Signals and Systems
Fall 2003
Lecture #19
18 November 2003
1,CT System Function Properties
2,System Function Algebra and
Block Diagrams
3,Unilateral Laplace Transform and
Applications
CT System Function Properties
2) Causality? h(t) right-sided signal? ROC of H(s) is a right-half plane
Question:
If the ROC of H(s) is a right-half plane,is the system causal?
|h(t)|dt<∞
∞
∞
∫
1) System is stableROC of H(s) includes jω axis
Ex.
H(s) =,system function”
Non-causal
Properties of CT Rational System Functions
a) However,if H(s) is rational,then
The system is causal? The ROC of H(s) is to the
right of the rightmost pole
jω-axis is in ROC
all poles are in LHP
b) If H(s) is rational and is the system function of a causal
system,then
The system is stable?
Checking if All Poles Are In the Left-Half Plane
Method #1,Calculate all the roots and see!
Method #2,Routh-Hurwitz – Without having to solve for roots.
Initial- and Final-Value Theorems
If x(t) = 0 for t < 0 and there are no impulses or higher order
discontinuities at the origin,then
Initial value
If x(t) = 0 for t < 0 and x(t) has a finite limit as t →∞,then
Final value
Applications of the Initial- and Final-Value Theorem
Initial value:
Final value
For
LTI Systems Described by LCCDEs
ROC =? Depends on,1) Locations of all poles.
2) Boundary conditions,i.e.
right-,left-,two-sided signals.
roots of numerator? zeros
roots of denominator? poles
System Function Algebra
Example,A basic feedback system consisting of causal blocks
ROC,Determined by the roots of 1+H
1
(s)H
2
(s),instead of H
1
(s)
More on this later
in feedback
Block Diagram for Causal LTI Systems
with Rational System Functions
— Can be viewed
as cascade of
two systems.
Example:
Example (continued)
Instead of
1
s
2
+ 3s+ 2
2s
2
+4s?6
H(s)
Notation,1/s — an integrator
We can construct H(s) using,
x(t) y(t)
Note also that
Lesson to be learned,There are many different ways to construct a
system that performs a certain function.
The Unilateral Laplace Transform
(The preferred tool to analyze causal CT systems
described by LCCDEs with initial conditions)
Note:
1) If x(t) = 0 for t < 0,then
2) Unilateral LT of x(t) = Bilateral LT of x(t)u(t-)
3) For example,if h(t) is the impulse response of a causal LTI
system,then
Same as Bilateral Laplace transform
4) Convolution property:If x
1
(t) = x
2
(t) = 0 for t < 0,then
Differentiation Property for Unilateral Laplace Transform
Note:
Derivation:
Initial condition!
Use of ULTs to Solve Differentiation Equations
with Initial Conditions
Example:
ZIR — Response for
zero input x(t)=0
ZSR — Response for zero state,
β=γ=0,initially at rest
Take ULT:
Example (continued)
Response for LTI system initially at rest (β = γ = 0)
Response to initial conditions alone (α = 0).
For example: