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Toeplitz, Circulant matrix and discrete convolution theorem

A topelitz matrix is a matrix with form of $$A=\begin{bmatrix} a & b & c & d & e \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ i & h & g & f & a\end{bmatrix}$$ Generally, the elements are denoted as $a_{i-j}$: $A_{ij}=A_{i+1, j+1}=a_{i-j}$. A circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element: $$C=\begin{bmatrix}c_{0}&c_{n-1}&\dots &c_{2}&c_{1}\\c_{1}&c_{0}&c_{n-1}&&c_{2}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{n-2}&&\ddots &\ddots &c_{n-1}\\c_{n-1}&c_{n-2}&\dots &c_{1}&c_{0}\\\end{bmatrix}$$ Circulant matrices are closely connected to discrete convolution: $$y=h\ast x=\begin{bmatrix}h_{1}&0&\cdots &0&0\\h_{2}&h_{1}&&\vdots &\vdots \\h_{3}&h_{2}&\cdots &0&0\\\vdots &h_{3}&\cdots &h_{1}&0\\h_{