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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$ .

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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University