2-Round Radix-2 IFFT

\(\vdots\)

\(P_0\)

\(P_{\frac{N}{2}}\)

\(P_0 + P_{\frac{N}{2}}\)

\(P_0 - P_{\frac{N}{2}}\)

\(P_2\)

\(P_{\frac{N}{2}+ 2}\)

\(P_2 + P_{\frac{N}{2}+2}\)

\(P_2 - P_{\frac{N}{2}+2}\)

\(P_{2i}\)

\(P_{\frac{N}{2}+2i}\)

\(P_{2i} + P_{\frac{N}{2}+2i}\)

\(P_{2i} - P_{\frac{N}{2}+2i}\)

\(P_{2i+2}\)

\(P_{\frac{N}{2}+2i+2}\)

\(P_{2i+2} + P_{\frac{N}{2}+2i+2}\)

\(P_{2i+2} - P_{\frac{N}{2}+2i+2}\)

\(\left(P_0 + P_{\frac{N}{2}}\right) + \left(P_2 + P_{\frac{N}{2}+2}\right)\)

\(\left(P_0 - P_{\frac{N}{2}}\right) + \omega_4^{-1}\left(P_2 - P_{\frac{N}{2}+2}\right)\)

\(\left(P_0 + P_{\frac{N}{2}}\right) - \left(P_2 + P_{\frac{N}{2}+2}\right)\)

\(\left(P_0 - P_{\frac{N}{2}}\right) - \omega_4^{-1}\left(P_2 - P_{\frac{N}{2}+2}\right)\)

\(\left(P_{2i} + P_{\frac{N}{2}+2i}\right) + \left(P_{2i+2} + P_{\frac{N}{2}+2i+2}\right)\)

\(\left(P_{2i} - P_{\frac{N}{2}+2i}\right) + \omega_4^{-1}\left(P_{2i+2} - P_{\frac{N}{2}+2i+2}\right)\)

\(\left(P_{2i} + P_{\frac{N}{2}+2i}\right) - \left(P_{2i+2} + P_{\frac{N}{2}+2i+2}\right)\)

\(\left(P_{2i} - P_{\frac{N}{2}+2i}\right) - \omega_4^{-1}\left(P_{2i+2} - P_{\frac{N}{2}+2i+2}\right)\)

\(\omega_4^{-1}\)

\(\omega_4^{0}\)

\(-1\)

Roots of Unity

\(1\)

\(\omega\)

\(\omega^2\)

\(\omega^3\)

\(\omega^4\)

\(\omega^5\)

\(\omega^6\)

\(\omega^7\)

\(\omega^8\)

\(\omega^9\)

\(\omega^{10}\)

\(\omega^{11}\)

\(\omega^{12}\)

\(\omega^{13}\)

\(\omega^{14}\)

\(\omega^{15}\)

\(= \omega'\)

\((\omega')^2 = \)

\((\omega')^3 = \)

2-Round Radix-2 IFFT for \(N=4n\)

\begin{aligned} p'_i &= \left(P_{2i} + P_{2n+2i}\right) + \left(P_{2i+2} + P_{2n+2i+2}\right) \\ p'_{i+1} &= \left(P_{2i} - P_{2n+2i}\right) + \omega_4^{-1}\left(P_{2i+2} - P_{2n+2i+2}\right) \\ p'_{i+2} &= \left(P_{2i} + P_{2n+2i}\right) - \left(P_{2i+2} + P_{2n+2i+2}\right)\\ p'_{i+3} &= \left(P_{2i} - P_{2n+2i}\right) - \omega_4^{-1}\left(P_{2i+2} - P_{2n+2i+2}\right) \end{aligned}

For \(i \in \{0,1,\dots, n-1\}\)

\begin{aligned} p'_{2n+i} &= \left(P_{2i+1} + P_{2n+2i+1}\right) + \left(P_{2i+3} + P_{2n+2i+3}\right) \\ p'_{2n+i+1} &= \left(P_{2i+1} - P_{2n+2i+1}\right) + \omega_4^{-1}\left(P_{2i+3} - P_{2n+2i+3}\right) \\ p'_{2n+i+2} &= \left(P_{2i+1} + P_{2n+2i+1}\right) - \left(P_{2i+3} + P_{2n+2i+3}\right)\\ p'_{2n+i+3} &= \left(P_{2i+1} - P_{2n+2i+1}\right) - \omega_4^{-1}\left(P_{2i+3} - P_{2n+2i+3}\right) \end{aligned}

\(0 \quad 1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7 \quad 8 \quad 9 \quad 10 \quad 11 \quad 12 \quad 13 \quad 14 \quad 15\)

2-Round Radix-2 IFFT for \(n=8\)

\(p'_0 = \left(P_0 + P_{4}\right) + \left(P_2 + P_{6}\right) = 4p_0 - p_2 + p_4 - p_6\)

\(p'_1 = \left(P_0 - P_{4}\right) + \omega^{-2}\left(P_2 - P_{6}\right) = 4(p_1+p_5) + (p_2 + p_4 + p_6)\)

\(p'_2 = \left(P_0 + P_{4}\right) - \left(P_2 + P_{6}\right)= 3p_2 + p_4 + 3p_6\)

\(p'_3 = \left(P_0 - P_{4}\right) - \omega^{-2}\left(P_2 - P_{6}\right) = 4(p_3+p_7) + (p_2 + p_4 + p_6)\)

\(p'_4 = \left(P_{1} + P_{5}\right) + \left(P_{3} + P_{7}\right) = 4(p_0 - p_4)\)

For simplicity, let \(\omega_8 = \omega\).

\(p'_5 = \left(P_{1} - P_{5}\right) + \omega^{-2}\left(P_{3} - P_{7}\right) = 4\omega (p_1 - p_5)\)

\(p'_6 = \left(P_{1} + P_{5}\right) - \left(P_{3} + P_{7}\right) = 4\omega^2 (p_2 - p_6)\)

\(p'_7 = \left(P_{1} - P_{5}\right) - \omega^{-2}\left(P_{3} - P_{7}\right) = 4 \omega^3(p_3 - p_7)\)