I have a question about using the permutation matrix for performing $iFFT$ for such matrix and then reshape it row-wise and column-wise way.
where $x_k$ represents the column $k$ of matrix $x$ with size of 32 for each column.
$1$- Let's take column wise $iFFT$ for matrix $x$ and then reshape it row-wise resulting matrix $Y$ as below matlab code
x = randn(32,8); y = ifft(x); Y = reshape(y.',,1);
$2$- let's manipulate that using another way of permutation matrix. first we can reshape the matrix $x$ column-wise into (256,1) as below:
$F$ is the Fourier transformation matrix
In order to get the equivalent matrix of Y representing the reshape of $iFFT(x)$ in row-wise, we can multiply the matrix Z with permutation matrix P resulting vector $Y2$ which should be similar to vector $Y$ in case $1$. That can be done by matlab as:
z = reshape(x,,1); % X = kron(eye(8),dftmtx(32)); Z = X*z; P = zeros(m*n); %Building the permutation matrix P col = 1; for i = 1:m for j = 1:n E = zeros(m,n); E(i,j) = 1; P(:,col) = E(:); col = col + 1; end end Y2 = P*Z
So, for $(1)$ and $(2)$, $Y$ should be similar to $Y2$. Is that right? but what I get is different results! My question is that calculation right? why do I get different results ?