Editing Discrete Fourier transform (section)

=== Orthogonality ===
The vectors <math>u_k = \left[\left. e^{ \frac{i 2\pi}{N} kn} \;\right|\; n=0,1,\ldots,N-1 \right]^\mathsf{T}</math>, for <math>k=0,1,\ldots,N-1</math>, form an [[orthogonal basis]] over the set of ''N''-dimensional complex vectors:

:<math>u^\mathsf{T}_k u_{k'}^*
 = \sum_{n=0}^{N-1} \left(e^{ \frac{i 2\pi}{N} kn}\right) \left(e^{\frac{i 2\pi}{N} (-k')n}\right)
 = \sum_{n=0}^{N-1} e^{ \frac{i 2\pi}{N} (k-k') n}
 = N~\delta_{kk'}
</math>

where <math>\delta_{kk'}</math> is the [[Kronecker delta]]. (In the last step, the summation is trivial if <math>k=k'</math>, where it is {{nowrap|1=1 + 1 + ⋯ = ''N'',}} and otherwise is a [[geometric series]] that can be explicitly summed to obtain zero.)  This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.