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

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 $\ensuremath{L_2}$ norm:

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

and $n,k \in [0,N-1]$ . The coefficient of projection of $x$ onto $\varphi_k$ is given by

$$\left<\varphi_k ,x\right> \isdef \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} x(n) e^{-j\omega_k n}$$

$$\isdef \hbox{NDFT}_{N,k}(x) \isdefs \frac{1}{\sqrt{N}}\hbox{DFT}_{N,k}(x) \isdefs \frac{X(\omega_k )}{\sqrt{N}}$$

and the expansion of $x$ in terms of the NDFT basis set is

$$x(n) = \sum_{k=0}^{N-1} \left<\varphi_k ,x\right> \varphi_k (n)$$

$$= \frac{1}{N} \sum_{k=0}^{N-1} X(k)e^{j\omega_k n}$$

$$\isdef \hbox{DFT}_{N,n}^{-1}(X)$$

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