Orthogonality of Sinusoids

• Sinusoids are orthogonal at different frequencies if their durations are infinite.
• For length $N$ sampled sinusoidal segments, orthogonality holds for the harmonics of the sampling rate divided by N, that is for frequencies
  $$f_k = k\frac{f_s}{N}, k=0,1,2,3,...,N-1.$$

• These are the only frequencies that have a whole number of periods in $N$ samples.
• The complex sinusoids corresponding to the frequencies $f_k$ are
  $$s_k(n)\triangleq e^{j\omega_k nT},\omega_k\triangleq k\frac{2\pi}{N}f_s, k=0,1,2,..., N-1.$$
  These sinusoids are generated by the $N$th roots of unity in the complex plane, so called since
  $$[e^{j\omega_k T}]^N = [e^{jk2\pi/N}]^N = e^{jk2\pi} = 1.$$