Line 57: | Line 57: | ||
| <math>\longrightarrow</math> | | <math>\longrightarrow</math> | ||
| <math> X[k] \ </math> | | <math> X[k] \ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Linearity | ||
+ | | <math> ax[n]+by[n] \ </math> | ||
+ | | | ||
+ | | <math> aX[k]+bY[k] \ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Circular Shift | ||
+ | | <math> x[((n-m))_N] \ </math> | ||
+ | | | ||
+ | | <math> X[k]e^{(-j\frac{2 \pi}{N}km)} \ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Duality | ||
+ | | <math> X[n] \ </math> | ||
+ | | | ||
+ | | <math> NX[((-k))_N] \ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | multiplication property | | align="right" style="padding-right: 1em;" | multiplication property |
Revision as of 11:37, 25 November 2011
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Discrete Fourier Transform
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Discrete Fourier Transform Pairs and Properties (info) | |
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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 property | $ x[n]y[n] \ $ | $ write DFT here $ | |
convolution property | $ x(t)*y(t) \! $ | $ X(f)Y(f) \! $ | |
time reversal | $ \ x(-t) $ | $ \ X(-f) $ |
Other Discrete Fourier Transform Properties | |
---|---|
property | $ type math here $ |