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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Derivation of the Discrete Fourier Transform (DFT)
       Orthogonality of the DFT Sinusoids

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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

\begin{eqnarray*} \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}} \end{eqnarray*}

where the last step made use of the closed-form expression for the sum of a geometric series (§6.1). 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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Next: Norm of the DFT Sinusoids

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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