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The Length 2 DFT

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

\begin{eqnarray*}
\underline{s}_0 &=& (1,1) \\
\underline{s}_1 &=& (1,-1)
\end{eqnarray*}

The DFT sinusoid $ \underline{s}_0$ is a sampled constant signal, while $ \underline{s}_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: Graphical interpretation of the length 2 DFT.
\includegraphics[width=\textwidth]{eps/dft2.eps}

Analytically, we compute the DFT to be

\begin{eqnarray*}
X(\omega_0) &=& \left<\underline{x},\underline{s}_0\right> = 6...
...underline{x},\underline{s}_1\right> = 6\cdot 1 + 2\cdot (-1) = 4
\end{eqnarray*}

and the corresponding projections onto the DFT sinusoids are

\begin{eqnarray*}
{\bf P}_{\underline{s}_0}(\underline{x}) &\isdef &
\frac{\left...
...-1)}{1^2 + (-1)^2} \underline{s}_0 = 2 \underline{s}_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 $ \{\underline{s}_0,
\underline{s}_1\}$, and they provide an orthogonal basis set which is rotated $ 45$ degrees relative to the time-domain basis vectors. Projecting orthogonally onto them gives $ X(\omega_0) = (4,4)$ and $ X(\omega_1)=(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*}
(6,2)&=& (4,4)+(2,-2)=4\cdot(1,1)+2\cdot(1,-1)\\
&=& 4\cdot\...
...\,\underline{s}_1\,\right\Vert^2}\underline{s}_1 = \underline{x}
\end{eqnarray*}


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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.

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