Given a signal , its DFT is defined by6.3
or, as it is most often written,
We may also refer to as the 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.,
The projection of onto is
Since the are orthogonal and span , using the main result of the preceding chapter, we have that the inverse DFT is given by the sum of the projections
or, as we normally write,
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.