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Signals and SystemsFall 2003

Lecture #18

6 November 2003

Inverse Laplace Transforms

Laplace Transform Properties The System Function of an LTI System

Geometric Evaluation of Laplace Transforms

and Frequency Responses

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Inverse Laplace Transform

But s = +j( fixed) ds = jd

Fix ROC and apply the inverse Fourier transform

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Inverse Laplace Transforms Via Partial Fraction

Expansion and Properties

Example:

Three possible ROCs corresponding to three differentsignals

Recall

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ROC I: Left-sided signal.

ROC III: Right-sided signal.

ROC II: Two-sided signal, has Fourier Transform.

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Properties of Laplace Transforms

For example:

Linearity

ROC at least the intersection of ROCs ofX1(s) and X2(s)

ROC can be bigger (due to pole-zero cancellation)

Many parallel properties of the CTFT, but for Laplace transforms

we need to determine implications for the ROC

ROC entire s-plane

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Time Shift

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Time-Domain Differentiation

ROC could be bigger than the ROC ofX(s), if there is pole-zero

cancellation. E.g.,

s

-Domain Differentiation

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Convolution Property

For

Then

ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs ofH(s) &X(s)

ROC could be empty if there is no overlap between the two ROCs

E.g.

ROC could be larger than the overlap of the two. E.g.

)t(ue)t(h),t(ue)t(x tt == and

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The System Function of an LTI System

The system function characterizes the system

System properties correspond to properties ofH(s) and its ROC

A first example:

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Geometric Evaluation of Rational Laplace Transforms

Example #1: A first-order zero

Graphic evaluation

of

Can reason about - vector length- angle w/ real axis

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First-Order System

Graphical evaluation ofH(j):

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Bode Plot of the First-Order System

-20 dB/decade

changes by -/2

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Second-Order System

0 < 1

Critically damped

2 poles on negative real axis

Overdamped

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Demo Pole-zero diagrams, frequency response, and step

response of first-order and second-order CT causal systems

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Bode Plot of a Second-Order System

-40 dB/decade

Top is flat when

= 1/2 = 0.707

a LPF for < n

changes by -

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Unit-Impulse and Unit-Step Response of a Second-

Order System

No oscillations when

1

Critically (=) andover (>) damped.

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First-Order All-Pass System

1. Two vectors have

the same lengths

2.

a

a