Toeplitz Matrix - Discrete Convolution

Discrete Convolution

The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of and can be formulated as:

y = h \ast x = \begin{bmatrix} h_1 & 0 & \ldots & 0 & 0 \\ h_2 & h_1 & \ldots & \vdots & \vdots \\ h_3 & h_2 & \ldots & 0 & 0 \\ \vdots & h_3 & \ldots & h_1 & 0 \\ h_{m-1} & \vdots & \ldots & h_2 & h_1 \\ h_m & h_{m-1} & \vdots & \vdots & h_2 \\ 0 & h_m & \ldots & h_{m-2} & \vdots \\ 0 & 0 & \ldots & h_{m-1} & h_{m-2} \\ \vdots & \vdots & \vdots & h_m & h_{m-1} \\ 0 & 0 & 0 & \ldots & h_m \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{bmatrix}
y^T = \begin{bmatrix} h_1 & h_2 & h_3 & \ldots & h_{m-1} & h_m \end{bmatrix} \begin{bmatrix} x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & 0& \ldots & 0 \\ 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & \ldots & 0 \\ 0 & 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \ldots & 0 \\ 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_n & \vdots \\ 0 & \ldots & 0 & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_n \end{bmatrix}.

This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.

Read more about this topic:  Toeplitz Matrix

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