The Laplace Transform

1. The Laplace Transform Montek Singh Thurs., Feb. 19, 20023:30-4:45 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, wheres =  + 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)