Digital Signal Processing 101

APPENDIX DZ-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:

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. ...