APPENDIX C Laplace Transform The Laplace transform is used for continuous or analog signals. Let's use x(t) and y(t) to represent the input and output signals to an analog filter, respectively. Rather than use delays, as digital filters use, we use the differentiator instead. The input and output of the analog filter have the following relationship: X X A i Á d i xðtÞ=dt i þ B i Á d i yðtÞ=dt i yðtÞ ¼ i¼0 to N i¼0 to M The first term is a sum of coefficient-weighted derivatives of the input, which is analogous to the FIR filter being a sum of weighted delayed inputs. The second term is a coefficient- weighted sum of the derivatives of the output, which implies feedback. These functions can be difficult to evaluate and characterize. Smart people long ago figured out a way to map things into the s-domain using the Laplace transform, where the math becomes much simpler. This is an introduction to that technique.