Laplace Transform Analysis of LTI Systems

Laplace transform is extremely useful to solve LTI system when it is unsolvable by Fourier series or Fourier transform.

1. Transfer function. For an LTI system with input x(t) and output y(t), the transfer function is defined as (zero-state response)

2. Transfer function and Differential equation. We look back to the property of the differential equation and transfer function. Assume that the system is at rest at t=0, and the input x(t)=0, for t<0. The LT with the use of the differentiation property.

Deatail explanation : Laplace transform and Formula

3.Stability

We focus on systems with a rational transfer function. We also assume M<N, and there are no common poles and zeros. At here we need to recall sufficient condition of BIBO stability. We know that a system is BIBO stable the output will be bounded for every input to the system that is bounded.

Best example for BIBO stable Detail explanation how stability works : Stability

4. Inverse System

The inverse of an LTI system H(s) is a second system H1 (s) will force with H(s) and yields the identity system.

Keep in mind for H1(s) system to be stable the M has to be greater than N. Because we are inverse a system that is why the require condition is opposite.

Practice Problems
Practice problem with differential equation: Question Solution

Practice problem with transfer function : Question Solution

Practice problem with stability : Question Solution

Extra Resources
Long detail explanation

Another long explanation

Third explanation