I have a Fourier matrix $F$ with size $N \times N$, such that $y = F \times x$, if I have the vector $x$ contains four identical parts, for example $x = [x_1, x_2,x_3,x_4]’$ and $x_1 = x_2 = x_3 = x_4$ can I simplify the multiplication $y = F \times x$ into smaller matrix to reduce the complexity? . For example use smaller matrix of Fourier matrix $F$ to be multiplied only with $x_1$ or any other way.
I am talking here about the complexity of radix-2 FFT, which is $O(N log_2N)$
I have the vector $x_1$ which is repeated four times yielding $x = [x_1, x_1,x_1,x_1]^T$, and I need to perform $y = F \times x$, can the complexity in that case be reduced compared if $x_1$ is not repeating?