Revision as of 12:33, 25 November 2011

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Discrete Fourier Transform

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Discrete Fourier Transform Pairs and Properties (info)
Definition CT Fourier Transform and its Inverse
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)
	$x[n] \$	$\longrightarrow$	$X[k] \$
name	$type signal here\$		$type transform here \! \$
name	$type signal here \$		$$ type transform here $$

Discrete Fourier Transform Properties
	$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] \$
time reversal	$\ x(-t)$		$\ X(-f)$

Other Discrete Fourier Transform Properties
Parseval's Theorem	$\sum_{n=0}^{N-1}\|x[n]\|^2 = \frac{1}{N} \sum_{k=0}^{N-1}\|X[k]\|^2$

@@ Line 73: / Line 73: @@
 | <math> NX[((-k))_N] \  </math>
 |-
-| align="right" style="padding-right: 1em;" | multiplication property
+| align="right" style="padding-right: 1em;" | Multiplication
-| <math>x[n]y[n] \ </math>
+| <math> x[n]y[n] \ </math>
 |
-| <math> write DFT here</math>
+| <math> \frac{1}{N} X[k]\circledast Y[k], \  \circledast \text{ denotes the circular convolution} </math>
 |-
-| align="right" style="padding-right: 1em;" | convolution property
+| align="right" style="padding-right: 1em;" | Convolution
-| <math>x(t)*y(t) \!</math>
+| <math>x(t) \circledast y(t) \ </math>
 |
-| <math> X(f)Y(f) \!</math>
+| <math> X[k]Y[k] \ </math>
 |-
 | align="right" style="padding-right: 1em;" | time reversal
@@ Line 93: / Line 93: @@
 ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Discrete Fourier Transform Properties
 |-
-| align="right" style="padding-right: 1em;" | property
+| align="right" style="padding-right: 1em;" | Parseval's Theorem
-| <math>type math here</math>
+| <math> \sum_{n=0}^{N-1}|x[n]|^2  = \frac{1}{N} \sum_{k=0}^{N-1}|X[k]|^2 </math>
 |}
 ----