1. Laplace Transform

2. Forward Laplace Transform
Decompose a signal f(t) into complex sinusoids of the form es t where s is complex: s = s + j2pf
Forward (bilateral) Laplace transform
f(t): complex-valued function of a real variable t
F(s): complex-valued function of a complex variable s
Bilateral means that the extent of f(t) can be infinite in both the positive t and negative t direction (a.k.a. two-sided)

3. Inverse (Bilateral) Transform
is a contour integral which represents integration over a complex region– recall that s is complex
c is a real constant chosen to ensure convergence of the integral
Notation
F(s) = L{f(t)} variable t implied for L
f(t) = L-1{F(s)} variable s implied for L-1

4. Laplace Transform Properties
Linear or nonlinear?
Linear operator L
F(s)
f(t)

5. Laplace Transform Properties
Time-varying or time-invariant?
This is an odd question to ask because the output is in a different domain than the input.

6. Example

7. Convergence
The condition Re{s} > -Re{a} is the region of convergence, which is the region of s for which the Laplace transform integral converges
Re{s} = -Re{a} is not allowed (see next slide)

8. Regions of Convergence
What happens to F(s) = 1/(s+a) at s = -a? (1/0)
-e-a t u(-t) and e-a t u(t) have same transform function but different regions of convergence
Im{s}
Re{s} = -Re{a}
Re{s}
f(t) 1 t
f(t)
f(t) = e-a t u(t)
causal t -1
f(t) = -e- a t u(-t)
anti-causal

9. Review of 0- and 0+
d(t) not defined at t = 0 but has unit area at t = 0
0- refers to an infinitesimally small time before 0
0+ refers to an infinitesimally small time after 0

10. Unilateral Laplace Transform
Forward transform: lower limit of integration is 0- (i.e. just before 0) to avoid ambiguity that may arise if f(t) contains an impulse at origin
Unilateral Laplace transform has no ambiguity in inverse transforms because causal inverse is always taken:
No need to specify a region of convergence
Disadvantage is that it cannot be used to analyze noncausal systems or noncausal inputs

11. Existence of Laplace Transform
As long as e-s t decays at a faster rate than rate f(t) explodes, Laplace transform converges
for some M and s0, there exists s0 > s to make the Laplace transform integral finite
We cannot always do this, e.g does not have a Laplace transform

12. Key Transform Pairs

13. Key Transform Pairs

14. Fourier vs. Laplace Transform Pairs
Assuming that Re{a} > 0