--- title: "Line spectra of 12 + 6sin(140πt+30°) - 9cos(80πt-70°)" subject: "Data Communication" chapter: 3 key: "semester_5/data_communication:line-spectrum-12-6sin" source: https://www.bctengineer.com/q/semester_5/data_communication/line-spectrum-12-6sin site: BCT Engineering --- # Line spectra of 12 + 6sin(140πt+30°) - 9cos(80πt-70°) Data Communication · Chapter 3: Signal Analysis - Asked in: 2076 Chaitra, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/data_communication/line-spectrum-12-6sin - Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3 ## Question Plot the line spectrums of X(t) = 12 + 6sin(140πt + 30°) - 9cos(80πt - 70°). ## Short answer Rewrite: 6sin(140πt+30°)=6cos(140πt-60°), -9cos(80πt-70°)=9cos(80πt+110°). Single-sided: DC 12, 40 Hz amplitude 9 phase 110°, 70 Hz amplitude 6 phase -60°. Two-sided: at ±40 Hz, amplitude 4.5 phases ±110°; at ±70 Hz, amplitude 3 phases ∓60°. ## Answer X(t) = 12 + 6 sin(140πt + 30°) - 9 cos(80πt - 70°) #### Step 1: write every term as a positive cosine - 6 sin(140πt + 30°) = 6 cos(140πt + 30° - 90°) = 6 cos(140πt - 60°) - -9 cos(80πt - 70°) = 9 cos(80πt - 70° + 180°) = 9 cos(80πt + 110°) #### Step 2: frequencies - 140π = 2π·70, so f = 70 Hz - 80π = 2π·40, so f = 40 Hz #### Step 3: table (single-sided) | f (Hz) | Amplitude | Phase | |---|---|---| | 0 (DC) | 12 | 0° | | 40 | 9 | +110° | | 70 | 6 | -60° | ```text Amplitude spectrum Phase spectrum 12 |* 110 | * 9 | | 0 +--+-----+------+--> f 6 | | | 0 40 70 | | | -60| * +--+-----+---> f 0 40 70 ``` #### Two-sided spectrum Each cosine of amplitude A gives lines of A/2 at ±f with phases ±φ. | f (Hz) | -70 | -40 | 0 | 40 | 70 | |---|---|---|---|---|---| | Magnitude | 3 | 4.5 | 12 | 4.5 | 3 | | Phase | +60° | -110° | 0 | +110° | -60° | The fundamental frequency is the gcd of 40 and 70, which is 10 Hz, so the period is 0.1 s. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. 6 sin(140πt+30°) becomes A. 6 cos(140πt + 120°) B. 6 cos(140πt + 30°) C. 6 cos(140πt - 60°) D. -6 cos(140πt+30°) Answer: C. Subtract 90°. (Key point: Conversion) 2. 140π corresponds to A. 140 Hz B. 35 Hz C. 140π Hz D. 70 Hz Answer: D. ω=2πf. (Key point: Frequency) 3. -9 cos(80πt-70°) becomes A. 9 cos(80πt + 110°) B. 9 cos(80πt - 110°) C. 9 cos(80πt - 70°) D. 9 cos(80πt + 70°) Answer: A. Add 180°. (Key point: Negative cosine) 4. Two-sided line at ±40 Hz has magnitude A. 9 B. 4.5 C. 3 D. 12 Answer: B. A/2. (Key point: Two-sided) ## Related in Data Communication - [Plotting line spectra of a CT signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aline-spectrum-plot) — Data Communication, chapter 3 - [Fourier transform of e^(-at)u(t) and spectra](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-exp-causal-spectra) — Data Communication, chapter 3, asked 2× - [Magnitude and phase spectra of a sum of sinusoids](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Amag-phase-5-sin12t) — Data Communication, chapter 3 ## Related in other subjects - [Three reservoirs: levels 100, 70 and 30 m](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Athree-reservoirs-fixed-levels-100-70-30) — Hydraulics, chapter 3, asked 2× - [Classify soil, No. 4 70%, No. 200 30%](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aclassify-soil-no4-no200-ll33) — Soil Mechanics, chapter 3 - [70 km, 40 MW line: regulation and zero-regulation pf](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3A70km-40mw-regulation) — Power System Analysis I, chapter 6 - [Capacitor for zero regulation, 70 km 220 kV line](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3A70km-220kv-zero-regulation-capacitor) — Power System Analysis I, chapter 6