## The Discrete Fourier Transform (DFT)

Given a signal
, its DFT is defined
by^{6.3}

*spectrum*of , and is the th

*sample*of the spectrum at frequency . Thus, the th sample of the spectrum of is defined as the inner product of with the th DFT sinusoid . This definition is times the

*coefficient of projection*of onto ,

*i.e.*,

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.

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Frequencies in the ``Cracks''

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An Orthonormal Sinusoidal Set