

- Trending Categories
Data Structure
Networking
RDBMS
Operating System
Java
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Signals & Systems – Properties of Continuous Time Fourier Series
The Fourier series representation of a periodic signal has various important properties which are useful for various purposes during the transformation of signals from one form to another.
Consider two periodic signals 𝑥1(𝑡) and 𝑥2(𝑡) which are periodic with time period T and with Fourier series coefficients 𝐶𝑛 and 𝐷𝑛 respectively. With this assumption, let us proceed and check the various properties of a continuoustime Fourier series.
Linearity Property
The linearity property of continuous-time Fourier series states that, if
$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}\: and\:x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$
Then
$$\mathrm{Ax_{1}(t)+Bx_{2}(t)\overset{FS}{\leftrightarrow}AC_{n}+BD_{n}}$$
Time Shifting Property
The time scaling property of Fourier series states that, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{x(t-t_{0})\overset{FS}{\leftrightarrow}e^{-jn\omega_{0}t_{0}}C_{n}}$$
Time Scaling Property
The time scaling property of Fourier series states that, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{x(at)\overset{FS}{\leftrightarrow}C_{n}\:with\:\omega_{0}\rightarrow a\omega_{0}}$$
Time Reversal Property
The time reversal property of the Fourier series states that, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{x(-t)\overset{FS}{\leftrightarrow}C_{-n}}$$
Time Differentiation Property
The time differentiation property of continuous-time Fourier series states that, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{\frac{dx(t)}{dt}\overset{FS}{\leftrightarrow}jn\omega_{0}t_{0}C_{n}}$$
Time Integration Property
The time integration property of continuous-time Fourier series states that, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{\int_{-\infty}^{t}x(\tau)d\tau\overset{FS}{\leftrightarrow}\frac{C_{n}}{jn\omega_{0}};\:C_{0}=0}$$
Convolution Property
The convolution theorem or convolution property of a continuous-time Fourier series states that “the convolution of two functions in time domain is equivalent to the multiplication of their Fourier coefficients in frequency domain.” Thus, if,
$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}\:\:and\:\:x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$
Then
$$\mathrm{x_{1}(t)*x_{2}(t)\overset{FS}{\leftrightarrow}TC_{n}D_{n}}$$
Multiplication or Modulation Property
The multiplication or modulation property of continuous-time Fourier series states that, if
$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}\:\:and\:\:x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$
Then
$$\mathrm{x_{1}(t).x_{2}(t)\overset{FS}{\leftrightarrow}\sum_{k=-\infty}^{\infty}C_{k}D_{n-k}}$$
Conjugation Property
The conjugation property of continuous-time Fourier series states that, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{x^*(t)\overset{FS}{\leftrightarrow}C_{-n}^{*}\:(for\:example\:x(t))}$$
Conjugate Symmetry Property
According to the conjugate symmetry property, if
$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$
Then
$$\mathrm{C_{-n}=C_{n}^{*}\:(for\:real\:x(t))}$$
Parseval’s Theorem
The Parseval’s theorem for Fourier series states that, if
$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}\:\:and\:\:x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}\:\:\:\:[for\:complex\:x_{1}(t)\& \: x_{2}(t)]}$$
Then
$$\mathrm{\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x_{1}(t)x_{2}^{*}(t)(dt)=\sum_{n=-\infty}^{\infty}C_nD_{n}^{*}\:\:\:\:[for\:complex\:x_{1}(t)\& \: x_{2}(t)]}$$
And, if
$$\mathrm{x_{1}(t)=x_{2}(t)=x(t)}$$
Then, the Parseval’s identity states that,
$$\mathrm{\frac{1}{T}\int_{t_{0}}^{t_{0}+T}|x(t)|^2dt=\sum_{n=-\infty}^{\infty}|C_{n}|^2}$$
- Related Questions & Answers
- Time Differentiation and Integration Properties of Continuous-Time Fourier Series
- Signals and Systems – Properties of Discrete-Time Fourier Transform
- Time Shifting, Time Reversal, and Time Scaling Properties of Continuous-Time Fourier Series
- Properties of Continuous-Time Fourier Transform (CTFT)
- Convolution Property of Continuous-Time Fourier Series
- Linearity and Conjugation Property of Continuous-Time Fourier Series
- Signals and Systems – Properties of Linear Time-Invariant (LTI) Systems
- Signals and Systems – Time-Reversal Property of Fourier Transform
- Signals and Systems – Time-Shifting Property of Fourier Transform
- Signals and Systems – Time Integration Property of Fourier Transform
- Signals and Systems – Fourier Transform of Periodic Signals
- Signals & Systems – Complex Exponential Fourier Series
- Multiplication or Modulation Property of Continuous-Time Fourier Series
- Signals and Systems – Properties of Even and Odd Signals
- Properties of Convolution in Signals and Systems