| Line 20: | Line 20: | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| <math> x[n] \ </math> | | <math> x[n] \ </math> | ||
| − | | <math>\longrightarrow</math> | + | | <math>\longrightarrow</math> |
| <math> \mathcal{X}(\omega) \ </math> | | <math> \mathcal{X}(\omega) \ </math> | ||
|- | |- | ||
| Line 62: | Line 62: | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| <math>x[n] \ </math> | | <math>x[n] \ </math> | ||
| − | | <math>\longrightarrow</math> | + | | <math>\longrightarrow</math> |
| <math> \mathcal{X}(\omega) \ </math> | | <math> \mathcal{X}(\omega) \ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | multiplication property | + | | align="right" style="padding-right: 1em;" | multiplication property |
| <math>x[n]y[n] \ </math> | | <math>x[n]y[n] \ </math> | ||
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|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Linearity | + | | align="right" style="padding-right: 1em;" | Linearity |
| − | | < | + | | <span class="texhtml">''a''''x''[''n''] + ''b''''y''[''n'']</span> |
| | | | ||
| − | | < | + | | <span class="texhtml">''a''''X''(ω) + ''b''''Y''(ω)</span> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Time Shifting | + | | align="right" style="padding-right: 1em;" | Time Shifting |
| − | | < | + | | <span class="texhtml">''x''[''n'' − ''n''<sub>0</sub>]</span> |
| | | | ||
| <math>e^{-j\omega n_0}X(\omega)</math> | | <math>e^{-j\omega n_0}X(\omega)</math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Frequency Shifting | + | | align="right" style="padding-right: 1em;" | Frequency Shifting |
| − | | <math>e^{j\omega_0 n}x[n]</math> | + | | <math>e^{j\omega_0 n}x[n]</math> |
| | | | ||
| − | | < | + | | <span class="texhtml">''X''(ω − ω<sub>0</sub>)</span> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Conjugation | + | | align="right" style="padding-right: 1em;" | Conjugation |
| − | | < | + | | <span class="texhtml">''x''<sup> * </sup>[''n'']</span> |
| | | | ||
| − | | < | + | | <span class="texhtml">''X''<sup> * </sup>( − ω)</span> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Time Expansion | + | | align="right" style="padding-right: 1em;" | Time Expansion |
| − | | <math>x_(k) [n]=\left\{\begin{array}{ll}x[n/k], & \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right.</math> | + | | <math>x_(k) [n]=\left\{\begin{array}{ll}x[n/k], & \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right.</math> |
| | | | ||
| − | | < | + | | <span class="texhtml">''X''(''k''ω)</span> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Differentiating in Time | + | | align="right" style="padding-right: 1em;" | Differentiating in Time |
| − | | < | + | | <span class="texhtml">''x''[''n''] − ''x''[''n'' − 1]</span> |
| | | | ||
| − | | < | + | | <span class="texhtml">(1 − ''e''<sup> − ''j''ω</sup>)''X''(ω)</span> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Accumulation | + | | align="right" style="padding-right: 1em;" | Accumulation |
| − | | <math>\sum^{n}_{k=-\infty} x[k]</math> | + | | <math>\sum^{n}_{k=-\infty} x[k]</math> |
| | | | ||
| − | | <math>\frac{1}{1-e^{-j\omega}X(\omega)</math> | + | | '''<math>\frac{1}{1-e^{-j\omega}}X(\omega)</math>''' |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | align="right" style="padding-right: 1em;" | Symmetry | + | | align="right" style="padding-right: 1em;" | Symmetry |
| − | | x[n] real and even | + | | x[n] real and even |
| | | | ||
| − | | < | + | | <span class="texhtml">''X''(ω)</span> real and even |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| − | | x[n] real and odd | + | | x[n] real and odd |
| | | | ||
| − | | < | + | | <span class="texhtml">''X''(ω)</span> purely imaginary and odd |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
Revision as of 13:13, 10 April 2011
| Discrete-time Fourier Transform Pairs and Properties | |
|---|---|
| DT Fourier transform and its Inverse | |
| DT Fourier Transform | $ \,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\, $ |
| Inverse DT Fourier Transform | $ \,x[n]=\mathcal{F}^{-1}(\mathcal{X}(\omega))=\frac{1}{2\pi} \int_{0}^{2\pi}\mathcal{X}(\omega)e^{j\omega n} d \omega\, $ |
| DT Fourier Transform Pairs | |||||||
|---|---|---|---|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | |||||
| DTFT of a complex exponential | $ e^{jw_0n} \ $ | $ \pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $ | |||||
| (info) DTFT of a rectangular window | $ w[n]= \ $ | add formula here | |||||
| $ a^{n} u[n], |a|<1 \ $ | $ \frac{1}{1-ae^{-j\omega}} \ $ | ||||||
| $ (n+1)a^{n} u[n], |a|<1 \ $ | $ \frac{1}{(1-ae^{-j\omega})^2} \ $ | ||||||
| $ \sin\left(\omega _0 n\right) u[n] \ $ | $ \frac{1}{2j}\left( \frac{1}{1-e^{-j(\omega -\omega _0)}}-\frac{1}{1-e^{-j(\omega +\omega _0)}}\right) $ | ||||||
| DT Fourier Transform Properties | |||||||
|---|---|---|---|---|---|---|---|
| $ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | |||||
| multiplication property | $ x[n]y[n] \ $ | $ \frac{1}{2\pi} \int_{2\pi} X(\theta)Y(\omega-\theta)d\theta $ | |||||
| convolution property | $ x[n]*y[n] \! $ | $ X(\omega)Y(\omega) \! $ | |||||
| time reversal | $ \ x[-n] $ | $ \ X(-\omega) $ | |||||
| Differentiation in frequency | $ \ nx[n] $ | $ \ j\frac{d}{d\omega}X(\omega) $ | |||||
| Linearity | a'x[n] + b'y[n] | a'X(ω) + b'Y(ω) | |||||
| Time Shifting | x[n − n0] | $ e^{-j\omega n_0}X(\omega) $ | |||||
| Frequency Shifting | $ e^{j\omega_0 n}x[n] $ | X(ω − ω0) | |||||
| Conjugation | x * [n] | X * ( − ω) | |||||
| Time Expansion | $ x_(k) [n]=\left\{\begin{array}{ll}x[n/k], & \text{ if n = multiple of k},\\ 0, & \text{else.}\end{array} \right. $ | X(kω) | |||||
| Differentiating in Time | x[n] − x[n − 1] | (1 − e − jω)X(ω) | |||||
| Accumulation | $ \sum^{n}_{k=-\infty} x[k] $ | $ \frac{1}{1-e^{-j\omega}}X(\omega) $ | |||||
| Symmetry | x[n] real and even | X(ω) real and even | |||||
| x[n] real and odd | X(ω) purely imaginary and odd | ||||||
| Other DT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \frac {1}{N} \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = $ |
