Signals and Systems
Fall 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
Inverse Laplace Transform
But s = σ + jω (σ fixed)? ds = jdω
Fix σ ∈ ROC and apply the inverse Fourier transform
Inverse Laplace Transforms Via Partial Fraction
Expansion and Properties
Example:
Three possible ROC’s — corresponding to three different signals
Recall
ROC I,— Left-sided signal.
ROC III:— Right-sided signal.
ROC II,— Two-sided signal,has Fourier Transform.
Properties of Laplace Transforms
For example:
Linearity
ROC at least the intersection of ROCs of X
1
(s) and X
2
(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
Time Shift
Time-Domain Differentiation
ROC could be bigger than the ROC of X(s),if there is pole-zero
cancellation,E.g.,
s-Domain Differentiation
Convolution Property
For
Then
ROC of Y(s) = H(s)X(s),at least the overlap of the ROCs of H(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
The System Function of an LTI System
The system function characterizes the system
System properties correspond to properties of H(s) and its ROC
A first example:
Geometric Evaluation of Rational Laplace Transforms
Example #1,A first-order zero
Graphic evaluation
of
Can reason about
- vector length
- angle w/ real axis
Example #2,A first-order pole
Example #3,A higher-order rational Laplace transform
Still reason with vector,but
remember to "invert" for poles
First-Order System
Graphical evaluation of H(jω):
Bode Plot of the First-Order System
-20 dB/decade
changes by -π/2
Second-Order System
0 <ζ <1? complex poles
ζ =1
— Underdamped
double pole at s =?ω
n
ζ >1
— Critically damped
2 poles on negative real axis
— Overdamped
Demo Pole-zero diagrams,frequency response,and step
response of first-order and second-order CT causal systems
Bode Plot of a Second-Order System
-40 dB/decade
Top is flat when
ζ= 1/√2 = 0.707
a LPF for
ω < ω
n
changes by -π
Unit-Impulse and Unit-Step Response of a Second-
Order System
No oscillations when
ζ ≥ 1
Critically (=) and
over (>) damped.
First-Order All-Pass System
1,Two vectors have
the same lengths
2,
a
a