Audio and DSP

The DFT:

(1)   \begin{align*}X_k = \sum_{n=0}^{N-1} x[n] e^{2\pi i k n /N}\end{align*}

where x[n] is an N-length sampled sequence, is a wonderful thing.

(2)   \begin{align*} X_k = \langle x[\cdot],\phi_k \rangle \end{align*}