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− | + | [[Category:Formulas]] | |
− | <center>''' | + | [[Category:Fourier transform]] |
− | </ | + | [[Category:ECE301]] |
+ | [[Category:ECE438]] | ||
+ | |||
+ | <center><font size= 4> | ||
+ | '''[[Collective_Table_of_Formulas|Collective Table of Formulas]]''' | ||
+ | </font size> | ||
+ | |||
+ | [[Discrete-time_Fourier_transform_info|Discrete-time (DT) Fourier Transforms]] Pairs and Properties | ||
+ | |||
+ | (used in [[ECE301]], [[ECE438]], [[ECE538]]) | ||
+ | |||
+ | </center> | ||
+ | |||
+ | ---- | ||
+ | |||
{| | {| | ||
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! colspan="2" style="background: #eee;" | DT Fourier transform and its Inverse | ! colspan="2" style="background: #eee;" | DT Fourier transform and its Inverse | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | DT Fourier Transform | + | | align="right" style="padding-right: 1em;" | [[Discrete-time Fourier transform info|DT Fourier Transform]] |
| <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math> | | <math>\,\mathcal{X}(\omega)=\mathcal{F}(x[n])=\sum_{n=-\infty}^{\infty}x[n]e^{-j\omega n}\,</math> | ||
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{| | {| | ||
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! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs | ! colspan="4" style="background: #eee;" | DT Fourier Transform Pairs | ||
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| <math>e^{jw_0n} \ </math> | | <math>e^{jw_0n} \ </math> | ||
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− | | <math>\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math> | + | | <math>\ 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
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| <math>w[n]= \ </math> | | <math>w[n]= \ </math> | ||
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− | | add formula here | + | | <math> \text{add formula here} \ </math> |
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | <math>\ | + | | <math>\sin\left(\omega _0 n\right) \ </math> |
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| <math>\frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k))</math> | | <math>\frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k))</math> | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | 1 | + | | <math> 1 \ </math> |
− | | | + | | |
| <math>2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k)</math> | | <math>2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k)</math> | ||
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| align="right" style="padding-right: 1em;" | DTFT of a Periodic Square Wave | | align="right" style="padding-right: 1em;" | DTFT of a Periodic Square Wave | ||
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− | <math>\left\{\begin{array}{ll}1, & |n|<N_1,\\ 0, & N_1<|n| | + | <math>\left\{\begin{array}{ll}1, & |n|<N_1,\\ 0, & N_1<|n|\leq\frac{N}{2}\end{array} \right. \text{ and } |
− | + | x[n+N]=x[n] </math> | |
− | and | + | |
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | < | + | | <math> \delta [n] \ </math> |
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− | | 1 | + | | <math> 1 \ </math> |
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | < | + | | <math> u[n] \ </math> |
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| <math>\frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k)</math> | | <math>\frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k)</math> | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | < | + | | <math> \delta[n - n_0] \ </math> |
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| <math>e^{-j\omega n_0}</math> | | <math>e^{-j\omega n_0}</math> | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | < | + | | <math> (n + 1)a^n u[n], \quad |a| < 1 </math> |
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| <math>\frac{1}{(1-ae^{-j\omega})^{2}}</math> | | <math>\frac{1}{(1-ae^{-j\omega})^{2}}</math> | ||
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|} | |} | ||
{| | {| | ||
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! colspan="4" style="background: #eee;" | DT Fourier Transform Properties | ! colspan="4" style="background: #eee;" | DT Fourier Transform Properties | ||
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| <math>x[n]y[n] \ </math> | | <math>x[n]y[n] \ </math> | ||
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− | | <math>\frac{1}{2\pi} \int_{ | + | | <math>\frac{1}{2\pi} \int_{-\pi}^{\pi} X(\theta)Y(\omega-\theta)d\theta</math> |
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| align="right" style="padding-right: 1em;" | convolution property | | align="right" style="padding-right: 1em;" | convolution property | ||
− | | <math>x[n]*y[n] \ | + | | <math>x[n]*y[n] \ </math> |
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| <math> X(\omega)Y(\omega) \!</math> | | <math> X(\omega)Y(\omega) \!</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 | ||
− | | < | + | | <math> ax[n]+by[n] \ </math> |
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− | | < | + | | <math> aX(\omega)+bY(\omega) \ </math> |
|- | |- | ||
| 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 | ||
− | | < | + | | <math> x[n - n_0] \ </math> |
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| <math>e^{-j\omega n_0}X(\omega)</math> | | <math>e^{-j\omega n_0}X(\omega)</math> | ||
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| <math>e^{j\omega_0 n}x[n]</math> | | <math>e^{j\omega_0 n}x[n]</math> | ||
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− | | < | + | | <math> X(\omega - \omega_0) \ </math> |
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| 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 | ||
− | | < | + | | <math> x^* [n] \ </math> |
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− | | < | + | | <math> X^* (-\omega) \ </math> |
|- | |- | ||
| 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> |
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− | | < | + | | <math> X(k\omega) \ </math> |
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| 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 |
− | | < | + | | <math> x[n] - x[n - 1] \ </math> |
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− | | < | + | | <math> (1 - e^{-j\omega}) X (\omega) \ </math> |
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
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| <math>\sum^{n}_{k=-\infty} x[k]</math> | | <math>\sum^{n}_{k=-\infty} x[k]</math> | ||
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− | | | + | | <math>\frac{1}{1-e^{-j\omega}}X(\omega)</math> |
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| 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 | + | | <math> x[n] \ \text{ real and even} \ </math> |
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− | | < | + | | <math> X(\omega) \ \text{ real and even} \ </math> |
|- | |- | ||
| 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 | + | | <math> x[n] \ \text{ real and odd} \ </math> |
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+ | | <math> X(\omega) \ \text{ purely imaginary and odd} \ </math> | ||
|} | |} | ||
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| align="right" style="padding-right: 1em;" | Parseval's relation | | align="right" style="padding-right: 1em;" | Parseval's relation | ||
− | | <math> | + | | <math>\sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega </math> |
|} | |} | ||
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---- | ---- | ||
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− | [[ | + | [[Collective Table of Formulas|Back to Collective Table]] |
Latest revision as of 21:05, 4 March 2015
Discrete-time (DT) Fourier Transforms 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} \ $ | $ \ 2\pi\sum_{l=-\infty}^{+\infty}\delta(w-w_0-2\pi l) \ $ | ||
(info) DTFT of a rectangular window | $ w[n]= \ $ | $ \text{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) $ | |||
$ \cos\left(\omega _0 n\right) \ $ | $ \pi \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)+\delta(\omega+\omega_0-2\pi k)) $ | |||
$ \sin\left(\omega _0 n\right) \ $ | $ \frac{\pi}{j} \sum^{\infty}_{k=-\infty} (\delta(\omega-\omega_0 + 2\pi k)-\delta(\omega+\omega_0-2\pi k)) $ | |||
$ 1 \ $ | $ 2\pi\sum^{\infty}_{k=-\infty}\delta(\omega-2\pi k) $ | |||
DTFT of a Periodic Square Wave |
$ \left\{\begin{array}{ll}1, & |n|<N_1,\\ 0, & N_1<|n|\leq\frac{N}{2}\end{array} \right. \text{ and } x[n+N]=x[n] $ |
$ 2\pi\sum^{\infty}_{k=-\infty}a_k\delta(\omega-\frac{2\pi k}{N}) $ | ||
$ \sum^{\infty}_{k=-\infty}\delta[n-kN] $ | $ \frac{2\pi}{N}\sum^{\infty}_{k=-\infty}\delta(\omega -\frac{2\pi k}{N}) $ | |||
$ \delta [n] \ $ | $ 1 \ $ | |||
$ u[n] \ $ | $ \frac{1}{1-e^{-j\omega}}+\sum^{\infty}_{k=-\infty}\pi\delta(\omega-2\pi k) $ | |||
$ \delta[n - n_0] \ $ | $ e^{-j\omega n_0} $ | |||
$ (n + 1)a^n u[n], \quad |a| < 1 $ | $ \frac{1}{(1-ae^{-j\omega})^{2}} $ |
DT Fourier Transform Properties | ||||
---|---|---|---|---|
$ x[n] \ $ | $ \longrightarrow $ | $ \mathcal{X}(\omega) \ $ | ||
multiplication property | $ x[n]y[n] \ $ | $ \frac{1}{2\pi} \int_{-\pi}^{\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 | $ ax[n]+by[n] \ $ | $ aX(\omega)+bY(\omega) \ $ | ||
Time Shifting | $ x[n - n_0] \ $ | $ e^{-j\omega n_0}X(\omega) $ | ||
Frequency Shifting | $ e^{j\omega_0 n}x[n] $ | $ X(\omega - \omega_0) \ $ | ||
Conjugation | $ x^* [n] \ $ | $ X^* (-\omega) \ $ | ||
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(k\omega) \ $ | ||
Differentiating in Time | $ x[n] - x[n - 1] \ $ | $ (1 - e^{-j\omega}) X (\omega) \ $ | ||
Accumulation | $ \sum^{n}_{k=-\infty} x[k] $ | $ \frac{1}{1-e^{-j\omega}}X(\omega) $ | ||
Symmetry | $ x[n] \ \text{ real and even} \ $ | $ X(\omega) \ \text{ real and even} \ $ | ||
$ x[n] \ \text{ real and odd} \ $ | $ X(\omega) \ \text{ purely imaginary and odd} \ $ |
Other DT Fourier Transform Properties | |
---|---|
Parseval's relation | $ \sum_{n=-\infty}^{\infty}\left| x[n] \right|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X( \omega )|^2d\omega $ |