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
.
Figure 6.4:
Graphical interpretation of the length 2 DFT.

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 timedomain 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 timedomain representation), or as the
vector sum of its projections onto the DFT sinusoids (a
frequencydomain representation of the timedomain 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:
Next Section: Matrix Formulation of the DFTPrevious Section: Normalized DFT