2-Round Radix-2 IFFT
\(\vdots\)
\(\vdots\)
\(P_0\)
\(P_{\frac{N}{2}}\)
\(P_0 + P_{\frac{N}{2}}\)
\(P_0 - P_{\frac{N}{2}}\)
\(P_2\)
\(P_{\frac{N}{2}+ 2}\)
\(P_2 + P_{\frac{N}{2}+2}\)
\(P_2 - P_{\frac{N}{2}+2}\)
\(P_{2i}\)
\(P_{\frac{N}{2}+2i}\)
\(P_{2i} + P_{\frac{N}{2}+2i}\)
\(P_{2i} - P_{\frac{N}{2}+2i}\)
\(P_{2i+2}\)
\(P_{\frac{N}{2}+2i+2}\)
\(P_{2i+2} + P_{\frac{N}{2}+2i+2}\)
\(P_{2i+2} - P_{\frac{N}{2}+2i+2}\)
\(\left(P_0 + P_{\frac{N}{2}}\right) + \left(P_2 + P_{\frac{N}{2}+2}\right)\)
\(\left(P_0 - P_{\frac{N}{2}}\right) + \omega_4^{-1}\left(P_2 - P_{\frac{N}{2}+2}\right)\)
\(\left(P_0 + P_{\frac{N}{2}}\right) - \left(P_2 + P_{\frac{N}{2}+2}\right)\)
\(\left(P_0 - P_{\frac{N}{2}}\right) - \omega_4^{-1}\left(P_2 - P_{\frac{N}{2}+2}\right)\)
\(\left(P_{2i} + P_{\frac{N}{2}+2i}\right) + \left(P_{2i+2} + P_{\frac{N}{2}+2i+2}\right)\)
\(\left(P_{2i} - P_{\frac{N}{2}+2i}\right) + \omega_4^{-1}\left(P_{2i+2} - P_{\frac{N}{2}+2i+2}\right)\)
\(\left(P_{2i} + P_{\frac{N}{2}+2i}\right) - \left(P_{2i+2} + P_{\frac{N}{2}+2i+2}\right)\)
\(\left(P_{2i} - P_{\frac{N}{2}+2i}\right) - \omega_4^{-1}\left(P_{2i+2} - P_{\frac{N}{2}+2i+2}\right)\)
\(\omega_4^{-1}\)
\(\omega_4^{0}\)
\(-1\)
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 \(\omega_8 = \omega\).
\(p'_5 = \left(P_{1} - P_{5}\right) + \omega^{-2}\left(P_{3} - P_{7}\right) = 4\omega (p_1 - p_5)\)
\(p'_6 = \left(P_{1} + P_{5}\right) - \left(P_{3} + P_{7}\right) = 4\omega^2 (p_2 - p_6)\)
\(p'_7 = \left(P_{1} - P_{5}\right) - \omega^{-2}\left(P_{3} - P_{7}\right) = 4 \omega^3(p_3 - p_7)\)
Radix-2 FFT
By Suyash Bagad
Radix-2 FFT
- 55
