<s>[INST] What is the Permutation Matrix in FFT DFT Factorization? Given: 
$$F_N = \frac{1}{\sqrt{N}} \begin{bmatrix}
1&1&1&1&\cdots &1 \\
1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\
1&\omega^2&\omega^4&\omega^6&\cdots&\omega^{2(N-1)}\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^{3(N-1)}\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
1&\omega^{N-1}&\omega^{2(N-1)}&\omega^{3(N-1)}&\cdots&\omega^{(N-1)(N-1)}\\
\end{bmatrix}, $$

where $D = $ diag($1,\omega,\omega^2,...,\omega^N$).  We note that $\omega^N = 1$, so $\omega^{N/2} = -1 $.
What is meant by odd-even permutation in this context?  
Available notes on FFT online are generally highly condensed, thus they are very difficult to decipher.  Using $F_4$ as a template, i've found that permutation matrices $P$ such that $P F$ is row sorted into odd powers of $\omega$ and then even powers of $\omega$, and visa versa, don't ostensibly work.   
I additionally am not sure how you can convert the consecutive $F_{N/2}$ matrices on the top of the second matrix into rows that span to all necessary powers of $\omega$.  
Thank you.
 [/INST] The permutation matrix P for FFT of $F_8$ is
$
P=\begin{bmatrix}
1&0&0&0&0&0&0&0\\
0&0&1&0&0&0&0&0\\
0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&1&0\\
0&1&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0\\
0&0&0&0&0&1&0&0\\
0&0&0&0&0&0&0&1
\end{bmatrix}
$
</s>