--- title: "Two-tone AM: spectrum, net index, power, efficiency" subject: "Communication Systems" chapter: 2 key: "semester_6/communication_systems:am-two-tone-100-30-40" source: https://www.bctengineer.com/q/semester_6/communication_systems/am-two-tone-100-30-40 site: BCT Engineering --- # Two-tone AM: spectrum, net index, power, efficiency Communication Systems · Chapter 2: Amplitude Modulation - Asked in: 2075 Bhadra (CS I), 3+2+2+1 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/am-two-tone-100-30-40 - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2 ## Question An Amplitude modulated wave is given by s(t) = 100cos(2π×10⁶t) + 30cos(2π×10⁶t)cos(2π×10³t) + 40cos(2π×10⁶t)cos(4π×10²t) Volt a) Draw the frequency spectrum of modulated wave b) Net modulation index c) Total modulated power d) Efficiency ## Short answer DSB-FC with two tones: μ₁=0.3 at 1 kHz, μ₂=0.4 at 200 Hz. Net modulation index μ = √(0.3²+0.4²) = 0.5. Carrier power Pc = 5000 W, total power Pt = 5625 W, efficiency η = 11.11%. Spectrum: carrier 100 V at 1 MHz; sidebands at ±1 kHz (15 V) and ±200 Hz (20 V). ## Answer Write in standard form: $$ s(t) = 100\left[1 + 0.3\cos(2\pi10^3t) + 0.4\cos(2\pi\,200\,t)\right]\cos(2\pi10^6t)\ \text{V} $$ since $30/100 = 0.3$, $40/100 = 0.4$ and $4\pi10^2t = 2\pi(200)t$. $A_c = 100$ V, $f_c = 1$ MHz; $\mu_1 = 0.3$ at 1 kHz; $\mu_2 = 0.4$ at 200 Hz. Assume $R = 1\ \Omega$. #### a) Frequency spectrum Side-frequency amplitudes $= \mu A_c/2$: $30/2 = 15$ V (1 kHz tone), $40/2 = 20$ V (200 Hz tone). | Frequency | Amplitude | |---|---| | 999 kHz | 15 V | | 999.8 kHz | 20 V | | 1000 kHz | 100 V | | 1000.2 kHz | 20 V | | 1001 kHz | 15 V | ```text A(V) 100 | | | | 20 | | | | 15 | | | | | | +---+----+-------+-------+----+---> f (kHz) 999 999.8 1000 1000.2 1001 ``` #### b) Net modulation index $$ \mu_t = \sqrt{\mu_1^2 + \mu_2^2} = \sqrt{0.09 + 0.16} = \sqrt{0.25} = 0.5 $$ #### c) Total power $$ \begin{aligned} P_c &= \frac{A_c^2}{2R} = \frac{100^2}{2} = 5000\ \text{W} \\ P_t &= P_c\left(1 + \frac{\mu_t^2}{2}\right) = 5000(1 + 0.125) = 5625\ \text{W} \end{aligned} $$ #### d) Efficiency $$ \eta = \frac{\mu_t^2}{2 + \mu_t^2} = \frac{0.25}{2.25} = 0.1111 = 11.11\% $$ **Answer:** $\mu_t = 0.5$; $P_t = 5625$ W; $\eta = 11.11\%$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are mu1 and mu2? A. 30 and 40 B. 0.3 and 0.4 C. 3.33 and 2.5 D. 0.15 and 0.2 Answer: B. 30/100 and 40/100. (Key point: Indices) 2. What is the net modulation index? A. 0.7 B. 0.25 C. 0.35 D. 0.5 Answer: D. sqrt(0.09 + 0.16) = 0.5. (Key point: Net index) 3. What is the total power? A. 5625 W B. 5000 W C. 10000 W D. 5125 W Answer: A. Pc = 100^2/2 = 5000 W and Pt = 5000(1 + 0.125). (Key point: Total power) 4. What is the efficiency? A. 25% B. 12.5% C. 11.11% D. 33.3% Answer: C. eta = mu_t^2/(2 + mu_t^2) = 0.25/2.25. (Key point: Efficiency) ## Related in Communication Systems - [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× - [Two-tone AM: spectrum, power, net index (50, 15, 20)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-50-15-20) — 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 - [Two-tone AM: index, power, spectrum (5cos, 10cos, 15cos)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-two-tone-5cos-10cos-15cos) — Communication Systems, chapter 2 - [AM transmitter 10 kW/11.8 kW: modulation index and power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-10kw-11-8kw) — Communication Systems, chapter 2 - [Multi-tone AM: spectrum, bandwidth, power efficiency](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aam-50-0-3-0-2-spectrum) — Communication Systems, chapter 2 ## Related in other subjects - [Modulation; AM and FM modulated waves](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Amodulation-am-fm-wave) — Basic Electronics Engineering, chapter 5 - [AM wave power and bandwidth numerical](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-40-carrier-power) — Data Communication, chapter 5