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== ECE438 Week9 Quiz Question6 Solution== | == ECE438 Week9 Quiz Question6 Solution== | ||
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The Duality Property of DFT is descriped below: | The Duality Property of DFT is descriped below: | ||
Revision as of 17:49, 20 October 2010
ECE438 Week9 Quiz Question6 Solution
The Duality Property of DFT is descriped below:
$ \text{If the DFT of }x[n]\text{ is denoted as }X(k)\text{ with a length of N} $
$ \text{Then the DFT of }X(n)\text{ is }Nx(-k) $
Derivation:
$ \begin{align} &x(n)=\frac{1}{N}\sum_{k=0}^{N-1}X(k)e^{j\frac{j2\pi kn}{N}} \\ &\text{Switch n and k} \\ \Leftrightarrow &x(k)=\frac{1}{N}\sum_{k=0}^{N-1}X(n)e^{j\frac{j2\pi kn}{N}} \\ \Leftrightarrow &Nx(-k)=\sum_{k=0}^{N-1}X(n)e^{-j\frac{j2\pi kn}{N}}=\text{ DFT of }X(n) \end{align} $
Conclusion:
if $ x(n)\xrightarrow{DFT} X(k) $
then $ X(n)\xrightarrow{DFT} Nx(-k) $
