Let \(x\in \mathbb{C}^n\). The components of the discrete Fourier transform \(\hat{x}\) can be calculated by

\[ \hat{x}(k)=\sqrt{n}\langle x, e_k^*\rangle \]

where \(\langle \cdot, \cdot\rangle\) and \(e_k^*\) are defined here .