My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, I wanted to clarify a conceptual point or proper mathematical nomenclature since most of these signal deconvolution problems are not written by mathematicians. Starting from convolution integral, the typical starting discussion is:
$$ C(E)=\int_{-\infty}^{\infty} W\left(E^{\prime}\right) R\left(E-E^{\prime}\right) \mathrm{d} E^{\prime} $$ with $$ \int_{-\infty}^{\infty} R(E) \mathrm{d} E=1 $$
where $E$ and $E^{\prime}$ denote the radiation energy; $C(E)$, $E(E)$ and $R(E)$ mean the observed spectrum, the intrinsic distribution and the response function, respectively. The goal is to obtain true distribution is obtained from deconvoluting the observed spectrum with the response function.
Most authors then switch to the following matrix equation, $$ \mathrm{C}=R W $$ where $R$ is a square matrix, and $C$ and $W$ are column vectors which correspond to the observed spectrum after background subtraction and to the intrinsic distribution, respectively.
(i) My understanding is that convolution by matrices is a circular convolution and this is done by circulant matrices. Is there a matrix formulation that is meant for linear convolution, or zero padding is the only way to obtain linear convolution out of circulant matrices? I mean is there a special matrix, besides circulant matrix, which performs linear convolution?
(ii) Sometimes the name Toepltiz matrix is used in deconvolution problems. Should we call $R$ a circulant matrix or a Toeplitz matrix?
I would appreciate a readible reference on this topic.
