DFT, Calculating Transform.

Cuthbert Nyack
Consider 2 sequences x1(n) and x2(n) given by:-
Using the Equation for the DFT, the real and imaginary parts of the transform of x1(n) and x2(n) are given by the equations below. The real parts are even and the imaginary parts are odd.
The sequence x1(n) + x2(n) = {0, 1, 1, 0, 0, 0, 1, 1} is even and its transform is real and even. On the other hand, the sequence x1(n) - x2(n) = {0, 1, 1, 0, 0, 0, -1, -1} is odd and its transform is imaginary and odd. Both of these are illustrated by the equations below.
Calculation of the DFT for 8 samples is shown in the applet below. Use is made of the symmetry of the W factors ie for n ³ 8, W(n) = W(n mod 8).

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COPYRIGHT 2007 Cuthbert Nyack.