Normalized DFT

The DFT Derived

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

``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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