Parseval’s Theorem & Parseval’s Identity of Fourier Transform

Signals and SystemsElectronics & ElectricalDigital Electronics

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,

$$\mathrm{\mathit{E\mathrm{=}\int_{-\infty }^{\infty }\left | x\left ( t \right ) \right |^{\mathrm{2}}dt\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }\left | X\left ( \omega \right ) \right |^{\mathrm{2}}d\omega\; \; \; \; \mathrm{\left ( Hence,Proved \right )}}}$$

raja
Published on 17-Dec-2021 07:38:13

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