As a well-known fact, a matrix is circulant if and only if it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. In other words, the class of matrices diagonalizable by FFT are circulant matrices.
Recall that the FFT algorithm has $\log_2(N)$ steps, where $N$ is the dimension of the input vector, i.e. $F$ can be written as the composition of
\begin{equation} F = F_1 \circ F_2 \circ \cdots \circ F_{\log(N) - 1} \circ F_{\log_2(N)} \end{equation}
By "generalized FFT", I mean something like
\begin{equation} F_\theta = F_1 \circ D_1 \circ P_1 \circ F_2 \circ \cdots \circ F_{\log_2(N) - 1} \circ D_{\log_2(N)-1} \circ P_{\log_2(N)-1} \circ F_{\log_2(N)} \end{equation}
where $D_i$ is a diagonal matrix whose diagonal elements all have magnitude 1 and $P_i$ is a permutation matrix for $1 \le i \le \log_2(N)-1$. Note that $F_\theta$ is orthogonal (because it's a product of multiple orthogonal matrices) and parameterized by $\theta = \{(D_i, P_i): 1 \le i \le \log_2(N)-1\}$.
Question: what is the class of matrices diagonalizable by "generalized FFT"? In other words, do matrices of the form $F_\theta^{-1} D F_\theta$ have any special structure, besides the fact that they encompass the class of circulant matrices?
