An Orthonormal Sinusoidal Set

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

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

The orthonormal sinusoidal basis signals satisfy

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

We call these the normalized DFT sinusoids.