Notes on Digital Signal Processing: Practical Recipes for Design, Analysis and Implementation

Note 38. Laplace Transform

The Laplace transform is a mathematical tool that is used primarily for analyzing linear analog systems such as filters. This transform is of interest in digital signal processing because analog filters having transfer functions defined in terms of Laplace transforms are often used as the starting point in the design of digital IIR filters. The characterization of analog filters is discussed in Note 39, and commonly used analog filter families are discussed in Notes 40 through 43.

The Laplace transform for a continuous-time function, x(t), is usually denoted as X(s) or L[x(t)], and is defined by Eq. (MB 38.1). The complex variable, s, is usually referred to as complex frequency, and can be put into the form σ + j