---
title: "Double-tone AM expression, bandwidth and modulation index"
subject: "Communication Systems"
chapter: 2
key: "semester_6/communication_systems:double-tone-am-bw-index"
source: https://www.bctengineer.com/q/semester_6/communication_systems/double-tone-am-bw-index
site: BCT Engineering
---

# Double-tone AM expression, bandwidth and modulation index

Communication Systems · Chapter 2: Amplitude Modulation

- Asked in: 2080 Chaitra, 5 marks
- Video: [Multiple Tone Amplitude Modulation (Definition, Basics, Derivation & Frequency Response) Explained](https://www.youtube.com/watch?v=JMq5T9fQqf0) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/double-tone-am-bw-index
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2

## Question

Derive the expression of double-tone AM, and define BW and modulation indices.

## Short answer

A double-tone AM signal is s(t) = A_c[1+μ₁cosω₁t+μ₂cosω₂t]cosω_c t. Its bandwidth is twice the highest modulating frequency, B = 2f_max. Modulation indices are μ₁=A_{m1}/A_c, μ₂=A_{m2}/A_c, and the net index is μ_t = √(μ₁²+μ₂²).

## Answer

**Double-tone AM** is amplitude modulation in which the message consists of two sinusoids:

$$
m(t)=A_{m1}\cos\omega_1t+A_{m2}\cos\omega_2t,\qquad f_1<f_2\ll f_c
$$

#### Expression
With carrier $A_c\cos\omega_ct$, the standard AM wave is

$$
\begin{aligned}
s(t)&=[A_c+A_{m1}\cos\omega_1t+A_{m2}\cos\omega_2t]\cos\omega_ct\\
&=A_c[1+\mu_1\cos\omega_1t+\mu_2\cos\omega_2t]\cos\omega_ct
\end{aligned}
$$

where $\mu_1=A_{m1}/A_c$ and $\mu_2=A_{m2}/A_c$ are the individual modulation indices. Expanding:

$$
\begin{aligned}
s(t)&=A_c\cos\omega_ct\\
&+\frac{\mu_1A_c}{2}\left[\cos(\omega_c+\omega_1)t+\cos(\omega_c-\omega_1)t\right]\\
&+\frac{\mu_2A_c}{2}\left[\cos(\omega_c+\omega_2)t+\cos(\omega_c-\omega_2)t\right]
\end{aligned}
$$

So the spectrum has five lines: the carrier at $f_c$, and side frequencies at $f_c\pm f_1$ and $f_c\pm f_2$.

```text
                         Ac
                         |
                         |
   mu2Ac/2   mu1Ac/2     |     mu1Ac/2   mu2Ac/2
      |         |        |        |         |
 -----+---------+--------+--------+---------+-----> f
    fc-f2     fc-f1      fc     fc+f1     fc+f2
```

#### Bandwidth
The bandwidth is set by the highest modulating frequency:

$$
B=(f_c+f_2)-(f_c-f_2)=2f_2=2f_{max}
$$

#### Modulation indices
- **Individual indices:** $\mu_1=\frac{A_{m1}}{A_c}$, $\mu_2=\frac{A_{m2}}{A_c}$.
- **Net (total, effective) index:** the single-tone index that gives the same total sideband power:

$$
P_t=P_c\left(1+\frac{\mu_1^2}{2}+\frac{\mu_2^2}{2}\right)=P_c\left(1+\frac{\mu_t^2}{2}\right)\ \Rightarrow\ \mu_t=\sqrt{\mu_1^2+\mu_2^2}
$$

For no over-modulation, $\mu_t\le1$ (strictly, $\mu_1+\mu_2\le1$ guarantees that the envelope never goes negative).

**Example:** $\mu_1=0.3$, $\mu_2=0.4$ gives $\mu_t=0.5$.

## Check yourself

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

1. A double-tone AM wave has how many spectral lines?
   A. Three
   B. Five: carrier and fc +/- f1, fc +/- f2
   C. Two
   D. Four without carrier
   Answer: B. Each tone adds an upper and lower side frequency to the carrier. (Key point: Spectrum lines)

2. The bandwidth of double-tone AM is:
   A. 2 f1
   B. 2 f2 (twice the highest modulating frequency)
   C. f1 + f2
   D. f2 - f1
   Answer: B. Highest side frequency fc+f2 and lowest fc-f2 give B = 2 f2. (Key point: Bandwidth)

3. If mu1 = 0.3 and mu2 = 0.4, the net modulation index is:
   A. 0.7
   B. 0.5
   C. 0.1
   D. 0.35
   Answer: B. mu_t = sqrt(mu1^2 + mu2^2) = sqrt(0.09+0.16) = 0.5. (Key point: Net index)

4. The individual modulation index mu1 is:
   A. Ac/Am1
   B. Am1/Ac
   C. Am1 Ac
   D. Am1 + Ac
   Answer: B. Each index is its tone amplitude divided by the carrier amplitude. (Key point: Individual index)

## 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
- [Two-tone AM: net modulation index, power, spectrum](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-50-20-12) — Communication Systems, chapter 2, asked 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
- [AM wave diagram, modulation index; why near 100%](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-diagram-index-100-percent) — 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×
- [Two-tone AM: equation, index, power, spectrum](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-10cos-15cos-20cos) — Communication Systems, chapter 2

## Related in other subjects

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- [Single-pulse PWM: modulation index and Fourier series](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Asingle-pulse-pwm-fourier) — Power Electronics, chapter 5
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- [Double ring infiltrometer, phi-index versus W-index](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Adouble-ring-phi-vs-w-index) — Engineering Hydrology, chapter 3