The DFT sinusoid is a sampled constant signal, while is a sampled sinusoid at half the sampling rate.
Figure 7.4 illustrates the graphical relationships for the length DFT of the signal .
Figure 7.4:Graphical interpretation of the length 2 DFT.
Analytically, we compute the DFT to be
Note the lines of orthogonal projection illustrated in the figure. The “time domain” basis consists of the vectors , and theorthogonal projections onto them are simply the coordinate 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 as the vector sum of its coordinate projections (a time-domain representation), or as the vector sum of its projections onto the DFT sinusoids (a frequency-domain representation). 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.”