... iid

The letters iid stand for independently and identically distributed meaning that each of the entries of X at a given position t are independent of X at any other position t' and that the distribution of X does not depend on position.

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... sparsifying

Data is sparse on a basis when this basis allows one to describe that signal with a small number of coefficients. This is a highly desirable property, since noise is not expected to be sparse at the same time in such a basis. Choosing a sparsifying basis thus allows one to enhance signal to noise ratio.

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