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Orthogonality of Sinusoids

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

$\displaystyle \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 durations must be infinity (in general).

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 (in Hz)

$\displaystyle 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

$\displaystyle 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.

Nth Roots of Unity

As introduced in §3.12, the complex numbers

$\displaystyle W_N^k \isdef e^{j\omega_k T} \isdef e^{j k 2\pi (f_s/N) T} = e^{j k 2\pi/N}, \quad k=0,1,2,\ldots,N-1, $

are called the $ N$th roots of unity because each of them satisfies

$\displaystyle \left[W_N^k\right]^N = \left[e^{j\omega_k T}\right]^N = \left[e^{j k 2\pi/N}\right]^N = e^{j k 2\pi} = 1. $

In particular, $ W_N$ is called a primitive $ N$th root of unity.6.2

The $ N$th roots of unity are plotted in the complex plane in Fig.6.1 for $ N=8$. It is easy to find them graphically by dividing the unit circle into $ N$ equal parts using $ N$ points, with one point anchored at $ z=1$, as indicated in Fig.6.1. When $ N$ is even, there will be a point at $ z=-1$ (corresponding to a sinusoid with frequency at exactly half the sampling rate), while if $ N$ is odd, there is no point at $ z=-1$.

Figure 6.1: The $ N$ roots of unity for $ N=8$.
\includegraphics[width=\twidth]{eps/dftfreqs}

DFT Sinusoids

The sampled sinusoids generated by integer powers of the $ N$ roots of unity are plotted in Fig.6.2. These are the sampled sinusoids $ (W_N^k)^n = e^{j 2 \pi k n / N} = e^{j\omega_k nT}$ used by the DFT. Note that taking successively higher integer powers of the point $ W_N^k$ on the unit circle generates samples of the $ k$th DFT sinusoid, giving $ [W_N^k]^n$, $ n=0,1,2,\ldots,N-1$. The $ k$th sinusoid generator $ W_N^k$ is in turn the $ k$th $ N$th root of unity ($ k$th power of the primitive $ N$th root of unity $ W_N$).

Figure 6.2: Complex sinusoids used by the DFT for $ N=8$.
\includegraphics[width=\twidth]{eps/dftsines}

Note that in Fig.6.2 the range of $ k$ is taken to be $ [-N/2,N/2-1] = [-4,3]$ instead of $ [0,N-1]=[0,7]$. This is the most ``physical'' choice since it corresponds with our notion of ``negative frequencies.'' However, we may add any integer multiple of $ N$ to $ k$ without changing the sinusoid indexed by $ k$. In other words, $ k\pm mN$ refers to the same sinusoid $ \exp(j\omega_k nT)$ for all integers $ m$.

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Orthogonality of the DFT Sinusoids
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