Simplification of Sequences of Operators

The output of a system is often expressed as a sequence of unary and/or binary operations on the input. Analysis or design considerations may require manipulation of the sequence of operators to obtain an equivalent sequence either mathematically simpler or easier to design. For this purpose, available operator algebra must be considerably expanded. Several new operators are introduced: the multiple operator which is a combination of unary operators, and the zero-convolution and zero-correlation operators. Three commutation tables are developed, one for unary and transform operators, one for binary operators, and one for binary operators operating on two functions already transformed by unary or transform operators. The commutation tables can be used to manipulate and simplify a sequence of operators. They should be useful in the analysis and design of complex systems. This becomes quite clear when one tries to repeat the examples of this paper without using the commutation tables. The simplification of sequence of operators procedure not only permits a faster simplification of multiple integral transformations, but it also provides such a clear picture of the operations to perform that the manipulations become obvious. This contrasts with the conventional technique of rearranging the integrals and performing changes of variables somewhat blindly. Since the purpose of this paper is to make available a practical tool, most of the derivations of the formulas were left out for conciseness.