---
title: "Properties of continuous-time Fourier series"
subject: "Data Communication"
chapter: 3
key: "semester_5/data_communication:fourier-series-properties"
source: https://www.bctengineer.com/q/semester_5/data_communication/fourier-series-properties
site: BCT Engineering
---

# Properties of continuous-time Fourier series

Data Communication · Chapter 3: Signal Analysis

- Asked in: 2074 Chaitra, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/data_communication/fourier-series-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/data_communication/chapter-3

## Question

State the properties of continuous time Fourier series.

## Short answer

Linearity, time shift (c_k e^{-jkω₀t₀}), frequency shift (c_{k-M}), time reversal (c_{-k}), conjugation (c*_{-k}), multiplication ↔ discrete convolution, differentiation (jkω₀ c_k), integration (c_k/(jkω₀) if c₀=0, Parseval Σ|c_k|². Real signals have even magnitude and odd phase; real even signals have real even coefficients.

## Answer

Let x(t) be periodic with period T₀ and ω₀ = 2π/T₀, with exponential Fourier coefficients cₖ (x(t) ↔ cₖ). Then:

| Property | Signal | Coefficients |
|---|---|---|
| Linearity | a·x(t) + b·y(t) | a·cₖ + b·dₖ |
| Time shift | x(t - t₀) | cₖ e^(-jkω₀t₀) |
| Frequency shift | e^(jMω₀t) x(t) | c_(k-M) |
| Time reversal | x(-t) | c_(-k) |
| Time scaling | x(αt) | same cₖ, fundamental frequency αω₀ |
| Conjugation | x*(t) | c*_(-k) |
| Multiplication | x(t) y(t) | Σ cₘ d_(k-m) (discrete convolution) |
| Periodic convolution | ∫ x(τ) y(t - τ) dτ over T₀ | T₀ cₖ dₖ |
| Differentiation | dx/dt | jkω₀ cₖ |
| Integration (if c₀ = 0) | ∫ x dt | cₖ / (jkω₀) |
| Parseval's theorem | (1/T₀) ∫\|x\|² dt | Σ \|cₖ\|² |

Symmetry properties:
- Real x(t): c_(-k) = cₖ*, so |cₖ| is even and the phase is odd.
- Real and even x(t): cₖ is real and even (only cosine terms).
- Real and odd x(t): cₖ is purely imaginary and odd (only sine terms).
- Half-wave symmetric x(t): only odd harmonics exist.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. x(t-t0) gives coefficients
   A. ck e^(jkω0t0)
   B. ck/t0
   C. ck e^(-jkω0t0)
   D. c(k-t0)
   Answer: C. Phase factor. (Key point: Time shift)

2. dx/dt gives
   A. ck/(jkω0)
   B. k ck
   C. ck
   D. jkω0 ck
   Answer: D. Property. (Key point: Differentiation)

3. Parseval equates average power to
   A. Sum |ck|^2
   B. Sum ck
   C. Sum |ck|
   D. Sum ck^2
   Answer: A. Property. (Key point: Parseval)

4. Real even x(t) has coefficients
   A. Imaginary and odd
   B. Real and even
   C. Complex
   D. Zero
   Answer: B. Only cosines. (Key point: Real even)

## Related in Data Communication

- [Fourier series coefficients of a square wave](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-coefficients-square-wave) — Data Communication, chapter 3
- [Properties of the Fourier transform](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-transform-properties) — Data Communication, chapter 3, asked 2×
- [Applications of Fourier series](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afourier-series-applications) — Data Communication, chapter 3
- [Fourier series of half-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-half-wave-rectified) — Data Communication, chapter 3
- [Fourier series of full-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-full-wave-rectified) — Data Communication, chapter 3
- [Fourier series vs Fourier transform](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-vs-ft) — Data Communication, chapter 3

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