In the theory of discrete time systems, there are two classes of “most elementary” operators, namely instantaneous or non-dynamic operators which affect only the current input and leave the state undisturbed, and “simple shifts” (unit delays). In our notation, the first class corresponds to diagonal transfer operators (block diagonal matrices, elements of D or matrices for which only the main diagonal is non-zero), whereas simple shifts are represented by Z: a matrix whose only non zero block-entries are identity matrices on the first off-diagonal. With these two basic components, we can set up a “diagonal algebra” which yields expressions that look like those of classical time-invariant system theory. Many results from that theory carry over straightforwardly as well: the notation is not just cosmetically interesting.
KeywordsHilbert Space Discrete Time System Diagonal Operator Simple Shift Slice Representation
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