We now show mathematically that the DFT sinusoids are exactly orthogonal. Let
denote the th DFT complex-sinusoid, for . Then
where the last step made use of the closed-form expression for the sum of a geometric series (§6.1). If , the denominator is nonzero while the numerator is zero. This proves
While we only looked at unit amplitude, zero-phase complex sinusoids, as used by the DFT, it is readily verified that the (nonzero) amplitude and phase have no effect on orthogonality.