Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Sponsor

DM6467T DaVinci video processor enables H.264 1080p decoding up to 60 fps/eight D1 channel encoding.
Details Here!

Chapters

See Also

Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Derivation of the Discrete Fourier Transform (DFT)
       Orthogonality of Sinusoids
          Nth Roots of Unity

Chapter Contents:

Search Mathematics of the DFT

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

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=\textwidth]{eps/dftfreqs}

Order a Hardcopy of Mathematics of the DFT

Previous: Orthogonality of Sinusoids
Next: DFT Sinusoids
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


Comments

No comments yet for this page


Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )