Normalized DFT
A more ``theoretically clean'' DFT is obtained by projecting onto the normalized DFT sinusoids (§6.5)
![$\displaystyle \tilde{s}_k(n) \isdef \frac{e^{j2\pi k n/N}}{\sqrt{N}}.
$](http://www.dsprelated.com/josimages_new/mdft/img1070.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$\displaystyle \tilde{X}(\omega_k) \isdef \left<x,\tilde{s}_k\right> = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x(n) e^{-j2\pi k n/N}
$](http://www.dsprelated.com/josimages_new/mdft/img1071.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ \tilde{s}_k$](http://www.dsprelated.com/josimages_new/mdft/img1072.png)
![$\displaystyle x(n) = \sum_{k=0}^{N-1}\tilde{X}(\omega_k) \tilde{s}_k(n)
= \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\tilde{X}(\omega_k)e^{j2\pi k n/N}.
$](http://www.dsprelated.com/josimages_new/mdft/img1073.png)
It can be said that only the NDFT provides a proper change of
coordinates from the time-domain (shifted impulse basis signals) to
the frequency-domain (DFT sinusoid basis signals). That is, only the
NDFT is a pure
rotation in , preserving both orthogonality and the unit-norm
property of the basis functions. The DFT, in contrast, preserves
orthogonality, but the norms of the basis functions grow to
. Therefore, in the present context, the DFT coefficients can be
considered ``denormalized'' frequency-domain coordinates.
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The Length 2 DFT
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Fourier Series Special Case