Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
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
Signals & Systems – Duality Property of Fourier Transform
Fourier Transform
For a continuous-time function x(t), the Fourier transform of x(t) can be defined as
$$\mathrm{X(\omega)= \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$
Duality Property of Continuous-Time Fourier Transform
Statement – If a function x(t) has a Fourier transform X(ω) and we form a new function in time domain with the functional form of the Fourier transform as X(t), then it will have a Fourier transform X(ω) with the functional form of the original time function, but it is a function of frequency.
Mathematically, the duality property of CTFT states that, if
$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$
Then, according to duality property,
$$\mathrm{X(t)\overset{FT}{\leftrightarrow}2\pi x(-\omega)}$$
Proof
From the definition of inverse Fourier transform, we have
$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega }$$
$$\mathrm{\Rightarrow 2\pi.x(t)=\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega}$$
By replacing 𝑡 = (−𝑡) in the above equation, we get,
$$\mathrm{\Rightarrow 2\pi.x(-t)=\int_{-\infty}^{\infty}X(\omega)e^{-j\omega t}d\omega}$$
Now, on interchanging t and ω, we get,
$$\mathrm{\Rightarrow 2\pi.x(-\omega)=\int_{-\infty}^{\infty}X(t)e^{-j\omega t}dt=F[X(t)]}$$
Therefore,
$$\mathrm{F[X(t)]=2\pi.x(-\omega)}$$
Or, it can also be represented as
$$\mathrm{X(t)\overset{FT}{\leftrightarrow}2\pi.x(-\omega)}$$
Also, for even functions,
$$\mathrm{x(-\omega)=x(\omega)}$$
Therefore, the duality property of Fourier transform for even functions states that
$$\mathrm{X(t)\overset{FT}{\leftrightarrow}2\pi x(\omega)}$$
Numerical Example
Using duality property of Fourier transform, find the Fourier transform of the following signal −
$$\mathrm{x(t)=\frac{1}{a^2+t^2}}$$
Solution
Given
$$\mathrm{x(t)=\frac{1}{a^2+t^2}}$$
The Fourier transform of a double-sided exponential function is defined as
$$\mathrm{F[e^{-a|t|}]=\frac{2a}{a^2+\omega^2}}$$
$$\mathrm{Now,\:by\: using\: duality\: property \:[i.e.,X(t)\overset{FT}{\leftrightarrow}2\pi.x(-\omega)].we\:have,}$$
$$\mathrm{F[\frac{2a}{a^2+t^2}]=2\pi e^{-a|-\omega|}}$$
$$\mathrm{\Rightarrow F[\frac{1}{a^2+t^2}]=\frac{1}{2a}.2\pi e^{-a|\omega|}}$$
Therefore, the Fourier transform of given signal is,
$$\mathrm{F[x(t)]=F[\frac{1}{a^2+t^2}]=\frac{\pi}{a}.e^{-a|\omega|}}$$
11K+ Views
