Leo Lam © Signals and Systems EE235

Leo Lam © Futile Q: What did the monserous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS FUTILE!"

Leo Lam © Today’s menu Laplace Transform

Ambiguous? Define it away! Bilateral Laplace Transform: Unilateral Laplace Transform (for causal system/signal): For EE, it’s mostly unilateral Laplace (any signal with u(t) is causal) Not all functions have a Laplace Transform (no ROC) Leo Lam ©

Inverse Laplace Example, find f(t) (assuming causal): Table: What if the exact expression is not in the table? –Hire a mathematician –Make it look like something in the table (partial fraction etc.) Leo Lam ©

Laplace properties (unilateral) Leo Lam © Linearity: f(t) + g(t) F(s) + G(s) Time-shifting: Frequency Shifting: Differentiation: and Time-scaling

Laplace properties (unilateral) Leo Lam © Multiplication in time Convolution in Laplace Convolution in time Multiplication in Laplace Initial value Final value Final value result Only works if All poles of sF(s) in LHP

Another Inverse Example Leo Lam © Example, find h(t) (assuming causal): Using linearity and partial fraction:

Another Inverse Example Leo Lam © Here is the reason:

Laplace & LTI Systems Leo Lam © If: Then LTI Laplace of the zero-state (zero initial conditions) response Laplace of the input

Laplace & Differential Equations Leo Lam © Given: In Laplace: –where So: Characteristic Eq: –The roots are the poles in s-domain, the “power” in time domain.

Laplace Stability Conditions Leo Lam © LTI – Causal system H(s) stability conditions: LTIC system is stable : all poles are in the LHP LTIC system is unstable : one of its poles is in the RHP LTIC system is unstable : repeated poles on the j-axis LTIC system is if marginally stable : poles in the LHP + unrepeated poles on the jaxis.

Laplace: Poles and Zeroes Leo Lam © Given: Roots are poles: Roots are zeroes: Only poles affect stability Example:

Laplace Stability Example: Leo Lam © Is this stable?

Laplace Stability Example: Leo Lam © Is this stable?

Laplace Stability Example: Leo Lam © Is this stable? Mathematically stable (all poles cancelled) In reality…explosive

Woohoo!!! Leo Lam ©