 # Leo Lam © 2010-2012 Signals and Systems EE235. Today’s menu Leo Lam © 2010-2012 Almost done! Laplace Transform.

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Leo Lam © 2010-2012 Signals and Systems EE235

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 © 2010-2011

Laplace properties (unilateral) Leo Lam © 2010-2011 Linearity: f(t) + g(t) F(s) + G(s) Time-shifting: Frequency Shifting: Differentiation: and Time-scaling

Laplace properties (unilateral) Leo Lam © 2010-2011 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 © 2010-2011

Another Inverse Example Leo Lam © 2010-2011 Example, find h(t) (assuming causal): Using linearity and partial fraction:

Another Inverse Example Leo Lam © 2010-2011 Here is the reason:

Another Inverse Example Leo Lam © 2010-2011 Example, find z(t) (assuming causal): Same degrees order for P(s) and Q(s) From table:

Inverse Example (Partial Fraction) Leo Lam © 2010-2011 Example, find x(t): Partial Fraction From table:

Inverse Example (almost identical!) Leo Lam © 2010-2011 Example, find x(t): Partial Fraction (still the same!) From table:

Output Leo Lam © 2010-2011 Example: We know: From table (with ROC):

All tied together LTI and Laplace So: Leo Lam © 2010-2012 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 © 2010-2011 If: Then LTI Laplace of the zero-state (zero initial conditions) response Laplace of the input

Laplace & Differential Equations Leo Lam © 2010-2011 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 © 2010-2011 Example (causal  LTIC): Cross Multiply and inverse Laplace:

Laplace Stability Conditions Leo Lam © 2010-2011 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 © 2010-2011 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 © 2010-2011 Given: Roots are poles: Roots are zeroes: Only poles affect stability Example:

Laplace Stability Example: Leo Lam © 2010-2011 Is this stable?

Laplace Stability Example: Leo Lam © 2010-2011 Is this stable?