# Interpreting FFT Coefficients from System Matrix That Is Originally Toeplitz (Not Circulant)

If I have a measured signal $y$, true signal $x$, and a convolution matrix $A$ that is a Toeplitz but not circulant matrix, I can write the convolution as

$$y = A x \ .$$

However, I would like to analyze the system using diagonalized coefficients of the Fourier transform. A re-formulation such as the one given below is often used to provide a means for fast computation: $$\left[\begin{array}{c} y\\ y' \end{array}\right] = C \left[\begin{array}{c} x\\ 0 \end{array}\right] = \left[\begin{array}{cc} A & B \\ B & A \end{array}\right] \left[\begin{array}{c} x\\ 0 \end{array}\right] = \left[\begin{array}{c} A x \\ B x \end{array}\right] \ ,$$ where $C$ is a circulant matrix into which the Toeplitz matrix $A$ is embedded, using an additional matrix $B$ derived from the values of $A$.

Diagnoalized values of $C$ can be obtained by multiplication with Discrete Fourier Transform matrices. But are these values relevant for interpreting the behavior of the original system, $y = Ax$ - for instance, for use in construction of a Wiener filter for the original system?