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
Parseval’s Theorem & Parseval’s Identity of Fourier Transform
Fourier Transform
For a continuous-time function $\mathrm{\mathit{x\left ( t \right )}}$ , the Fourier transform of $\mathrm{\mathit{x\left ( t \right )}}$ can be defined as,
$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt }}$$
And the inverse Fourier transform is defined as,
$$\mathrm{\mathit{x\left ( t \right )\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )e^{j\omega t}d\omega }} $$
Parseval’s Theorem of Fourier Transform
Statement – Parseval’s theorem states that the energy of signal $\mathrm{\mathit{x\left ( t \right )}}$ [if $\mathrm{\mathit{x\left ( t \right )}}$ is aperiodic] or power of signal $\mathrm{\mathit{x\left ( t \right )}}$ [if $\mathrm{\mathit{x\left ( t \right )}}$ is periodic] in the time domain is equal to the energy or power in the frequency domain.
Therefore, if,
$$\mathrm{\mathit{x_{\mathrm{1}}\left ( t \right )\overset{FT}{\leftrightarrow} X_{\mathrm{1}}\left ( \omega \right )\;and \;x_{\mathrm{2}} \left ( t \right )\overset{FT}{\leftrightarrow} X_{\mathrm{2}}\left ( \omega \right )}}$$
Then, Parseval’s theorem of Fourier transform states that
$$\mathrm{\mathit{\int_{-\infty }^{\infty }x_{\mathrm{1}}\left ( t \right )x_{\mathrm{2}}^{*}\left ( t \right )dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X_{\mathrm{1}}\left ( \omega \right )X_{\mathrm{2}}^{*}\left ( \omega \right )d\omega}} $$
Where, $\mathrm{\mathit{x_{\mathrm{1}}\left ( t \right )}}$ and $\mathrm{\mathit{x_{\mathrm{2}}\left ( t \right )}}$ are complex functions.
Proof
Parseval’s relation is given by,
$$\mathrm{\mathit{\int_{-\infty }^{\infty }x_{\mathrm{1}}\left ( t \right )x_{\mathrm{2}}^{*}\left ( t \right )dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X_{\mathrm{1}}\left ( \omega \right )X_{\mathrm{2}}^{*}\left ( \omega \right )d\omega}}$$
From the definition of inverse Fourier transform, we have,
$$\mathrm{LHS \mathrm{=} \mathit{\int_{-\infty }^{\infty }x_{\mathrm{1}}\left ( t \right )x_{\mathrm{2}}^{*}\left ( t \right )dt\mathrm{=}\int_{-\infty }^{\infty }\left [ \frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X_{\mathrm{1}}\left ( \omega \right )e^{j\omega t}d\omega \right ]x_{\mathrm{2}}^{*}\left ( t \right )dt}} $$
By interchanging the order of integration in RHS of the above expression, we get,
$$\mathrm{\mathit{\int_{-\infty }^{\infty }x_{\mathrm{1}}\left ( t \right )x_{\mathrm{2}}^{*}\left ( t \right )dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X_{\mathrm{1}}\left ( \omega \right )\left [ \int_{-\infty }^{\infty }x_{\mathrm{2}}^{*}\left ( t \right )e^{j\omega t}dt \right ]d\omega }}$$
$$\mathrm{\Rightarrow \mathit{\int_{-\infty }^{\infty }x_{\mathrm{1}}\left ( t \right )x_{\mathrm{2}}^{*}\left ( t \right )dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X_{\mathrm{1}}\left ( \omega \right )\left [ \int_{-\infty }^{\infty }x_{\mathrm{2}}\left ( t \right )e^{-j\omega t}dt \right ]^{*}d\omega }}$$
$$\mathrm{\therefore \mathit{\int_{-\infty }^{\infty }x_{\mathrm{1}}\left ( t \right )x_{\mathrm{2}}^{*}\left ( t \right )dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X_{\mathrm{1}}\left ( \omega \right )X_{\mathrm{2}}^{*}\left ( \omega \right )d\omega \mathrm{=}\mathrm{RHS}}}$$
Parseval’s Identity of Fourier Transform
The Parseval’s identity of Fourier transform states that the energy content of the signal $\mathit{x\left ( t \right )}$ is given by,
$$\mathrm{ \mathit{E\mathrm{=}\int_{-\infty }^{\infty }\left | x\left ( t \right ) \right |^{\mathrm{2}}dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }\left | X\left ( \omega \right ) \right |^{\mathrm{2}}d\omega}}$$
The Parseval’s identity is also called energy theorem or Rayleigh’s energy theorem.
The quantity $\mathrm{\mathit{\left [ \left | X\left ( \omega \right ) \right |^{\mathrm{2}}\right ]}}$ is called the energy density spectrum of the signal $\mathit{x\left ( t \right )}$.
Proof
If $\mathrm{\mathit{x_{\mathrm{1}}\left ( t \right )}}$ = $\mathrm{\mathit{x_{\mathrm{2}}\left ( t \right )}}$ = $\mathit{x\left ( t \right )}$; then the energy of the signal is given by,
$$\mathrm{\mathit{E\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )x^{*}\left ( t \right )dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )X^{*}\left ( \omega \right )d\omega}}$$
$$\mathrm{\mathit{\because x\left ( t \right )x^{*}\left ( t \right )\mathrm{=}\left |x\left ( t \right ) \right |^{\mathrm{2}}\: \: and\: \: X\left ( \omega \right )X^{*}\left ( \omega \right )\mathrm{=}\left |X\left ( \omega \right ) \right |^{\mathrm{2}}}}$$
Therefore,
$$\mathrm{\mathit{E\mathrm{=}\int_{-\infty }^{\infty }\left | x\left ( t \right ) \right |^{\mathrm{2}}dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }\left | X\left ( \omega \right ) \right |^{\mathrm{2}}d\omega\; \; \; \; \mathrm{\left ( Hence,Proved \right )}}}$$
31K+ Views
