Background
I have a signal $X \in \mathbb{R}^d$. Let's define the following terms:
- $F_X \in \mathbb{C}^d$ is the discrete Fourier transform of $X$. Each element is a complex number with a Fourier coefficient.
- $F_{X,i}$ is the $i^{th}$ element of $F_X$. $\overline{F_{X,i}}$ is its complex conjugate
- $A_X \in \mathbb{R}^d$ is the amplitude spectrum of $X$
- $A_{X,i}$ is the $i^{th}$ element of $A_{X}$. We know that $A_{X,i}=\sqrt{\overline{F_{X,i}}F_{X,i}}$ (Eq.1).
We also have a constant vector (a filter), $\mathbf{f}\in \mathbb{R}^d$, with all the same notation for its Fourier representation and amplitude spectrum.
Problem
The problem that I want to solve requires that I find $(A_X \cdot A_\mathbf{f})^2$ (the square of the inner product of the two amplitude spectra), with a formula of the following form:
$(A_{X}\cdot A_{\mathbf{f}})^2 = X^T B X$ (Eq.2)
where $B$ is a symmetric matrix. In summary, as the title says, I want to calculate the result of a dot product of Fourier amplitude spectra as a quadratic form in signal space. I am not sure if there is a solution to this problem, so first of all I'd be interested in hearing if there is demonstrably no solution.
Attempted solution:
We expand $(A_{X}\cdot A_{\mathbf{f}})^2 = A_X^T A_{\mathbf{f}} A_{\mathbf{f}}^T A_X = A_X^T B' A_X$, where $B'= A_{\mathbf{f}} A_{\mathbf{f}}^T$.
The Discrete Fourier Transform can be represented by a complex matrix $D$, so that $F_X = D\cdot X$. We also know that $\overline{D}D = I$, and that with $D_i$ being the $i^{th}$ row of $D$,we have (Eq.3) $F_{X,i} = D_i X$ (important to note that $D_i$ is a row vector).
Expanding (Eq.2):
$A_X^T B' A_X = \sum_{i,j} B_{i,j} A_{X,i} A_{X,j}$ (Eq.4)
Substituting (Eq.3) into (Eq.1) and introducing the result into (Eq.4) we obtain:
$A_X^T B' A_X = \sum_{i,j} B_{i,j} \sqrt{X^t \overline{D_i}D_i X} \sqrt{X^t \overline{D_j}D_j X}$
(it is important to note that $\overline{D_i}D_i$ is a square symmetric matrix, because $D_i$ is a row vector). When $i=j$, we obtain terms of the desired form, $B_{i,i} X^t \overline{D_i}D_i X$, which are quadratic forms of $X$.
Now I have to deal with the cases where $i\neq j$. It seems like for these we will not have nice quadratic forms? I tried to see if the $B_{i,j}$ terms are 0 in these cases, but it doesn't look like they are. I thought about grouping the terms in a way that I recover quadratic forms, but that doesn't seem to work.
Any help will be highly appreciated. Thanks!
(PS. In case anyone is interested, I need to do this because I want to the dot product of amplitude spectra in a formula like the one I describe in this other question of mine https://stats.stackexchange.com/questions/605979/expected-value-of-rayleigh-quotient-non-centered-gaussian-vector).