# 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,

$$\mathrm{\frac{1}{T}\int_{t_{0}}^{t_{0}+T}|x(t)|^2dt=\sum_{n=-\infty}^{\infty}|C_{n}|^2}$$