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,

denote the $ k$th DFT complex-sinusoid, for $ k=0:N-1$. Then

\left<s_k,s_l\right> &\isdef & \sum_{n=0}^{N-1}s_k(n) \overlin...
...i (k-l) n /N}
= \frac{1 - e^{j2\pi (k-l)}}{1-e^{j2\pi (k-l)/N}}

where the last step made use of the closed-form expression for the sum of a geometric series6.1). If $ k\neq l$, the denominator is nonzero while the numerator is zero. This proves

$\displaystyle \zbox {s_k \perp s_l, \quad k \neq l.}

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.

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Norm of the DFT Sinusoids
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Orthogonality of Sinusoids