Matrix Formulation of the DFT
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# The Length 2 DFT

The length DFT is particularly simple, since the basis sinusoids are real:

The DFT sinusoid is a sampled constant signal, while is a sampled sinusoid at half the sampling rate.

Figure 6.4 illustrates the graphical relationships for the length DFT of the signal .

Analytically, we compute the DFT to be

and the corresponding projections onto the DFT sinusoids are

Note the lines of orthogonal projection illustrated in the figure. The time domain'' basis consists of the vectors , and the orthogonal projections onto them are simply the coordinate axis projections and . The frequency domain'' basis vectors are , and they provide an orthogonal basis set which is rotated degrees relative to the time-domain basis vectors. Projecting orthogonally onto them gives and , respectively. The original signal can be expressed either as the vector sum of its coordinate projections (0,...,x(i),...,0), (a time-domain representation), or as the vector sum of its projections onto the DFT sinusoids (a frequency-domain representation of the time-domain signal ). Computing the coefficients of projection is essentially taking the DFT'' and constructing as the vector sum of its projections onto the DFT sinusoids amounts to taking the inverse DFT.''

In summary, the oblique coordinates in Fig. 6.4 are interpreted as follows:

Matrix Formulation of the DFT
Normalized DFT
The DFT Derived   Index   Search

Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.

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