APPENDIX D Z-Transform The z-transform is used for sampled signals. It is analogous to the s-transform used with continuous or analog signals. Let's use x k and y k to represent the input and output signals to a digital filter, respectively. Rather than use differentiators, as analog filters use, we use the delay function instead. The input and output of the IIR (recursive) digital filter have the following relationship: X X C i x kÀi þ D i y kÀi y k ¼ i¼0 to NÀ1 i¼1 to NÀ1 The first term is a sum of coefficient-weighted delayed versions of the input (this is the familiar FIR filter expression). The second term is a coefficient-weighted sum of the delayed versions of the output, which implies feedback. The feedback part prevents us from using our earlier method of FIR design analysis, described in Appendix B. But again, some smart people figured out a way to map things into the z-domain, where the math becomes simpler. This is an introduction to that technique.