A more ``theoretically clean'' DFT is obtained by projecting onto the
normalized DFT sinusoids (§6.5)

In this case, the normalized DFT (NDFT) of is

which is also precisely the coefficient of projection of onto
.
The inverse normalized DFT is then more simply

While this definition is much cleaner from a ``geometric signal theory''
point of view, it is rarely used in practice since it requires slightly 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.

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.