--- title: "Plotting line spectra of a CT signal" subject: "Data Communication" chapter: 3 key: "semester_5/data_communication:line-spectrum-plot" source: https://www.bctengineer.com/q/semester_5/data_communication/line-spectrum-plot site: BCT Engineering --- # Plotting line spectra of a CT signal Data Communication · Chapter 3: Signal Analysis - Asked in: 2078 Bhadra, 8 marks - Video: [Time & Frequency domain representation,Line Spectrum,amplitude & phase spectrum,Spectrum Definition](https://www.youtube.com/watch?v=dAYlm20XE0Y) — Competitive Shiksha - 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-plot - Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3 ## Question Explain how we plot line spectrums of a continuous time signal and illustrate an example. ## Short answer Convert the signal to a sum of positive cosines: x(t)=A₀+Σ Aₖcos(2πfₖt+φₖ). Plot amplitude Aₖ as vertical lines versus frequency fₖ for single-sided spectrum; phase φₖ similarly. Example: x(t)=4+6cos(20πt+30°)+2sin(60πt) gives DC 4, 10 Hz (6,30°), 30 Hz (2,-90°). ## Answer A line spectrum shows the frequency content of a **periodic** continuous-time signal as vertical lines at the discrete frequencies present (the harmonics). There are two plots: the **amplitude (magnitude) spectrum** and the **phase spectrum**. #### Procedure 1. Express the signal as a sum of cosines (or a Fourier series): x(t) = A₀ + Σ Aₖ cos(2πfₖt + φₖ). Convert sines using sin θ = cos(θ - 90°), and make all amplitudes positive by adjusting the phase by ±180°. 2. Identify each frequency fₖ, its amplitude Aₖ and its phase φₖ. 3. Plot amplitude Aₖ against frequency as a vertical line at each fₖ (single-sided), with DC at f = 0. 4. Plot phase φₖ against frequency in the same way. 5. For a two-sided spectrum, use the complex form: each cosine gives lines of height Aₖ/2 at +fₖ and -fₖ, with phases +φₖ and -φₖ; the DC value stays A₀. #### Example x(t) = 4 + 6 cos(2π·10t + 30°) + 2 sin(2π·30t) Rewrite 2 sin(2π·30t) = 2 cos(2π·30t - 90°). | f (Hz) | Amplitude | Phase | |---|---|---| | 0 | 4 | 0° | | 10 | 6 | 30° | | 30 | 2 | -90° | ```text Amplitude (single-sided) Phase 6 | | 30 | * 4 | | | 0 +-----+-----+--> f 2 | | | | 10 30 +-+---+------+--> f -90 | * 0 10 30 ``` The two-sided spectrum has lines of height 3 at ±10 Hz and 1 at ±30 Hz, with DC 4. The spacing of the lines is the fundamental frequency (here 10 Hz); a periodic signal always gives a discrete (line) spectrum. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Line spectra apply to A. Aperiodic signals B. Only noise C. Periodic signals D. Only impulses Answer: C. Discrete harmonics. (Key point: Applicability) 2. sin θ is rewritten as A. cos(θ + 90°) B. -cos θ C. cos θ D. cos(θ - 90°) Answer: D. Standard identity. (Key point: Sine conversion) 3. A negative amplitude is made positive by A. Adding or subtracting 180° to the phase B. Dropping the sign C. Changing frequency D. Doubling amplitude Answer: A. Keeps the same signal. (Key point: Negative amplitude) 4. In the two-sided spectrum a cosine of amplitude A gives A. Lines of A at ±f B. Lines of A/2 at ±f C. Line of 2A at f D. Line of A/2 at f only Answer: B. Complex form. (Key point: Two-sided) ## Related in Data Communication - [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 - [Properties of continuous-time Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-properties) — 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 - [Fourier transform of a DC signal](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aft-dc-signal) — Data Communication, chapter 3 ## Related in other subjects - [Fourier transform of a CT signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aft-unspecified-ct-signal) — Signals and Systems, chapter 3 - [Z-transform of ramp-and-hold continuous signal](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Awaveform-ramp-to-1-at-6) — Digital Control System, chapter 2 - [Line spectrum of a given modulated signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aline-spectrum-modulated-signal) — Communication Systems, chapter 2 - [Discrete vs continuous signal; convert (25.25)₁₀](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Adiscrete-continuous-convert-25-25) — Digital Logic, chapter 1