Leo Lam © Signals and Systems EE235 Lecture 31

Leo Lam © Today’s menu Laplace Transform!

We have done… Laplace intro Region of Convergence Causality Existence of Fourier Transform Leo Lam ©

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.) 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

Laplace transform table Leo Lam ©

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:

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

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

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

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

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

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

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 Stability Conditions Leo Lam © Generally: system H(s) stability conditions: The system’s ROC includes the jaxis Stable? Causal? σ jωjω x x x Stable+CausalUnstable+Causal σ jωjω x x x x σ jωjω x x x Stable+Noncausal

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?

Standard Laplace question Find the Laplace Transform, stating the ROC. So: Leo Lam © ROC extends from to the right of the most right pole ROC xxo

Inverse Laplace Example (2 methods!) Find z(t) given the Laplace Transform: So: Leo Lam ©

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: Leo Lam ©

Inverse Laplace Example (Diffy-Q) Find the differential equation relating y(t) to x(t), given: Leo Lam ©

Laplace for Circuits! Don’t worry, it’s actually still the same routine! Leo Lam © Time domain inductor resistor capacitor Laplace domain Impedance!

Laplace for Circuits! Find the output current i(t) of this ugly circuit! Then KVL: Solve for I(s): Partial Fractions: Invert: Leo Lam © R L +-+- Given: input voltage And i(0)=0 Step 1: represent the whole circuit in Laplace domain.

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! Leo Lam © LTIC

A “strange signal” example Find the Laplace transform of this signal: What is x(t)? We know these pairs: So: Leo Lam © x(t)

Leo Lam © And we are DONE!