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1

Assuming we know how to solve: $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| C \boldsymbol{x} - \boldsymbol{b} \right\|}_{2}^{2} + {\left\| E \boldsymbol{x} \right\|}_{1} $$ For any matrix $ E $ one could see that: $$ \boldsymbol{w} \circ D \boldsymbol{x} = \operatorname{Diag} \left( \boldsymbol{w} \right) D \boldsymbol{x} = E \boldsymbol{x} $$ Where $ \...


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Here's an answer from me in a video form: Circular vs. Linear Convolution on YouTube and a blog post: Circular vs. Linear Convolution on TheWolfSound.com


3

The way to build the matrix is playing with indices of the signal data and the convolution kernel. For example: function [ mK ] = CreateConvMtx1D( vK, numElements, convShape ) % ----------------------------------------------------------------------------------------------- % % [ mK ] = CreateConvMtx1D( vK, numElements, convShape ) % Generates a Convolution ...


1

A convolution matrix is really just a diagonal band-structure matrix, where every row is all zeros but for the elements around the diagonal, which are identical (but shifted) for every row: the elements of the kernel.


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Doing direct division in Frequency Domain means you are assuming cyclic / periodic boundary conditions for your data which I don't think is the proper assumption for your data. The model I'd pursue would be as I described in my answer to Deconvolution of Synthetic 1D Signals - How To? As one can see in that question the marked answer by @Hilmar is wrong. ...


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