1. Signals and Systems
EE235
Lecture 31
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2. Today’s menu
Laplace Transform!
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3. We have done… Laplace intro Region of Convergence Causality
Existence of Fourier Transform
No Laplace Transform since there is no overlapped ROC!
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4. Inverse Laplace Example, find f(t) (given causal): Table:
What if the exact expression is not in the table?
Hire a mathematician
Make it look like something in the table (partial fraction etc.)
Hire a Mathematician! Or write F(s) in recognisable terms and use the table (using Laplace Properties)
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5. Laplace properties (unilateral)
Linearity: f(t) + g(t) F(s) + G(s)
Time-shifting:
Frequency
Shifting:
Differentiation:
and
Hire a Mathematician! Or write F(s) in recognisable terms and use the table (using Laplace Properties)
Time-scaling
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6. Laplace properties (unilateral)
Multiplication in Laplace
Convolution in time
Multiplication in time
Convolution in Laplace
Initial value
Final value
Final value result
Only works if
All poles of
sF(s) in LHP
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7. Laplace transform table
Laplace Table
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8. Another Inverse Example
Example, find h(t) (assuming causal):
Using linearity and partial fraction:
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9. Another Inverse Example
Here is the reason:
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10. Another Inverse Example
Example, find z(t) (assuming causal):
Same degrees order for P(s) and Q(s)
From table:
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11. Inverse Example (Partial Fraction)
Example, find x(t):
Partial Fraction
From table:
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12. Inverse Example (almost identical!)
Example, find x(t):
Partial Fraction (still the same!)
From table:
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13. Output
Example:
We know:
From table (with ROC):
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14. All tied together Y(s) H(s)= X(s) LTI and Laplace So: LTI x(t)
y(t) = x(t)*h(t)
Laplace
Inverse Laplace
LTI
X(s)
Y(s)=X(s)H(s)
Multiply
H(s)=
X(s)
Y(s)
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15. Laplace & LTI Systems LTI LTI If: Then
Laplace of the zero-state (zero initial
conditions) response
Laplace of the input
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16. Laplace & Differential Equations
Given:
In Laplace:
where
So:
Characteristic Eq:
The roots are the poles in s-domain, the “power” in time domain.
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17. Laplace & Differential Equations
Example (causal  LTIC):
Cross Multiply and inverse Laplace:
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18. Laplace Stability Conditions
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 jw-axis
LTIC system is if marginally stable : poles in the LHP + unrepeated poles on the jw-axis.
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19. Laplace Stability Conditions
Generally: system H(s) stability conditions:
The system’s ROC includes the jw-axis
Stable? Causal?
Stable+Causal
Unstable+Causal
Stable+Noncausal σ jω x σ jω x σ jω x
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20. Laplace: Poles and Zeroes
Given:
Roots are poles:
Roots are zeroes:
Only poles affect stability
Example:
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21. Laplace Stability Example:
Is this stable?
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22. Laplace Stability Example:
Is this stable?
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23. Standard Laplace question
Find the Laplace Transform, stating the ROC.
So:
ROC extends from to the right of the most right pole
ROC x o
Laplace transform not uniquely invertible without region of convergence
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24. Inverse Laplace Example (2 methods!)
Find z(t) given the Laplace Transform:
So:
Laplace transform not uniquely invertible without region of convergence
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25. Inverse Laplace Example (2 methods!)
Find z(t) given the Laplace Transform (alternative method):
Re-write it as:
Then:
Substituting back in to z(t) and you get the same answer as before:
Laplace transform not uniquely invertible without region of convergence
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26. Inverse Laplace Example (Diffy-Q)
Find the differential equation relating y(t) to x(t), given:
Laplace transform not uniquely invertible without region of convergence
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27. Laplace for Circuits!
Don’t worry, it’s actually still the same routine!
Time domain
Laplace domain
inductor
resistor
capacitor
Laplace transform not uniquely invertible without region of convergence
Impedance!
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28. Laplace for Circuits! L R + -
Find the output current i(t) of this ugly circuit!
Then KVL:
Solve for I(s):
Partial Fractions:
Invert: L R
Given: input voltage
And i(0)=0 + -
Step 1: represent
the whole circuit in
Laplace domain.
Laplace transform not uniquely invertible without region of convergence
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29. Step response example Find the transfer function H(s) of this system:
We know that:
We just need to convert both the input and the output and divide!
LTIC
Laplace transform not uniquely invertible without region of convergence
LTIC
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30. A “strange signal” example
Find the Laplace transform of this signal:
What is x(t)?
We know these pairs:
So:
x(t) 2 1
Laplace transform not uniquely invertible without region of convergence
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31. And we are DONE!
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