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Signals and Systems – Properties of Discrete-Time Fourier Transform
Discrete Time Fourier Transform
The discrete time Fourier transform is a mathematical tool which is used to convert a discrete time sequence into the frequency domain. Therefore, the Fourier transform of a discrete time signal or sequence is called the discrete time Fourier transform (DTFT).
Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time sequence, then the discrete time Fourier transform of the sequence is defined as −
$$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{-\mathit{j\omega n}}}$$
Properties of Discrete-Time Fourier Transform
Following table gives the important properties of the discrete-time Fourier transform −
| Property | Discrete-Time Sequence | DTFT |
|---|---|---|
| Notation | $\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega}\right)}}$ |
| $\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega}\right)}}$ | |
| $\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega}\right)}}$ | |
| Linearity | $\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left( \mathit{n}\right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left( \mathit{\omega }\right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega}\right)}}$ |
| Time Shifting | $\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$ | $\mathrm{\mathit{e}^{\mathit{-j\omega k}}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$ |
| Frequency Shifting | $\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{\mathit{j\omega} _{\mathrm{0}}\mathit{n}}}$ | $\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega -\omega _{\mathrm{0}}}\right)}}$ |
| Time Reversal | $\mathrm{\mathit{x}\mathrm{\left(\mathit{-n}\right)}}$ | $\mathrm{\mathit{X}\mathrm{\left(\mathit{-\omega}\right)}}$ |
| Frequency Differentiation | $\mathrm{\mathit{n}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{j}\frac{\mathit{d}}{\mathit{d\omega}}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$ |
| Time Convolution | $\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\:*\:\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega }\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega }\right)}}$ |
| Frequency Convolution (Multiplication in time domain) | $\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega }\right)}\:*\:\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega }\right)}}}$ |
| Correlation | $\mathrm{\mathit{R}_{\mathit{x}_{\mathrm{1}}\mathit{x}_{\mathrm{2}}}\mathrm{\left(\mathit{l}\right )}}$ | $\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega }\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{-\omega }\right)}}$ |
| Modulation Property | $\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{cos}\mathit{\omega _{\mathrm{0}}\mathit{n}}}$ | $\mathrm{\frac{1}{2}\mathrm{\left[\mathit{X}\mathrm{\left(\mathit{\omega \:\mathrm{+}\:}\omega _{\mathrm{0}}\right)}\:\mathrm{+}\: \mathit{X}\mathrm{\left(\mathit{\omega \:\mathrm{-}\:}\omega _{\mathrm{0}}\right)} \right ]}}$ |
| Parseval’s Relation | $\mathrm{\sum_{\mathit{n=-\infty}}^{\infty}\left|\mathit{x}\mathrm{\left(\mathit{n}\right)} \right|^{\mathrm{2}}}$ | $\mathrm{\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\mathit{X}\mathrm{\left(\mathit{\omega }\right)} \right|^{\mathrm{2}}\:\mathit{d\omega}}$ |
| Conjugation | $\mathrm{\mathit{x}^{*}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}\mathrm{\left(\mathit{-\omega}\right)}}$ |
| $\mathrm{\mathrm{\mathit{x}^{*}\mathrm{\left(\mathit{-n}\right)}}}$ | $\mathrm{\mathrm{\mathit{X}^{*}\mathrm{\left(\mathit{\omega}\right)}}}$ | |
| Symmetry Properties | $\mathrm{\mathit{x}_{\mathit{R}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}_{\mathit{e}}\mathrm{\left(\mathit{\omega }\right)}}$ |
| $\mathrm{\mathit{j}\:\mathit{x}_{\mathit{I}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}_{\mathrm{0}}\mathrm{\left(\mathit{\omega }\right)}}$ | |
| $\mathrm{\mathit{x}_{\mathit{e}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{X}_{\mathit{R}}\mathrm{\left(\mathit{\omega }\right)}}$ | |
| $\mathrm{\mathit{x}_{\mathrm{0}}\mathrm{\left(\mathit{n}\right)}}$ | $\mathrm{\mathit{j}\:\mathit{X}_{\mathit{I}}\mathrm{\left(\mathit{\omega}\right)}}$ |
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