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 𝐶_𝑛 and 𝐷_𝑛 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,

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