Signals and Systems EE235 Leo Lam © 2010-2012

Today’s menu Almost done! Laplace Transform Leo Lam © 2010-2012

Laplace & LTI Systems LTI LTI If: Then Laplace of the zero-state (zero initial conditions) response Laplace of the input Leo Lam © 2010-2012

Laplace & Differential Equations Given: In Laplace: where So: Characteristic Eq: The roots are the poles in s-domain, the “power” in time domain. Leo Lam © 2010-2012

Laplace & Differential Equations Example (causal  LTIC): Cross Multiply and inverse Laplace: Leo Lam © 2010-2012

Laplace Stability Conditions 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 jw-axis LTIC system is if marginally stable : poles in the LHP + unrepeated poles on the jw-axis. Leo Lam © 2010-2012

Laplace Stability Conditions Generally: system H(s) stability conditions: The system’s ROC includes the jw-axis Stable? Causal? Stable+Causal Unstable+Causal Stable+Noncausal σ jω x σ jω x σ jω x Leo Lam © 2010-2012

Laplace: Poles and Zeroes Given: Roots are poles: Roots are zeroes: Only poles affect stability Example: Leo Lam © 2010-2012

Laplace Stability Example: Is this stable? Leo Lam © 2010-2012

Laplace Stability Example: Is this stable? Leo Lam © 2010-2012

Standard Laplace question Find the Laplace Transform, stating the ROC. So: ROC extends from to the right of the most right pole ROC x o Laplace transform not uniquely invertible without region of convergence Leo Lam © 2010-2012

Inverse Laplace Example (2 methods!) Find z(t) given the Laplace Transform: So: Laplace transform not uniquely invertible without region of convergence Leo Lam © 2010-2012

Inverse Laplace Example (2 methods!) Find z(t) given the Laplace Transform (alternative method): Re-write it as: Then: Substituting back in to z(t) and you get the same answer as before: Laplace transform not uniquely invertible without region of convergence Leo Lam © 2010-2012

Inverse Laplace Example (Diffy-Q) Find the differential equation relating y(t) to x(t), given: Laplace transform not uniquely invertible without region of convergence Leo Lam © 2010-2012

Laplace for Circuits! Don’t worry, it’s actually still the same routine! Time domain Laplace domain inductor resistor capacitor Laplace transform not uniquely invertible without region of convergence Impedance! Leo Lam © 2010-2012

Laplace for Circuits! L R + - Find the output current i(t) of this ugly circuit! Then KVL: Solve for I(s): Partial Fractions: Invert: L R Given: input voltage And i(0)=0 + - Step 1: represent the whole circuit in Laplace domain. Laplace transform not uniquely invertible without region of convergence Leo Lam © 2010-2012

Step response example Find the transfer function H(s) of this system: We know that: We just need to convert both the input and the output and divide! LTIC Laplace transform not uniquely invertible without region of convergence LTIC Leo Lam © 2010-2012

A “strange signal” example Find the Laplace transform of this signal: What is x(t)? We know these pairs: So: x(t) 1 2 3 2 1 Laplace transform not uniquely invertible without region of convergence Leo Lam © 2010-2012