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Spectral Audio Signal Processing
    Multirate Filter Banks
       Wavelet Filter Banks
          Geometric Signal Theory
             Normalized DFT Basis for

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Normalized DFT Basis for

The Normalized Discrete Fourier Transform (NDFT) (introduced in Book I [248]) 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}, $

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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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