--- title: "Double-tone AM; DSB vs SSB" subject: "Communication Systems" chapter: 2 key: "semester_6/communication_systems:double-tone-am-dsb-vs-ssb" source: https://www.bctengineer.com/q/semester_6/communication_systems/double-tone-am-dsb-vs-ssb site: BCT Engineering --- # Double-tone AM; DSB vs SSB Communication Systems · Chapter 2: Amplitude Modulation - Asked in: 2079 Chaitra, 6+2 marks - Video: [DSBSC SSB and VSB Modulation](https://www.youtube.com/watch?v=h82Z-b2UooY) — CourseKart - 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-dsb-vs-ssb - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-2 ## Question Derive the expression for double tone AM. How DSB is different from SSB signal? ## Short answer Double-tone AM: s(t)=A_c[1+μ₁cosω_{m1}t+μ₂cosω_{m2}t]cosω_c t, producing carrier and four sidebands. DSB transmits both sidebands using bandwidth 2f_m; SSB transmits only one sideband, using half the bandwidth (f_m) and saving power by eliminating the redundant sideband and carrier. ## Answer #### Double tone AM In **double tone (two-tone) AM** the carrier is modulated by a message that contains two sinusoids of different frequencies. Let the message and carrier be: $$ m(t) = A_{m1}\cos\omega_{m1}t + A_{m2}\cos\omega_{m2}t, \qquad c(t) = A_c\cos\omega_c t $$ The conventional (DSB-FC) AM wave is $$ \begin{aligned} s(t) &= [A_c + m(t)]\cos\omega_c t \\ &= A_c\left[1 + \frac{A_{m1}}{A_c}\cos\omega_{m1}t + \frac{A_{m2}}{A_c}\cos\omega_{m2}t\right]\cos\omega_c t \\ &= A_c\left[1 + \mu_1\cos\omega_{m1}t + \mu_2\cos\omega_{m2}t\right]\cos\omega_c t \end{aligned} $$ where $\mu_1 = A_{m1}/A_c$ and $\mu_2 = A_{m2}/A_c$ are the individual modulation indices. Expanding with $\cos A\cos B = \tfrac12[\cos(A+B)+\cos(A-B)]$: $$ \begin{aligned} s(t) = {} & A_c\cos\omega_c t \\ & + \frac{\mu_1A_c}{2}\cos(\omega_c+\omega_{m1})t + \frac{\mu_1A_c}{2}\cos(\omega_c-\omega_{m1})t \\ & + \frac{\mu_2A_c}{2}\cos(\omega_c+\omega_{m2})t + \frac{\mu_2A_c}{2}\cos(\omega_c-\omega_{m2})t \end{aligned} $$ So the spectrum has the carrier plus **two upper and two lower side frequencies**. ```text amplitude | Ac | | | | | u2Ac/2 u1Ac/2 | u1Ac/2 u2Ac/2 | | | | | | ---+-----+---------+-------+--------+---------+----> f fc-fm2 fc-fm1 fc fc+fm1 fc+fm2 (fm1 < fm2) ``` **Power** (load $R$): $$ P_t = \frac{A_c^2}{2R}\left[1 + \frac{\mu_1^2}{2} + \frac{\mu_2^2}{2}\right] = P_c\left(1 + \frac{\mu_t^2}{2}\right) $$ with the **net (total) modulation index** $$ \mu_t = \sqrt{\mu_1^2 + \mu_2^2} $$ For no over-modulation, $\mu_t \le 1$. Bandwidth $= 2f_{m,\max}$ (twice the higher message frequency). #### DSB vs SSB | Point | DSB (DSB-SC / DSB-FC) | SSB | |---|---|---| | Sidebands sent | Both USB and LSB | Only one (USB or LSB) | | Bandwidth | $2f_m$ | $f_m$ | | Power | More (two sidebands, plus the carrier in DSB-FC) | Less; all power in one sideband | | Generation | Simple (product/balanced modulator) | Complex (sharp filter or phase-shift method) | | Detection | Envelope (DSB-FC) or coherent | Coherent only | | Use | AM broadcast; DSB-SC in stereo FM | Point-to-point HF radio, telephony | ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For two-tone AM with indices mu1 and mu2, what is the net modulation index? A. mu1 + mu2 B. sqrt(mu1^2 + mu2^2) C. mu1 x mu2 D. (mu1 + mu2)/2 Answer: B. Power adds from each tone, so mu_t^2 = mu1^2 + mu2^2 and mu_t = sqrt(mu1^2 + mu2^2). (Key point: Net modulation index) 2. How many spectral components does a two-tone DSB-FC AM wave have? A. Three: a carrier and one pair of side frequencies B. Four: only the side frequencies C. Two: one upper and one lower sideband only D. Five: a carrier, two upper and two lower side frequencies Answer: D. Each tone produces an upper and a lower side frequency, plus the single carrier, giving 5 lines. (Key point: Two-tone spectrum) 3. What is the total power of a two-tone AM wave? A. Pc(1 + mu_t^2/2) B. Pc(1 + mu_t) C. Pc(1 + mu_t^2) D. Pc mu_t^2/2 Answer: A. Total power is carrier power plus sideband power, Pc(1 + mu1^2/2 + mu2^2/2) = Pc(1 + mu_t^2/2). (Key point: Total power) 4. How do DSB and SSB differ in bandwidth and detection? A. DSB needs fm and SSB needs 2fm; both use envelope detection B. DSB needs 2fm and SSB needs 2fm, but SSB is cheaper C. DSB needs 2fm and can use envelope detection (with carrier); SSB needs fm and needs coherent detection D. DSB needs fm and coherent detection; SSB needs 2fm and envelope detection Answer: C. SSB sends one sideband so it halves the bandwidth, but with no carrier it must be detected coherently. (Key point: DSB vs SSB bandwidth) ## Related in Communication Systems - [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 - [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 - [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 - [Advantages of SSB over DSB-FC; SSB expression](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-advantages-derive) — Communication Systems, chapter 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 - [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 ## Related in other subjects - [Double-tone AM](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Adouble-tone-am) — Data Communication, chapter 5, asked 2×