Transfer Functions

For LTI systems, the general input/output behavior can be characterized using complex exponentials.

Complex Exponentials in LTI Systems

• Drive an LTI system with a complex exponential:
  • The output is the same complex exponential, scaled by a complex number
• Transfer function:
  • i.e. how do we find , and how do we use it for inputs that aren’t complex exponentials?

Laplace’s Idea

• Write any input as a weighted sum of complex exponentials at different (complex) frequencies:
  • Transform:
  • Inverse transform:
  • This is a weighted sum of complex exponentials , with weights
• Since each just gets scaled by , by linearity:
• As such:

Transfer Function Response

Corrected from the notes

• The quick notes wrote the inverse transform as ; the inverse Laplace transform integrates over (along a vertical line in the complex plane) with the factor, as written above

• The Laplace properties used throughout the course are collected in Laplace Transform Review; the MIMO version of is the Transfer Function Matrix

Laplace Does Not Work for LTV Systems

• Importantly:
  • The simplicity of Laplace is enabled by the special role of complex exponentials in LTI systems
  • Laplace does not work for LTV systems:

WARNING

• We cannot just extend Laplace to time-varying systems
• Instead, we approach the problem from a different angle (state space, fundamental solutions, the state transition matrix) and recover the Laplace results as the LTI special case (see LTI Systems)