4. The Complex Domain

In this chapter we generalize both the DT and the CT Fourier transforms to complex inputs instead of (the real) frequencies.

We will still be looking at LTI systems, therefor the question what another view on systems and signals may add to our understanding is a reasonable one. We have seen that an LTI system is completely characterized with its impulse response and also with its frequency response. What could another representation add to this?

It turns out that the frequency domain view on LTI systems is geared towards the analysis of such systems given more or less periodic time varying input signals.

The Laplace Transform for CT signals and systems and the Z-Transform for DT systems and signals will make it easier to analyze sudden changes in the input signal. For instance what will happen when we feed the system with a step response.

The Laplace transform of CT systems and signals lead to the representation of such systems in the \(s\)-domain. The Z-transform of DT systems and signals lead to the representation of systems and signals in the \(z\)-domain. Both domains are mathematically equivalent with the complex plane. Their interpretation in the analysis of LTI systems is different though.

• 4.1. The S-Domain
  • 4.1.1. The Laplace Transform
    • 4.1.1.1. Definition
    • 4.1.1.2. Eigenfunctions of an LTI system
    • 4.1.1.3. The Laplace Transform and the Fourier Transform
  • 4.1.2. Properties of the Unilateral Laplace transform
  • 4.1.3. Pairs of the (Unilateral) Laplace Transform
  • 4.1.4. Differential Equations and the Laplace Transform
    • 4.1.4.1. Poles and Zeros in the S-plane
    • 4.1.4.2. An Electronic Example
    • 4.1.4.3. A Mechanical Example
  • 4.1.5. Exercises
• 4.2. The Z-Domain
  • 4.2.1. The \(\ZT\)-Transform
    • 4.2.1.1. Definition
    • 4.2.1.2. Eigenfunctions of an LTI System
    • 4.2.1.3. Finite and Infinite Signals
    • 4.2.1.4. The Z-Transform and the Fourier Transform
  • 4.2.2. \(\ZT\)-Transform Properties
  • 4.2.3. Pairs of Z-Transforms
  • 4.2.4. Difference Equations in the Z-domain
    • 4.2.4.1. From Difference Equation to Transfer Function
    • 4.2.4.2. Poles and Zeros
  • 4.2.5. The Z-Operator
  • 4.2.6. Exercises