The Danielson-Lanczos Lemma Expanded to 8 Terms
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Equation 10
After the expansion above you can plug in for N = 8 samples, and k=0, since all summation would be unity when N=8.
Equation 11
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A DFT and FFT TUTORIAL
A DFT is a "Discrete Fourier Transform". An FFT is a "Fast Fourier Transform". An FFT is a DFT, but is much faster for calculations. The whole point of the FFT is speed in calculating a DFT.