Orthogonality of Sinusoids

A key property of sinusoids is that they are orthogonal at different frequencies. That is,

$$\omega_1 \neq \omega_2 \implies A_1\sin(\omega_1 t + \phi_1) \perp A_2\sin(\omega_2 t + \phi_2)$$

This is true whether they are complex or real, and whatever amplitude and phase they may have. All that matters is that the frequencies be different. Note, however, that the sinusoidal durations must be infinity.

For length $N$ sampled sinusoidal signal segments, such as used by the DFT, exact orthogonality holds only for the harmonics of the sampling rate divided by $N$, i.e., only for the frequencies

$$f_k = k \frac{f_s}{N}, \quad k=0,1,2,3,\ldots,N-1.$$

These are the only frequencies that have a whole number of periods in $N$ samples (depicted in Fig. 6.2 for $N=8$).^6.1

The complex sinusoids corresponding to the frequencies $f_k$ are

$$s_k(n) \isdef e^{j\omega_k nT}, \quad \omega_k \isdef k \frac{2\pi}{N}f_s, \quad k = 0,1,2,\ldots,N-1.$$

These sinusoids are generated by the $N$th roots of unity in the complex plane.