Mathematics of the Discrete Fourier Transform (DFT)



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Orthogonality

The vectors (signals) $x$ and $y$ are said to be orthogonal if $\, denoted $x\. That is to say

\

Note that if $x$ and $y$ are real and orthogonal, the cosine of the angle between them is zero. In plane geometry ($N=2$), the angle between twoperpendicular lines is $\, and $\, as expected. More generally, orthogonality corresponds to the fact that two vectors in$N$-space intersect at a right angle and are thus perpendiculargeometrically.

Example ($N=2$):

Let $x=[1,1]$ and $y=[1,-1]$, as shown in Fig. 6.8.

Figure 6.8:Example of two orthogonal vectors for $N=2$.
\

The inner product is $\. This shows that the vectors are orthogonal. As marked in the figure, the lines intersect at a right angle and are therefore perpendicular.

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