# Mathematics of the DFTDerivation of the Discrete Fourier Transform (DFT)The Length 2 DFT

Chapter Contents:

Search Mathematics of the DFT

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

## 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 that 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:

Previous: Normalized DFT
Next: Matrix Formulation of the DFT

#### Order a Hardcopy of Mathematics of the DFT

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 Associate Professor (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 http://ccrma.stanford.edu/~jos/ for details.