Signals and Systems EE235 Lecture 31 Leo Lam © 2010-2012

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

We have done… Laplace intro Region of Convergence Causality Existence of Fourier Transform No Laplace Transform since there is no overlapped ROC! Leo Lam © 2010-2012

Inverse Laplace Example, find f(t) (given 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.) Hire a Mathematician! Or write F(s) in recognisable terms and use the table (using Laplace Properties) Leo Lam © 2010-2012

Laplace properties (unilateral) Linearity: f(t) + g(t) F(s) + G(s) Time-shifting: Frequency Shifting: Differentiation: and Hire a Mathematician! Or write F(s) in recognisable terms and use the table (using Laplace Properties) Time-scaling Leo Lam © 2010-2012

Laplace properties (unilateral) Multiplication in Laplace Convolution in time Multiplication in time Convolution in Laplace Initial value Final value Final value result Only works if All poles of sF(s) in LHP Leo Lam © 2010-2012

Laplace transform table Laplace Table Leo Lam © 2010-2012

Another Inverse Example Example, find h(t) (assuming causal): Using linearity and partial fraction: Leo Lam © 2010-2012

Another Inverse Example Here is the reason: Leo Lam © 2010-2012

Another Inverse Example Example, find z(t) (assuming causal): Same degrees order for P(s) and Q(s) From table: Leo Lam © 2010-2012

Inverse Example (Partial Fraction) Example, find x(t): Partial Fraction From table: Leo Lam © 2010-2012

Inverse Example (almost identical!) Example, find x(t): Partial Fraction (still the same!) From table: Leo Lam © 2010-2012

Output Example: We know: From table (with ROC): Leo Lam © 2010-2012

All tied together Y(s) H(s)= X(s) LTI and Laplace So: LTI x(t) y(t) = x(t)*h(t) Laplace Inverse Laplace LTI X(s) Y(s)=X(s)H(s) Multiply H(s)= X(s) Y(s) 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

And we are DONE! Leo Lam © 2010-2012