone AM expressions](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amodulation-need-am-single-tone) — Communication Systems, chapter 2, asked 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