--- title: "Magnitude and phase spectra of a sum of sinusoids" subject: "Data Communication" chapter: 3 key: "semester_5/data_communication:mag-phase-5-sin12t" source: https://www.bctengineer.com/q/semester_5/data_communication/mag-phase-5-sin12t site: BCT Engineering --- # Magnitude and phase spectra of a sum of sinusoids Data Communication · Chapter 3: Signal Analysis - Asked in: 2076 Asoj, 4 marks - Video: [The Spectrum: Representing Signals as a Function of Frequency](https://www.youtube.com/watch?v=xh5pLY3f6H8) — Barry Van Veen - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/data_communication/mag-phase-5-sin12t - Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3 ## Question Plot the magnitude and phase spectra of X(t) = 5 + sin(12t + 20°) - cos(16t - 60°) + cos(20t + 40°). ## Short answer Convert: sin(12t+20°)=cos(12t-70°), -cos(16t-60°)=cos(16t+120°). Signal =5+cos(12t-70°)+cos(16t+120°)+cos(20t+40°). Magnitudes: DC=5, amplitude 1 at ω=12,16,20 rad/s; phases -70°, 120°, 40°. Plot vertical lines at those ω. Two-sided gives lines of 0.5 at ±12, ±16, ±20 with phases odd-symmetric. ## Answer X(t) = 5 + sin(12t + 20°) - cos(16t - 60°) + cos(20t + 40°) #### Step 1: convert to positive cosines - sin(12t + 20°) = cos(12t + 20° - 90°) = cos(12t - 70°) - -cos(16t - 60°) = cos(16t - 60° + 180°) = cos(16t + 120°) - cos(20t + 40°) is already in the right form. #### Step 2: table (ω in rad/s; f = ω/2π) | ω (rad/s) | f (Hz) | Amplitude | Phase | |---|---|---|---| | 0 (DC) | 0 | 5 | 0° | | 12 | 1.91 | 1 | -70° | | 16 | 2.55 | 1 | +120° | | 20 | 3.18 | 1 | +40° | #### Single-sided plots ```text Magnitude Phase (deg) 5 |* 120 | * | 40 | * 1 | | | | 0 +--+---+----+----+--> w +---+----+----+--> w 0 12 16 20 0 12 16 20 -70| * ``` #### Two-sided plots Lines of height 0.5 at ω = ±12, ±16, ±20 and 5 at ω = 0, with phases odd-symmetric: -70°, +120°, +40° at positive ω and +70°, -120°, -40° at negative ω. The magnitude spectrum is even and the phase spectrum is odd for a real signal. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. sin(12t+20°) becomes A. cos(12t + 110°) B. cos(12t + 20°) C. cos(12t - 70°) D. -cos(12t - 70°) Answer: C. Subtract 90°. (Key point: sin conversion) 2. -cos(16t - 60°) becomes A. cos(16t - 120°) B. cos(16t - 60°) C. cos(16t + 60°) D. cos(16t + 120°) Answer: D. Add 180°. (Key point: Negative cosine) 3. 12 rad/s corresponds to A. 1.91 Hz B. 12 Hz C. 75.4 Hz D. 0.5 Hz Answer: A. 12/2π. (Key point: Frequency) 4. Two-sided magnitude of each unit cosine is A. 1 B. 0.5 C. 2 D. 5 Answer: B. A/2. (Key point: Two-sided) ## Related in Data Communication - [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× - [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 - [Line spectra of 12 + 6sin(140πt+30°) - 9cos(80πt-70°)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aline-spectrum-12-6sin) — Data Communication, chapter 3 - [AM numerical: 20 sin2π500t on 60 sin2π10⁶t](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-20sin500) — Data Communication, chapter 5 - [AM numerical: 10 sin1000πt on 50 sin 2π×10⁵t](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aam-numeric-10sin1000pi) — Data Communication, chapter 5 ## Related in other subjects - [Periodicity of sin 6πt, cos 8t and their sum](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aperiodicity-sin6pit-cos8t) — Signals and Systems, chapter 1 - [Periodicity of sin 5n and cos sum](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aperiodicity-sin5n-cos-sum) — Signals and Systems, chapter 1 - [DTFS of 1 + sin + 3cos + cos(4πn/N + π/2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-coefficients-with-phase-term) — Signals and Systems, chapter 2, asked 3× - [H(e^(jω)), magnitude plot and h[n] for y[n] = 0.5y[n−1] + x[n]](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Afirst-order-difference-plot) — Signals and Systems, chapter 6, asked 9×