Mathematics of the Discrete Fourier Transform (DFT)



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Vector Addition

Given two vectors in ${\, say $\ and $\, the vector sum is defined byelementwise addition. If we denote the sum by $w\, then we have $w(n) = x(n)+y(n)$ for $n=0,1,2,\.

The vector diagram for the sum of two vectors can be found using the parallelogram rule, as shown in Fig. 6.2 for $N=2$, $x=(2, 3)$, and$y=(4,1)$.

Figure 6.2:Geometric interpretation of a length 2 vector sum.
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Also shown are the lighter construction lines which complete the parallelogram started by $x$ and $y$, indicating where the endpoint of the sum $x+y$ lies. Since it is a parallelogram, the two construction lines are congruent to the vectors $x$ and $y$. As a result, the vector sum is often expressed as a triangle by translating the origin of one member of the sum to the tip of the other, as shown in Fig. 6.3.

Figure 6.3:Vector sum with translation of one vector to the tip of the other.
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In the figure, $x$ was translated to the tip of $y$. It is equally valid to translate $y$ to the tip of $x$, because vector addition is commutative, i.e., $x+y$ = $y+x$.

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