Normalized DFT

A more ``theoretically clean'' DFT is obtained by projecting onto the normalized DFT sinusoids

$${\tilde s}_k(n) \isdef \frac{e^{j2\pi k n/N}}{\sqrt{N}}.$$

In this case, the normalized DFT of $x$ is

$${\tilde X}(\omega_k) \isdef \left<x,{\tilde s}_k\right> = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x(n) e^{-j2\pi k n/N}$$

which is also precisely the coefficient of projection of $x$ onto ${\tilde s}_k$. The inverse normalized DFT is then more simply

$$x(n) = \sum_{k=0}^{N-1}X(\omega_k) {\tilde s}_k(n) = \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X(\omega_k)e^{j2\pi k n/N}$$

While this definition is much cleaner from a ``geometric signal theory'' point of view, it is rarely used in practice since it requires more computation than the typical definition. However, note that the only difference between the forward and inverse transforms in this case is the sign of the exponent in the kernel.