Discrete-Time LTI Systems Beyond Convolution

Examples for LTI systems are found in the literature that cannot be represented as a convolution. Their outputs can be approximated by outputs of FIR filters and considered as generalized convolution systems. These examples illustrate that impulse and frequency response provide no complete description of the system. In this paper, a general theory for discrete-time LTI systems is represented. LTI systems are defined on a signal space, which is a vector space, closed with respect to a shift operation. Signals are not necessarily bounded and need not belong to a normed vector space. Vector space concepts like dependent and independent vectors are transferred to signal spaces in order to define arbitrary LTI systems. A first method, defining LTI systems by independent input signals, shows that impulse and frequency response can be defined independently from each other. According to another method, the signal space is extended by a new input. An equation is given describing all possible outputs belonging to the new input. Extending the signal space to all discrete-time signals reveals universal LTI systems, which are not even generalized convolution systems.