- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# What is Fourier Spectrum? – Theory and Example

The graph plotted between the Fourier coefficients of a periodic function $x(t)$ and the frequency (ω) is known as the ** Fourier spectrum** of a periodic signal.

The Fourier spectrum of a periodic function has two parts −

**Amplitude Spectrum**− The amplitude spectrum of the periodic signal is defined as the plot of amplitude of Fourier coefficients versus frequency.**Phase Spectrum**− – The plot of the phase of Fourier coefficients versus frequency is called the*phase spectrum*of the signal.

The amplitude spectrum and phase spectrum together are known as * Fourier frequency spectra* of the periodic signal $x(t)$. This type of representation of a periodic signal is known as

*frequency domain representation*.

The Fourier frequency spectra exists only at discrete frequencies, i.e., at , where, n = 0, 1, 2, 3,… . Therefore, the Fourier frequency spectra is also known as *discrete spectra or line spectra.* The envelope of the Fourier frequency spectra depends only upon shape of the pulse, but not upon the period of repetition.

The trigonometric Fourier series representation of a periodic function $x(t)$ contains both sine and cosine terms with positive and negative amplitude coefficients $a_{n}$ and $b_{n}$but does not have phase angles.

## Single-Sided Spectra

The cosine Fourier series representation of a periodic signal x(t) has only positive amplitude coefficients $A_{n}$ with phase angle $\theta_{n}$. Hence, we can plot amplitude spectrum ($A_{n}$ versus $\omega$) and the phase spectrum ($\theta_{n}$ versus $\omega$ ).

In the cosine representation, the Fourier coefficients exist only for positive frequencies. Therefore, this spectra is called the * single-sided spectra*.

Figure-1 represents the spectrum of a trigonometric (cosine) Fourier series extending from 0 to ∞, producing a one sided spectrum because no negative frequencies exist in this representation.

## Two-Sided Spectra

The exponential Fourier series representation of a periodic function $x(t)$ has amplitude coefficients $C_{n}$ which are complex and can be represented by magnitude and phase. Hence, we can plot the amplitude spectrum ($|C_{n}|$ versus $\omega$) and the phase spectrum ($\angle C_{n}\:versus\:\omega$).

As in the exponential representation, the spectra can be plotted for both positive and negative frequencies. Therefore, this spectra is known as * two-sided spectra*.

Figure-2 represents the spectrum of a complex exponential Fourier series extending from (−∞ to ∞),producing a * two-sided spectrum*.

Also, if $C_{n}$ is complex number, then

$$\mathrm{C_{n}=|C_{n}|e^{j\theta_{n}}}$$

$$\mathrm{C_{-n}=|C_{n}|e^{-j\theta_{n}}}$$

And

$$\mathrm{C_{n}=|C_{-n}|}$$

Therefore, the amplitude spectrum of the exponential Fourier series is symmetrical about the vertical axis passing through the origin, i.e., the magnitude spectrum exhibits even symmetry and the phase spectrum is antisymmetrical about the vertical axis passing through the origin, i.e., the phase spectrum exhibits odd symmetry.

Also, when the periodic signal $x(t)$ is real, then

$$\mathrm{C_{-n}=C_{n}^{*}}$$

i.e.$C_{-n}$ is the complex conjugate of the exponential coefficient $C_{n}$.

## Numerical Example

The exponential Fourier series of a periodic function is given by,

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}\frac{2A}{\pi(1-4n^{2})}e^{j2nt}=\frac{2A}{\pi}+\frac{2A}{\pi}\sum_{\substack{n=-\infty\ n

eq 0}}^{\infty}\left ( \frac{e^{j2nt}}{1-4n^{2}} \right ) }$$

Plot the frequency spectrum for the given function.

**Solution **

The given exponential Fourier series is,

$$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}\frac{2A}{\pi(1-4n^{2})}e^{j2nt}=\frac{2A}{\pi}+\frac{2A}{\pi}\sum_{\substack{n=-\infty\ n

eq 0}}^{\infty}\left ( \frac{e^{j2nt}}{1-4n^{2}} \right )}$$

Here, the exponential Fourier coefficients are −

$$\mathrm{C_{0}=\frac{2A}{\pi}\:\:and\:\:C_{n}=\frac{2A}{\pi(1-4n^{2})}}$$

Therefore,

$$\mathrm{C_{1}=C_{-1}=\frac{2A}{\pi[1-4(1)^{2}]}=-\frac{2A}{3\pi}}$$

$$\mathrm{C_{2}=C_{-2}=\frac{2A}{\pi[1-4(2)^{2}]}=-\frac{2A}{15\pi}}$$

$$\mathrm{C_{3}=C_{-3}=\frac{2A}{\pi[1-4(3)^{2}]}=-\frac{2A}{35\pi}}$$

$$\mathrm{C_{4}=C_{-4}=\frac{2A}{\pi[1-4(4)^{2}]}=-\frac{2A}{63\pi}}$$

And so on…

Hence, the frequency spectrum of the given function can be plotted as shown in Figure-3.

- Related Articles
- What is electromagnetic spectrum?
- What is spectrum? Draw a labelled diagram to show the formation of spectrum. Name the various colours of spectrum.
- What is a Fourier Analysis?
- What is quantum theory ?
- What is Cell Theory?
- What is agency theory?
- Difference between Fourier Series and Fourier Transform
- What is Dalton's Atomic Theory?
- What is Arbitrage Pricing Theory?
- What is the theory of parallel universe and wormhole?
- (a) What is spectrum? What is the name of glass shape used to produce a spectrum?(b) How many colours are there in a full spectrum of white light? Write the various colours of spectrum in the order, starting with red.
- What is the theory of computation?
- Expectation Theory, Liquidity Premium Theory, and Segmented Market Theory
- What is Kiting, Definition and Example?
- What is the Scientific Theory of Management?