Non-Existence of Convolution Sum System Representations

Convolution sum system representations are commonly used in signal processing. It is known that the convolution sum, treated as the limit of its partial sums, can be divergent for certain continuous signals and stable linear time-invariant (LTI) systems, even when the convergence of the partial sums is treated in a distributional setting. In this paper, we ask a far more general question: is it at all possible to define a generalized convolution sum with natural properties that works for all absolutely integrable continuous signals that vanish at infinity and all stable LTI systems? We prove that the answer is 'no.' Further, for certain subspaces, we give a sufficient and necessary condition for uniform convergence. Finally, we discuss the implications of our results on the effectiveness of window functions in the convolution sum.