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

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

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

Some Useful Equalities

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

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:

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

Example 2: The Unit Step Function

Some Useful Transform Pairs

Properties of the Laplace Transform (1)

Properties of the Laplace Transform (2)