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 6.4: Graphical interpretation of the length 2 DFT.

Analytically, we compute the DFT to be

$$\begin{eqnarray*} X(\omega_0) &=& \left<\underline{x},\underline{s}_0\right> = 6\cdot 1 + 2\cdot 1 = 8\\ X(\omega_1) &=& \left<\underline{x},\underline{s}_1\right> = 6\cdot 1 + 2\cdot (-1) = 4 \end{eqnarray*}$$

and the corresponding projections onto the DFT sinusoids are

$$\begin{eqnarray*} {\bf P}_{\underline{s}_0}(\underline{x}) &\isdef & \frac{\left<\underline{x},\underline{s}_0\right>}{\left<\underline{s}_0,\underline{s}_0\right>} \underline{s}_0 = \frac{6\cdot 1 + 2 \cdot 1}{1^2 + 1^2} \underline{s}_0 = 4 \underline{s}_0 = (4,4),\mbox{ and}\\ {\bf P}_{\underline{s}_1}(\underline{x}) &\isdef & \frac{\left<\underline{x},\underline{s}_1\right>}{\left<\underline{s}_1,\underline{s}_1\right>} \underline{s}_1 = \frac{6\cdot 1 + 2 \cdot (-1)}{1^2 + (-1)^2} \underline{s}_1 = 2 \underline{s}_1 = (2,-2). \end{eqnarray*}$$

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 that is rotated $45$ degrees relative to the time-domain basis vectors. Projecting orthogonally onto them gives ${\bf P}_{\underline{s}_0}(\underline{x}) = (4,4)$ and ${\bf P}_{\underline{s}_1}(\underline{x}) =(2,-2)$ , 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:

$$\begin{eqnarray*} \underline{x}\;=\; (6,2)&=& (4,4)+(2,-2)=4\cdot(1,1)+2\cdot(1,-1)\\ &=& 4\cdot\mbox{dc}+2\cdot\mbox{sinusoid-at-half-the-sampling-rate}\\ &=& \frac{X(\omega_0)}{\left\Vert\,\underline{s}_0\,\right\Vert^2}\underline{s}_0 + \frac{X(\omega_1)}{\left\Vert\,\underline{s}_1\,\right\Vert^2}\underline{s}_1 \end{eqnarray*}$$