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− | < | + | <center><font size= 4> |
− | + | '''[[Collective_Table_of_Formulas|Collective Table of Formulas]]''' | |
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− | </ | + | |
+ | Discrete Fourier transforms (DFT) Pairs and Properties | ||
− | + | click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]] | |
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+ | </center> | ||
+ | |||
+ | ---- | ||
{| | {| | ||
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− | ! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | Discrete Fourier Transform Pairs and Properties [[ | + | ! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | Discrete Fourier Transform Pairs and Properties [[Discrete Fourier Transform|(info)]] |
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− | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition | + | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition Discrete Fourier Transform and its Inverse |
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− | | align="right" style="padding-right: 1em;" | [[Discrete Fourier Transform|Discrete Fourier Transform]] | + | | Let x[n] be a periodic DT signal, with period N. |
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | N-point [[Discrete Fourier Transform|Discrete Fourier Transform]] | ||
| <math>X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, </math> | | <math>X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, </math> | ||
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− | | align="right" style="padding-right: 1em;" | Inverse Discrete Fourier Transform | + | | align="right" style="padding-right: 1em;" | Inverse Discrete Fourier Transform |
| <math>\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,</math> | | <math>\,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \,</math> | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | <math> x[n] \ </math> | + | | <math> x[n] \ \text{ (period } N) </math> |
− | | <math>\longrightarrow</math> | + | | <math>\longrightarrow </math> |
− | | <math> | + | | <math> X_N[k] \ \ (N \text{ point DFT)}</math> |
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− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math> | + | | <math>\ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right.</math> |
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− | | <math> | + | | <math>\ 1 \text{ (period } N) </math> |
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− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math> | + | | <math>\ 1 \text{ (period } N) </math> |
+ | | | ||
+ | | <math>\ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right.</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ e^{j2\pi k_0 n} </math> | ||
+ | | | ||
+ | | <math>\ N\delta[((k - k_0))_N] </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ \cos(\frac{2\pi}{N}k_0n) </math> | ||
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− | | <math> | + | | <math>\ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) </math> |
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| <math> X[k]Y[k] \ </math> | | <math> X[k]Y[k] \ </math> | ||
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− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math>\ x | + | | <math>\ x^*[n] </math> |
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− | | <math>\ X(- | + | | <math>\ X^*[((-k))_N] </math> |
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ x^*[((-n))_N] </math> | ||
+ | | | ||
+ | | <math>\ X^*[k] </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ \Re\{x[n]\} </math> | ||
+ | | | ||
+ | | <math>\ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ j\Im\{x[n]\} </math> | ||
+ | | | ||
+ | | <math>\ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} </math> | ||
+ | | | ||
+ | | <math>\ \Re\{X[k]\} </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} </math> | ||
+ | | | ||
+ | | <math>\ j\Im\{X[k]\} </math> | ||
|} | |} | ||
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|} | |} | ||
---- | ---- | ||
+ | [[ECE438|Go to Relevant Course Page: ECE 438]] | ||
+ | |||
+ | [[ECE538|Go to Relevant Course Page: ECE 538]] | ||
+ | |||
[[Collective_Table_of_Formulas|Back to Collective Table]] | [[Collective_Table_of_Formulas|Back to Collective Table]] | ||
[[Category:Formulas]] | [[Category:Formulas]] | ||
+ | [[Category:discrete Fourier transform]] | ||
+ | [[Category:Fourier transform]] | ||
+ | [[Category:ECE438]] |
Latest revision as of 15:28, 23 April 2013
Discrete Fourier transforms (DFT) Pairs and Properties
click here for more formulas
Discrete Fourier Transform Pairs and Properties (info) | |
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Definition Discrete Fourier Transform and its Inverse | |
Let x[n] be a periodic DT signal, with period N. | |
N-point Discrete Fourier Transform | $ X [k] = \sum_{n=0}^{N-1} x[n]e^{-j 2\pi \frac{k n}{N}} \, $ |
Inverse Discrete Fourier Transform | $ \,x [n] = (1/N) \sum_{k=0}^{N-1} X[k] e^{j 2\pi\frac{kn}{N}} \, $ |
Discrete Fourier Transform Pairs (info) | |||
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$ x[n] \ \text{ (period } N) $ | $ \longrightarrow $ | $ X_N[k] \ \ (N \text{ point DFT)} $ | |
$ \ \sum_{k=-\infty}^\infty \delta[n+Nk] = \left\{ \begin{array}{ll} 1, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $ | $ \ 1 \text{ (period } N) $ | ||
$ \ 1 \text{ (period } N) $ | $ \ N\sum_{m=-\infty}^\infty \delta[k+Nm] = \left\{ \begin{array}{ll} N, & \text{ if } n=0, \pm N, \pm 2N , \ldots\\ 0, & \text{ else.} \end{array}\right. $ | ||
$ \ e^{j2\pi k_0 n} $ | $ \ N\delta[((k - k_0))_N] $ | ||
$ \ \cos(\frac{2\pi}{N}k_0n) $ | $ \ \frac{N}{2}(\delta[((k - k_0))_N] + \delta[((k+k_0))_N]) $ |
Discrete Fourier Transform Properties | |||
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$ x[n] \ $ | $ \longrightarrow $ | $ X[k] \ $ | |
Linearity | $ ax[n]+by[n] \ $ | $ aX[k]+bY[k] \ $ | |
Circular Shift | $ x[((n-m))_N] \ $ | $ X[k]e^{(-j\frac{2 \pi}{N}km)} \ $ | |
Duality | $ X[n] \ $ | $ NX[((-k))_N] \ $ | |
Multiplication | $ x[n]y[n] \ $ | $ \frac{1}{N} X[k]\circledast Y[k], \ \circledast \text{ denotes the circular convolution} $ | |
Convolution | $ x(t) \circledast y(t) \ $ | $ X[k]Y[k] \ $ | |
$ \ x^*[n] $ | $ \ X^*[((-k))_N] $ | ||
$ \ x^*[((-n))_N] $ | $ \ X^*[k] $ | ||
$ \ \Re\{x[n]\} $ | $ \ X_{ep}[k] = \frac{1}{2}\{X[((k))_N] + X^*[((-k))_N]\} $ | ||
$ \ j\Im\{x[n]\} $ | $ \ X_{op}[k] = \frac{1}{2}\{X[((k))_N] - X^*[((-k))_N]\} $ | ||
$ \ x_{ep}[n] = \frac{1}{2}\{x[n] + x^*[((-n))_N]\} $ | $ \ \Re\{X[k]\} $ | ||
$ \ x_{op}[n] = \frac{1}{2}\{x[n] - x^*[((-n))_N]\} $ | $ \ j\Im\{X[k]\} $ |
Other Discrete Fourier Transform Properties | |
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Parseval's Theorem | $ \sum_{n=0}^{N-1}|x[n]|^2 = \frac{1}{N} \sum_{k=0}^{N-1}|X[k]|^2 $ |
Go to Relevant Course Page: ECE 438