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I am reading the paper "Algebraic Signal Processing Theory: Foundation and 1-D Time". In page one, the first paragraph of second column, I did not get why shift invariance forces the algebra $A$ to be commutative? Is it taking about a particular signal model $(A,M,\Phi)$ or it is considered a general case?

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The central notion is that the considered algebras are generated by a shift operator $\mathcal{S}$. Every element (filter) of the algebra should be (by definition) generated by a (finite) number of algebraic operations (addition, product, scalar multiplication) involving the generator and its inverse. This is explained in more details in Section II-A, around Equation (30), and footnote 19. So if the shift commutes with any filter, since any filter is made of a combination of shifts and inverse shifts, then any filter commutes with any filter.

To me, this is really related to the subset of shift generated algebras they consider (and not on the module of signals).

For instance, with standard discrete filters this unit element would be the Kronecker $\delta_0$ (the "unit" algebra element relative to the multiplicative $\cdot$ operation of the algebra, here the convolution). A (finite support) filter would be written as sums and multiplies of shifts, with scalars corresponding to filter taps: $$h_0S^0(\delta_0)+h_1S(\delta_0)+h_2S^2(\delta_0)+\cdots$$ and commute with basic shifts.

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  • $\begingroup$ Thanks for your attention. Actually, I am positive with your idea. Just some points need to be noticed. 1) Every singly generated algebra is commutative. So, I do not get why it is emphasized in the paper that, when the operator is shift-invariance then the algebra $A$ is commutative. 2) In abstract case, in an algebra, what is meaning of shift operator? shift-invariance operator?!! $\endgroup$ Dec 14 '20 at 6:32
  • $\begingroup$ I shall have a deeper look, my familiarity with algebras has vanished over the years. However, I understand that they use an algebraic framework to 1) unify concepts that are generally taught separately 2) unveil connections and tools inherited from mathematical structures 3) devise novel algorithmic frameworks. More to come $\endgroup$ Dec 14 '20 at 8:48

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