Characterization of discrete linear shift-invariant systems (original) (raw)

Linear time-invariant (LTI) systems are of fundamental importance in classical digital signal processing. LTI systems are linear operators commuting with the time-shift operator. For N-periodic discrete time series the time-shift operator is a circulant N × N permutation matrix. Sandryhaila and Moura developed a linear discrete signal processing framework and corresponding tools for datasets arising from social, biological, and physical networks. In their framework, the circulant permutation matrix is replaced by a network-specific N × N matrix A, called a shift matrix, and the linear shift-invariant (LSI) systems are all N × N matrices H over C commuting with the shift matrix: HA = AH. Sandryhaila and Moura described all those H for the non-degenerate case, in which all eigenspaces of A are one-dimensional. Then the authors reduced the degenerate case to the non-degenerate one. As we show in this paper this reduction does, however, not generally hold, leaving open one gap in the pr...