The Length 2 DFT
The length DFT is particularly simple, since the basis sinusoids
are real:
![\begin{eqnarray*}
\sv_0 &=& (1,1) \\
\sv_1 &=& (1,-1)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1075.png)
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
![\begin{eqnarray*}
X(\omega_0) &=& \left<\underline{x},\sv_0\right> = 6\cdot 1 + ...
...=& \left<\underline{x},\sv_1\right> = 6\cdot 1 + 2\cdot (-1) = 4
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1078.png)
and the corresponding projections onto the DFT sinusoids are
![\begin{eqnarray*}
{\bf P}_{\sv_0}(\underline{x}) &\isdef &
\frac{\left<\underlin...
...6\cdot 1 + 2 \cdot (-1)}{1^2 + (-1)^2} \sv_1 = 2 \sv_1 = (2,-2).
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1079.png)
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:
![\begin{eqnarray*}
\underline{x}\;=\; (6,2)&=& (4,4)+(2,-2)=4\cdot(1,1)+2\cdot(1,...
..._0
+ \frac{X(\omega_1)}{\left\Vert\,\sv_1\,\right\Vert^2}\sv_1
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1087.png)
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Matrix Formulation of the DFT
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Normalized DFT