--- title: "Need for modulation; single tone AM expressions" subject: "Communication Systems" chapter: 2 key: "semester_6/communication_systems:modulation-need-am-single-tone" source: https://www.bctengineer.com/q/semester_6/communication_systems/modulation-need-am-single-tone site: BCT Engineering --- # Need for modulation; single tone AM expressions Communication Systems · Chapter 2: Amplitude Modulation - Asked in: 2071 Bhadra (CS I), 2+2+3 marks; 2069 Bhadra (CS I), 2+2+6 marks (asked 2 times) - Video: [Mathematical Expression of Single Tone Amplitude Modulation | Analog Com | R K Classes | Lec-9 |](https://www.youtube.com/watch?v=AwKeCO98Oi0) — R K Classes - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/modulation-need-am-single-tone - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2 ## Question What do you understand by modulation? Why modulation is needed? Find the time and frequency domain expressions for standard AM wave for single tone message signal. ## Short answer Modulation varies carrier amplitude, frequency, or phase. It is needed for practical antenna size, multiplexing, efficient radiation. Single-tone AM: s(t) = A_c[1+μ cos ω_m t] cos ω_c t; spectrum has carrier at f_c and sidebands at f_c±f_m with amplitude μA_c/2. ## Answer #### Modulation **Modulation** is the process in which some characteristic (amplitude, frequency or phase) of a high-frequency **carrier** wave is varied in accordance with the instantaneous value of the message (modulating) signal. In **amplitude modulation (AM)** the carrier amplitude is varied. #### Need for modulation 1. **Practical antenna height:** antenna length $\approx\lambda/4$; for 10 kHz audio it would be 7.5 km, but for a 1 MHz carrier only 75 m. 2. **Multiplexing:** different stations use different carriers, so many signals share the same channel without interference. 3. **Efficient radiation and long range:** high-frequency signals radiate and propagate much better. 4. **Narrowbanding:** reduces the ratio of highest to lowest frequency, so one antenna and amplifier design works. 5. **Noise reduction:** wideband schemes (FM) improve SNR. 6. **Channel matching:** shifts the signal into the frequency band suited to the medium. #### Time-domain expression of single-tone AM Let the message be $m(t)=A_m\cos\omega_mt$ and carrier $c(t)=A_c\cos\omega_ct$, with $f_c\gg f_m$. In standard AM the envelope is $A_c+m(t)$: $$ \begin{aligned} s(t)&=[A_c+A_m\cos\omega_mt]\cos\omega_ct\\ &=A_c[1+\mu\cos\omega_mt]\cos\omega_ct,\qquad \mu=\frac{A_m}{A_c} \end{aligned} $$ where $\mu$ is the **modulation index** ($0\le\mu\le1$). Using $\cos A\cos B=\frac{1}{2}[\cos(A+B)+\cos(A-B)]$: $$ s(t)=\underbrace{A_c\cos\omega_ct}_{\text{carrier}}+\underbrace{\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t}_{\text{USB}}+\underbrace{\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t}_{\text{LSB}} $$ ```text Ac(1+mu) .-. envelope .-. / \ / \ / \ .-. / \ <- carrier inside envelope \ / \ / '- '- Ac(1-mu) ``` #### Frequency-domain expression Using $\cos2\pi f_0t\leftrightarrow\frac{1}{2}[\delta(f-f_0)+\delta(f+f_0)]$: $$ \begin{aligned} S(f)&=\frac{A_c}{2}\left[\delta(f-f_c)+\delta(f+f_c)\right]\\ &+\frac{\mu A_c}{4}\left[\delta(f-f_c-f_m)+\delta(f+f_c+f_m)\right]\\ &+\frac{\mu A_c}{4}\left[\delta(f-f_c+f_m)+\delta(f+f_c-f_m)\right] \end{aligned} $$ ```text S(f) Ac/2 Ac/2 muAc/4 | muAc/4 muAc/4 | muAc/4 | | | | | | --+----+----+---+---+----+----+--> f -fc-fm -fc -fc+fm 0 fc-fm fc fc+fm ``` #### Key results - **Bandwidth:** $B=(f_c+f_m)-(f_c-f_m)=2f_m$. - **Power:** $P_c=\frac{A_c^2}{2}$, each sideband $\frac{\mu^2A_c^2}{8}$, so total $P_t=P_c\left(1+\frac{\mu^2}{2}\right)$. - **Efficiency:** $\eta=\frac{\mu^2}{2+\mu^2}$; maximum 33.3% at $\mu=1$. **Example:** $A_c=10$ V, $\mu=0.5$, $f_c=1$ MHz, $f_m=1$ kHz gives lines of 10 V at 1 MHz and 2.5 V at 0.999 MHz and 1.001 MHz; bandwidth 2 kHz. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is modulation needed for radiation? A. Practical antenna length is about lambda/4, impractical for low audio frequencies B. It makes audio louder C. It removes noise completely D. It reduces bandwidth to zero Answer: A. A 10 kHz signal needs a 7.5 km antenna, but a 1 MHz carrier needs only 75 m. (Key point: Antenna length) 2. The single-tone AM wave is: A. Ac cos wct + Am cos wm t B. Ac[1 + mu cos wm t] cos wct C. Ac cos(wct + mu cos wm t) D. Am cos wm t cos wct Answer: B. Envelope is Ac + Am cos wm t = Ac(1 + mu cos wm t), multiplied by the carrier. (Key point: Single-tone AM) 3. The AM spectrum of a single tone has: A. Only the carrier B. Carrier plus USB and LSB, each of amplitude mu Ac/2 C. Two sidebands, no carrier D. A carrier at fm Answer: B. Expansion gives lines at fc and fc +/- fm with sideband amplitude mu Ac/2; bandwidth is 2fm. (Key point: Spectrum) 4. Total AM power is: A. Pc(1 + mu^2) B. Pc(1 + mu^2/2) C. Pc mu^2/2 D. 2Pc Answer: B. Each sideband carries mu^2 Pc/4, so Pt = Pc(1 + mu^2/2). (Key point: Power) ## Related in Communication Systems - [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 - [Angle modulation; single tone FM time and frequency domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aangle-mod-single-tone-fm-time-freq) — Communication Systems, chapter 3 - [SQNR of delta modulation for a single-tone signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Adm-sqnr-single-tone) — Communication Systems, chapter 4 - [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 - [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 - [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