Leo Lam © Signals and Systems EE235

Today’s menu Leo Lam © Laplace Transform

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 theorem Final value theorem 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?