Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 40E / d

Here ${\omega }_4=i$. Splitting into even and odd pairs giveswhose two-point inverse discrete Fourier transforms are $(2,0)$ and $(-2,0)$. The butterfly therefore yields

Now $y^{(E)}=(1,-1)$ and $y^{(O)}=(1,-1)$, and both two-point transforms equal $(0,2)$. The twiddle factors $1,i$ give