Orthogonality of the DFT Sinusoids
We now show mathematically that the DFT sinusoids are exactly orthogonal. Let
![$\displaystyle s_k(n) \isdef e^{j\omega_k nT} = e^{j2\pi k n /N} = \left[W_N^k\right]^n,
\quad n=0,1,2,\ldots,N-1,
$](http://www.dsprelated.com/josimages_new/mdft/img1025.png)



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

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Norm of the DFT Sinusoids
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Orthogonality of Sinusoids