DFT Sinusoids

• The $N$th roots of unity are plotted below for $N=8$.
  
• The sampled sinusoids corresponding to the $N$ roots of unity are given by $(e^{j\omega_kT})^n = e^{j2\pi kn/N}$, and are used by the DFT.
• Taking successively higher integer powers of the root $e^{j\omega_kT}$ on the unit circle, generates samples of the $k$th DFT sinusoid.
• Since each sinusoid is of a different frequency and each is a harmonic of the sampling rate divided by $N$, the DFT sinusoids are orthogonal.