System with complex-valued inputs and outputs

What we have defined above is a real-valued system; that is, one that has real-valued inputs and outputs. Presumably such a system internally manipulates real-valued signals.

Let's don't change the system, but define the output of the system assuming that the input is complex-valued:

It is very convenient to have a definition of the system for complex-valued inputs, even though in practice we don't plan to use complex-valued inputs. The utility is similar to the notion of complex-valued eigenvalues for a matrix, even though the matrix is real-valued.

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