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Normalized DFT Basis for $ {\bf C}^N$

The Normalized Discrete Fourier Transform (NDFT) (introduced in Book I [264]) projects the signal $ x$ onto $ N$ discrete-time sinusoids of length $ N$ , where the sinusoids are normalized to have unit $ L2$ norm:

$\displaystyle \varphi_k (n) \isdefs \frac{e^{j\omega_k n}}{\sqrt{N}}, \quad \omega_k \isdef k\frac{2\pi}{N},$ (12.112)

and $ n,k \in [0,N-1]$ . The coefficient of projection of $ x$ onto $ \varphi_k $ is given by
$\displaystyle \left<\varphi_k ,x\right>$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} x(n) e^{-j\omega_k n}$  
  $\displaystyle \isdef$ $\displaystyle \hbox{NDFT}_{N,k}(x) \isdefs \frac{1}{\sqrt{N}}\hbox{DFT}_{N,k}(x) \isdefs \frac{X(\omega_k )}{\sqrt{N}}$  

and the expansion of $ x$ in terms of the NDFT basis set is
$\displaystyle x(n)$ $\displaystyle =$ $\displaystyle \sum_{k=0}^{N-1} \left<\varphi_k ,x\right> \varphi_k (n)$  
  $\displaystyle =$ $\displaystyle \frac{1}{N} \sum_{k=0}^{N-1} X(k)e^{j\omega_k n}$  
  $\displaystyle \isdef$ $\displaystyle \hbox{DFT}_{N,n}^{-1}(X)$  

for $ n=0,1,2,\ldots,N-1$ .

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