1. Signals and Systems
EE235
Leo Lam ©

2. Today’s menu Fourier Transform table posted Laplace Transform
Leo Lam ©

3. Laplace Transform
Definition:
Where
Inverse:
Leo Lam ©

4. Region of Convergence Example: Find the Laplace Transform of:
We have a problem: the first term for t=∞ doesn’t always vanish!
Leo Lam ©

5. 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!
Leo Lam ©

6. 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.
Leo Lam ©

7. 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
Leo Lam ©

8. 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).
Leo Lam ©

9. Laplace Example No Laplace Example: ROC: Re(s)>-1 ROC: Re(s)<-3
Combined:
ROC: None!
No Laplace Transform since there is no overlapped ROC!
Leo Lam ©

10. Laplace and Fourier
If the ROC includes the jw –axis, then the signal has a Fourier Transform where s= jw
Caution: If the ROC doesn't quite include the jw-axis (if poles lie on the jw-axis), then it might still have a Fourier transform, but it is not given by s=jw. σ jw
ROC –a
No Laplace Transform since there is no overlapped ROC!
Leo Lam ©

11. Laplace and Fourier No Fourier Transform Example:
ROC exists: Laplace ok
ROC does not include jw-axis, Fourier Transform is not F(jw). (In fact, here it does not exist).
ROC: Re(s)>-3
ROC: Re(s)<-1
Combined:
-3<ROC<-1
No Laplace Transform since there is no overlapped ROC!
Leo Lam ©

12. Finding ROC Example Example: Find the Laplace Transform of:
From table:
Thus:
With ROC:
ROC: Re(s)<-2
ROC: Re(s)>-3
Combined:
ROC: -3<Re(s)<-2 x o
Causal signal: Right-sided ROC (at the roots).
Leo Lam ©

13. Poles and Zeros (the X’s and O’s)
H(s) is almost always rational for a physical system:
Rational = Can be expressed as a polynomial
ZEROs = where H(s)=0, which is
POLES = where H(s)=∞, which is
Example:
Leo Lam ©

14. Plotting Poles and Zeros
H(s) is almost always rational for a physical system:
Plot is in the s-plane (complex plane) σ jω x o
Leo Lam ©

15. Plotting Poles and Zeros
What does it look like?
Leo Lam ©

16. ROC Properties (Summary)
All ROCs are parallel to the jw –axis
Casual signal  right-sided ROC and vice versa
Two-sided signals appear either as a strip or no ROC exist (no Laplace).
For a rational Laplace Transform, the ROC is bounded by poles or ∞.
If ROC includes the jw-axis, Fourier Transform of the signal exists = F(jw).
If it has poles on the jw-axis, FT can still exist. However, it is no longer s=jw, almost always something else.
Leo Lam ©

17. Laplace and Fourier
Very similar (Fourier for Signal Analysis, Laplace for Control, Circuits 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
If it has poles on the jw-axis, FT can still exist. However, it is no longer s=jw, almost always something else.
Leo Lam ©

18. Ambiguous? Define it away!
Bilateral Laplace Transform:
Unilateral Laplace Transform (for causal system/signal):
For EE, it’s mostly unilateral Laplace (any signal with u(t) is causal)
Not all functions have a Laplace Transform (no ROC)
Laplace transform not uniquely invertible without region of convergence
Leo Lam ©

19. Inverse Laplace Example, find f(t) (assuming 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)
Leo Lam ©

20. 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
Leo Lam ©

21. 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
Leo Lam ©

22. Another Inverse Example
Example, find h(t) (assuming causal):
Using linearity and partial fraction:
Leo Lam ©

23. Another Inverse Example
Here is the reason:
Leo Lam ©

24. Summary Laplace intro Region of Convergence Causality
Existence of Fourier Transform & relationships
Leo Lam ©