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The Length 2 DFT

The length $ 2$ DFT is particularly simple, since the basis sinusoids are real:

\begin{eqnarray*}
\sv_0 &=& (1,1) \\
\sv_1 &=& (1,-1)
\end{eqnarray*}

The DFT sinusoid $ \sv_0$ is a sampled constant signal, while $ \sv_1$ is a sampled sinusoid at half the sampling rate.

Figure 6.4 illustrates the graphical relationships for the length $ 2$ DFT of the signal $ \underline{x}=[6,2]$ .

Figure 6.4: Graphical interpretation of the length 2 DFT.
\includegraphics[width=\twidth]{eps/dft2}

Analytically, we compute the DFT to be

\begin{eqnarray*}
X(\omega_0) &=& \left<\underline{x},\sv_0\right> = 6\cdot 1 + 2\cdot 1 = 8\\
X(\omega_1) &=& \left<\underline{x},\sv_1\right> = 6\cdot 1 + 2\cdot (-1) = 4
\end{eqnarray*}

and the corresponding projections onto the DFT sinusoids are

\begin{eqnarray*}
{\bf P}_{\sv_0}(\underline{x}) &\isdef &
\frac{\left<\underline{x},\sv_0\right>}{\left<\sv_0,\sv_0\right>} \sv_0 =
\frac{6\cdot 1 + 2 \cdot 1}{1^2 + 1^2} \sv_0 = 4 \sv_0 = (4,4),\mbox{ and}\\
{\bf P}_{\sv_1}(\underline{x}) &\isdef &
\frac{\left<\underline{x},\sv_1\right>}{\left<\sv_1,\sv_1\right>} \sv_1 =
\frac{6\cdot 1 + 2 \cdot (-1)}{1^2 + (-1)^2} \sv_1 = 2 \sv_1 = (2,-2).
\end{eqnarray*}

Note the lines of orthogonal projection illustrated in the figure. The ``time domain'' basis consists of the vectors $ \{\underline{e}_0,\underline{e}_1\}$ , and the orthogonal projections onto them are simply the coordinate axis projections $ (6,0)$ and $ (0,2)$ . The ``frequency domain'' basis vectors are $ \{\sv_0,
\sv_1\}$ , and they provide an orthogonal basis set that is rotated $ 45$ degrees relative to the time-domain basis vectors. Projecting orthogonally onto them gives $ {\bf P}_{\sv_0}(\underline{x}) = (4,4)$ and $ {\bf P}_{\sv_1}(\underline{x}) =(2,-2)$ , respectively. The original signal $ \underline{x}$ 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 $ \underline{x}$ ). Computing the coefficients of projection is essentially ``taking the DFT,'' and constructing $ \underline{x}$ 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,-1)\\
&=& 4\cdot\mbox{dc}+2\cdot\mbox{sinusoid-at-half-the-sampling-rate}\\
&=& \frac{X(\omega_0)}{\left\Vert\,\sv_0\,\right\Vert^2}\sv_0
+ \frac{X(\omega_1)}{\left\Vert\,\sv_1\,\right\Vert^2}\sv_1
\end{eqnarray*}


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA