1. Signals and Systems
EE235
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2. Today’s menu
Almost done!
Laplace Transform
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3. Laplace & LTI Systems LTI LTI If: Then
Laplace of the zero-state (zero initial
conditions) response
Laplace of the input
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4. 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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5. Laplace & Differential Equations
Example (causal  LTIC):
Cross Multiply and inverse Laplace:
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6. 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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7. 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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8. Laplace: Poles and Zeroes
Given:
Roots are poles:
Roots are zeroes:
Only poles affect stability
Example:
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9. Laplace Stability Example:
Is this stable?
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10. Laplace Stability Example:
Is this stable?
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11. 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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12. 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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13. 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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14. 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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15. 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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16. 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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17. 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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18. 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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