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Join us for an insightful session on the Laplace Transform, focusing on complex frequencies and frequency domain representation of signals. We will explore the manipulation of signals, the significance of complex exponential functions, and useful equalities. You will learn how the Laplace Transform facilitates simplified operations in the s-domain, allowing for easier handling of convolution, differentiation, and integration. Through examples such as the Unit Impulse and Unit Step functions, we aim to clarify the correspondence between signals and their transforms.
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The Laplace Transform Montek Singh Thurs., Feb. 19, 20023:30-4:45 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, wheres = + j Examples: <0, =0 >0, =0 =0, =0 Re(est) <0 =0 >0
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!
