An Orthonormal Sinusoidal Set

We can normalize the DFT sinusoids to obtain an orthonormal set:

$\displaystyle \tilde{s}_k(n) \isdef \frac{s_k(n)}{\sqrt{N}} = \frac{e^{j2\pi k n /N}}{\sqrt{N}}

The orthonormal sinusoidal basis signals satisfy

$\displaystyle \left<\tilde{s}_k,\tilde{s}_l\right> = \left\{\begin{array}{ll}
1, & k=l \\ [5pt]
0, & k\neq l. \\
\end{array} \right.

We call these the normalized DFT sinusoids. In §6.10 below, we will project signals onto them to obtain the normalized DFT (NDFT).

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The Discrete Fourier Transform (DFT)
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Norm of the DFT Sinusoids