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# Poles < # Zeros

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

1

Real Part

ImaginaryPart

POLE / ZERO PLOT

j

1 Pole at s =

Inverse Laplace Transform

Impulse Response:

h(t) = Inv FT{ H(f) }

h(t) = Inv LT{ H(s) }

Output Signal for Given Input Signal (Relaxed)

y(t) = h(t) * x(t)LaplaceTransform

Inverse Laplace Transform

Definition:

Inverse Laplace Transform

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Inverse Laplace Transform

Rational Functions:

Can get analytical form of h(t) by

Partial Fraction Expansion

and inverse transform using known forms

Partial Fraction Expansion:Requirements:1. Function to be expanded is a "strictly proper" fraction

Order of numerator polynomial < denom. polynomial order

2. Need each partial fraction term to resemble a knowntransform term (or one modified by a L-T property,such as shifting, scaling by e-t, etc.)

Convert to a sum of individual/repeated Pole terms:

and inverse LTusing tables

Partial Fraction Expansion:Method:1. Factor the denominator of the function (determine the pole terms).

2. Determine each partial fraction numerator constant "K" bymultiplying the remaining rational expression by the denominator

term for its pole, and evaluate at the pole's value of "s".