4.1. The S-Domain

The Laplace transform takes a continuous time signal and transforms it to the \(s\)-domain.

The Laplace transform is a generalization of the CT Fourier Transform. Let \(X(s)\) be the Laplace transform of \(x(t)\), then the Fourier transform of \(x\) is found as \(X(j\omega)\).

For most engineers (and many fysicists) the Laplace transform is just a mathematical trick to easily solve a class of partial differential equations. The Laplace transforms a constant coefficient linear differential equation relating the input \(x(t)\) with the output \(y(t)\) into an algebraic polynomial in \(s\). We will look at this class of constant coefficient linear partial differential equations.

The dynamical behaviour of many (physical) systems are described with a PDE in this class (like the inverted pendulum). This is no class where you learn to model systems with PDE’s. We just give some examples.

In a later chapter on Control Theory the Laplace transform and its use in solving these type of differential equations will be used quite a lot.

• 4.1.1. The Laplace Transform
• 4.1.2. Relation with the Fourier transform
• 4.1.3. The pulse, step and exponential function
• 4.1.4. Derivatives and Integrals
• 4.1.5. Convolutions
• 4.1.6. Properties of the unilateral Laplace transform
• 4.1.7. Table of Selected Laplace Transforms
• 4.1.8. Differential Equations
• 4.1.9. Exercises