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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 + ...
...=& \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<\underlin...
...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,...
..._0
+ \frac{X(\omega_1)}{\left\Vert\,\sv_1\,\right\Vert^2}\sv_1
\end{eqnarray*}


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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.


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