The Length 2 DFT

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The Length 2 DFT

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The Length 2 DFT

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

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

Figure 7.4 illustrates the graphical relationships for the length$2$ DFT of the signal $\.

Figure 7.4:Graphical interpretation of the length 2 DFT.
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Analytically, we compute the DFT to be

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Note the lines of orthogonal projection illustrated in the figure. The “time domain” basis consists of the vectors $\, and theorthogonal projections onto them are simply the coordinate projections$(6,0)$ and $(0,2)$. The “frequency domain” basis vectors are $\, 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(\ and $X(\, respectively. The original signal $\ can be expressed as the vector sum of its coordinate projections (a time-domain representation), or as the vector sum of its projections onto the DFT sinusoids (a frequency-domain representation). 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.”

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