A more ``theoretically clean'' DFT is obtained by projecting onto the normalized DFT sinusoids (§6.5)
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.
The Length 2 DFT
Fourier Series Special Case