In this paper, we investigate impulse response sequences over the integers by presenting their generating functions and expressions. We also establish some of the corresponding identities. In addition, we give the relationship between an impulse response sequence and all linear recurring sequences satisfying the same linear recurrence relation, which can be used to transfer the identities among different sequences. Finally, we discuss some applications of impulse response sequences to the structure of Stirling numbers of the second kind, the Wythoff array, and the Boustrophedon transform.