Inverse DFT

The inverse DFT is given by

$$x(t_n)=\sum_{k=0}^{N-1}\frac{\left<x,s_k\right>}{\left\Vert\,s_k\,\right\Vert^2} s_k(t_n) = \frac{1}{N}\sum_{k=0}^{N-1}X(\omega_k)e^{j\omega_kt_n} \hspace{1cm}$$

It can be interpreted as the superposition of the projections, i.e., the sum of the sinusoidal basis signals weighted by their respective coefficients of projection:

$$\zbox{x = \sum_k \frac{\left<x,s_k\right>}{\left\Vert\,s_k\,\right\Vert^2} s_k}$$

The inverse normalized DFT is cleaner:

$$\zbox{x = \sum_k \left<x,\tilde{s}_k\right>\tilde{s}_k}$$