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The Discrete Fourier Transform (DFT)

Given a signal $ x(\cdot)\in{\bf C}^N$, its DFT is defined by6.3

$\displaystyle X(\omega_k) \isdef \left<x,s_k\right> \isdef \sum_{n=0}^{N-1}x(n) \overline{s_k(n)},
\quad k=0,1,2,\ldots,N-1,


$\displaystyle s_k(n)\isdef e^{j\omega_k t_n},
\quad t_n\isdef nT,
\quad \omega_k\isdef 2\pi\frac{k}{N}f_s,
\quad f_s\isdef \frac{1}{T},

or, as it is most often written,

$\displaystyle \zbox {X(\omega_k) \isdef \sum_{n=0}^{N-1}x(n) e^{-j\frac{2\pi k n}{N}},
\quad k=0,1,2,\ldots,N-1.}

We may also refer to $ X$ as the spectrum of $ x$, and $ X(\omega_k)$ is the $ k$th sample of the spectrum at frequency $ \omega_k$. Thus, the $ k$th sample $ X(\omega_k)$ of the spectrum of $ x$ is defined as the inner product of $ x$ with the $ k$th DFT sinusoid $ s_k$. This definition is $ N$ times the coefficient of projection of $ x$ onto $ s_k$, i.e.,

$\displaystyle \frac{\left<x,s_k\right>}{\left\Vert\,s_k\,\right\Vert^2} = \frac{X(\omega_k)}{N}.

The projection of $ x$ onto $ s_k$ is

$\displaystyle {\bf P}_{s_k}(x) = \frac{X(\omega_k)}{N} s_k.

Since the $ \{s_k\}$ are orthogonal and span $ {\bf C}^N$, using the main result of the preceding chapter, we have that the inverse DFT is given by the sum of the projections

$\displaystyle x = \sum_{k=0}^{N-1}\frac{X(\omega_k)}{N} s_k,

or, as we normally write,

$\displaystyle \zbox {x(n) = \frac{1}{N} \sum_{k=0}^{N-1}X(\omega_k) e^{j\frac{2\pi k n}{N}}, \quad n=0,1,\ldots,N-1.} \protect$ (6.1)

In summary, the DFT is proportional to the set of coefficients of projection onto the sinusoidal basis set, and the inverse DFT is the reconstruction of the original signal as a superposition of its sinusoidal projections. This basic ``architecture'' extends to all linear orthogonal transforms, including wavelets, Fourier transforms, Fourier series, the discrete-time Fourier transform (DTFT), and certain short-time Fourier transforms (STFT). See Appendix B for some of these.

We have defined the DFT from a geometric signal theory point of view, building on the preceding chapter. See §7.1.1 for notation and terminology associated with the DFT.

Previous: An Orthonormal Sinusoidal Set
Next: Frequencies in the ``Cracks''

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About the Author: 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 (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 for details.


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