The Laplace Transform Montek Singh Thurs Feb 19

The Laplace Transform Montek Singh Thurs. , Feb. 19, 2002 3: 30 -4: 45 pm, SN 115 1

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

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

Some Useful Equalities 4

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

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

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

Example 2: The Unit Step Function 8

Some Useful Transform Pairs 9

Properties of the Laplace Transform (1) 10

Properties of the Laplace Transform (2) 11