1. Signals and Systems
EE235
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2. Futile
Q: What did the monserous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS FUTILE!"
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3. Today’s menu
Laplace Transform
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4. Laplace Transform Focus on: Doing (Definitions and properties)
Understanding its possibilities (ROC)
Poles and zeroes (overlap with EE233)
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5. Laplace Transform Definition: Where Inverse:
Good news: We don’t need to do this, just use the tables.
Fourier Series coefficient dk differs from its F(w) equivalent by 2pi
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6. Laplace Transform Definition: Where Inverse:
Good news: We don’t need to do this, just use the tables.
Fourier Series coefficient dk differs from its F(w) equivalent by 2pi
Inverse Laplace expresses f(t) as sum of exponentials with fixed s  has specific requirements
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7. Region of Convergence Example: Find the Laplace Transform of:
Fourier Series coefficient dk differs from its F(w) equivalent by 2pi
We have a problem: the first term for t=∞ doesn’t always vanish!
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8. Region of Convergence Example: Continuing… In general: for
In our case if: then
For what value of s does:
Pole at s=-3. Remember this result for now!
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9. Region of Convergence
A very similar example: Find Laplace Transform of:
For what value does:
This time: if then
Same result as before!
Note that both cases have the region dissected at s=-3, which is the ROOT of the Laplace Transform.
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10. Region of Convergence Laplace transform not uniquely
Comparing the two:
Laplace transform not uniquely
invertible without region of convergence
ROC -3
ROC -3
Non-casual,
Left-sided
Casual,
Right-sided
Laplace transform not uniquely invertible without region of convergence
s-plane
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11. Finding ROC Example Example: Find the Laplace Transform of:
From table:
ROC: Re(s)>-6
ROC: Re(s)>-2
Combined:
ROC: Re(s)>-2
Causal signal: Right-sided ROC (at the roots).
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12. Laplace and Fourier
Very similar (Fourier for Signal Analysis, Laplace for Control and System Designs)
ROC includes the jw-axis, then Fourier Transform = Laplace Transform (with s=jw)
If ROC does NOT include jw-axis but with poles on the jw-axis, FT can still exist!
Example:
But Fourier Transform still exists:
No Fourier Transform if ROC is Re(s)<0 (left of jw-axis)
ROC: Re(s) > 0
Not including jw-axis
Laplace transform not uniquely invertible without region of convergence
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13. Laplace and Fourier No Fourier Transform Example:
ROC exists: Laplace ok
ROC does not include jw-axis, no Fourier Transform
ROC: Re(s)>-3
ROC: Re(s)<-1
Combined:
-3<ROC<-1
No Laplace Transform since there is no overlapped ROC!
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14. Laplace and Fourier No Laplace Example: ROC: Re(s)>-1
Combined:
ROC: None!
No Laplace Transform since there is no overlapped ROC!
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