1. Montek Singh Thurs., Feb. 19, 2002 3:30-4:45 pm, SN115
The Laplace Transform
Montek Singh
Thurs., Feb. 19, :30-4:45 pm, SN115

2. What we will learn The notion of a complex frequency
Representing a signal in the frequency domain
Manipulating signals in the frequency domain

3. Complex Exponential Functions
Complex exponential = est, where s =  + j
Examples:
<0, =0
>0, =0
=0, =0
Re(est)
=0
<0
>0

4. Some Useful Equalities

5. The Laplace Transform: Overview
Key Idea:
Represent signals as sum of complex exponentials
since all exponentials have the form Aest, it suffices to know the value of A for each s, to completely represent the original signal
i.e., representation transformed from “t” to “s” domain
Benefits:
Complex operations in the time domain get transformed into simpler operations in the s-domain
e.g., convolution, differentiation and integration in time  algebraic operations in the s-domain!
Even fairly complex differential equations can be transformed into algebraic equations

6. The Laplace Transform F(s) = Laplace Transform of f(t):
1-to-1 correspondence between a signal and its Laplace Transform
Frequently, only need to consider time t > 0:

7. Example 1: The Unit Impulse Function
F(s) = 1 everywhere!

8. Example 2: The Unit Step Function

9. Some Useful Transform Pairs

10. Properties of the Laplace Transform (1)

11. Properties of the Laplace Transform (2)